AO—A fl3 362 STANFORD UNIV CALIF DEPT OF COMPUTER SCIENCE F/S 9/2COMPLEXITY OF COMBINATORIAL ALGORITHMS. CU)APR 77 R E TARJAN N0001e—76—C—0668

UNCLASSIFIED STAN—CS— 77—609 NL

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~ ~1iNATIONAL BUREAU OF STANDARDS

M~CROCOP~ RESOLUTI ON lEST CH A RT

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COMPLEX ITY OF COM B I NATOR IAL ALGOR ITHMS

by

Robert E. Tarja n,

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C O M P U T E R S C I E N C E D E P A R T M E N TSc hool of Huma nities and Sciences

STANFORD UNIVERSITY

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SECURITY C L A S S I F I C A T~~~N OF THIS P A G E II~~~, t) o~~~~~~ r r e f

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__________________________________ —

C omplexity of Combinatorial ~ i c ~ r I t I c : :—

Robert ~Indro Tarj afl~ —’

Computer Sciene~ Cep art omor tStanford ‘.Jiiivers.I ty

Stanford 5 California oL3-)5

A b ct r a ct .

T~H s i a i cr exanines r oLont wo rk -:a the c. :: loxi ty 01

combina tor ia l al g: r i thm :, high l ight ing the a : . r n: of thc ’ w o r t , the

ra t .h emat i ca . tools used , an i the i:o~ or tant results . Included a rc

oec’t i no di :cussing way: t o ::ca: are the complexity :f an aigor itboc .

caclh -do for r -v in g tha cert a in r ob lecos are very hard to colvo ,

tools useful in the der ign H good ~~g - r itbms , and re:u’nt c-o r ve:.c r st s

in alg~ ri tb r.: L r solving t or i rd reoentotive ~ r.b~ e:oo. Tho o ln a l :-oct I

suggests some ii rcct~ ons for future research .

I.

-

I .1 ‘rf . ’ —

Based on a talk presented at the Cyn i. osiuns in Honor of thu ~~ t,c:Anniversary of the Office of Naval Research, C lAN 197 Fai l ~ec t I u : : ,At lanta, Georgia , October 1 ‘ - C - , 197:-.

Research partially :‘u~ o r t t c : by National Cci on: Foundat on ~rcHM CC75 _ flf l °7t and by the o r f : c e of Naval Research c n t r a c~N )C1L_ 7 ..r _

~. . ~~ . Re Tr cj u ctj j n in wh le r in arl ‘- ° I e r r

for any purpose of the United tt . a t u s Government.

1

5 - --.5 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~ ‘ .

intr od.uct .LuIl.

In recent years there has been an explosive growth in research

sealing -~dth the development and complexity analysis of combinatori al

JC.~rithms. While much of thi s research is theoretical in nature, many

Ut the newly discovered algori thms are very practical. These algorit.hrr .c

~id the dat a manipulation techniques they use are valuable in both

o:r:binatorial and numeric computing. Some pr -oblLercac which at :ir:t

glance are entirely numeri c in character require for their efficient

.- - olution not only the proper numeric techniques but also the 1 r ’ - 1 1 ~~~ als I c e ol

data structures and of data manipulation methods. An example of such a

cublein is the solution 01’ a system of linear equations when thu coefficient

. ‘ L r l x contains mostly zeros (Tewar:-:, n [ 1973]) .

in this paper I shall survey :ome of the recent rocui t,: on

c omplexity of combinatorial algori thms, examine some of the I dear behind

hem , and suggest possible directions for future research . Cection 2 of

ho pai er discusses ways to r ::ca:u.re the complexity

:1 algorithms . Though several dif ferent measures are u :eI ’ui in di f fe r e n t

circumstances, I shall concentrate upon one ::.ecc: cre , the wor ;~ - - ,ca ,o e

running time of the algorithm as a function of the input slo e . Cect ion 3

t o :cu:ses techniques for proving that certain combinatorial problems are

very hard to solve. The results in this area are a natural ex~ cc i : ion ,

erhaps more relevant for real-world computing, of the incomp leteness

and undecidability results of Gödel, Turing and others. ect ioI~ 14 r eoerst :

a sma.U collection of genera]. techni ques wh ich are useful

in the construction of efficient combinatorial algorithms. Cection 5

JJ: cioreo efficient algorithms for solving ten representative problems.I .

-7, - ~- 2 )~_____ c c ‘< ‘CLi 7_ _ _ _ ~~~ ~~~~~~~~~~~~

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Ther e robHoccc: .1 1 lu:tr:ci.c the imp - rtance of the cc Lb iii: in Cacti ~n 14, an

thoy in clude :ooic , but certainly not all , of the corucbinat tr ial sr obl ercc c

c r whi ch good a l g o r i thm s are known. Section ~ suggests come un:’clved

c c- l ens and directj.nc for future research . The acr e n d I x c ri ta i ri: a

list or t orm in o i ’ogy for th :e unfamil ia r with gra~ 1i theory.

2

i - - - ‘ r - o Ht~ Ioato cn - oL1rr~ 1LirTiTT~ti~ IdtL’I~i~~~~ -oio~tL - , -~ T ~~~~ - , —- ~~- ~~~~~~~ ______

2. Machi flc ’ Mt H. 1: and Io::1~ f e x I Ly M :a: ’ures

tn the early year: of conic ut-ing ( b e f o re c oci] afar so lence am::

recognizable as an academi c disci i iine~ , an individual confronted with a

computat i onal ~r cb lecr: was likely to ~r to ed in the fo l l ow In g wal . He -or

shed would ~-ori dor the problem for a whi Ic , f ,~~ uIa te an algorith-o c r

i ts r lut i on , and. a r , te a cc. -::c ~uter :r ogra ~o which w cold hoc e fro l ly .i rrj - erc ent

his :tl ~~ r I t h o r . To Lest, thu al guri tbm ’ s correctness , he a U.L J run the

program on several sets- of dat a, “d ’ -:bugging ” the pr - gras: until it

~roduced c r-r ect output f-o r u : ,H i srI. of sanl le inc ut . To f o r t the

algorithm ’ s u f f i c i - n i cy , he would :r:ea: u1—~ the t in e and storad e :~

r i .c o tled by hi s c ! - orcs: t~ c r car s the sample da l i, f i t these measurements

t ‘urvu : (b y ‘;‘~~~~, by J o a . :t— :quaros f i t , or by some ::tbor :c :Hh d ) , and

claim that L }i ’:se ourocs measured the e f f ic iency of th e algor I h . .

The -t r :ca ’, a c k : c I t t hI s ~-cio~ ir ical a ~roaoh are obv’l d i : . The d- v e l o : c : -n

O f V’ -:r7 : ‘Lrgu 0 fr a : :. such as conr~ ii err and operating :y:t oco : , re ,i o . res

a .roi”h r n - c r y s t - r o c H i c method - o f checking c or r ec tn i : s. This need has led

crc co t :-r scien ~ -: St s i o deal se meth -o .i . : fo r 1crOving the c : r r U :~ n ess (an o

o t h : r çr ’ ; er t .e s) of ~: gras:: (Floyd [ 10971, Manna 1 ~, 11 -a ra- [10 ] ) .

These methods use rcci t b :’natical i n d u i r t i o c i to establish that cur l -e n invar:ac1l

ru l at ions ti - h U t whenever cer ta in -°i ni s in t he rogra” . are r ‘ ci. 1cc i . I To r,:c ut e r

SCie r I t i ~~t s have aIm r ’ s : ~ -ooi math - dr (such arc “ structured j’r ogra ’mning”

r cc n : t r u o t ing eacy-t- -understand and easy .tr - v ur i f y r .grrc s ( I c rchJ ,

D ij k~:tr :i , and Hoare [1972]), and hav e formul ated new rograrncning I uigria ~ e:

t. mac c t h ose :cct:th od : uasy to a~ ~- oy (w] rth [1-1 7 ] .]) . The ] circs: t of t h i s

research is to - 1 - m motr a l ,e that deal ng an algorithm and Ut er i sing a r o t ’

of it s cor rec tne ss are in : ar abic art ; of t sacs- J r - ce::. Ierb . - i: s— — ___________________________________

~~ H r r ~-efor th I shal l use ‘h e ” to lo u t o any I nd kr i -iii-h , male d r i ’ uncah

L ‘

~~~~~~~~~~~~~~~~~~~~~~ - ~~~~~~~~~~~~~~~~~~~~~~~ ~~~

- —

~~~~~~~~

-~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~ -~~~~

- ~~~~~~~~~~~~~ -.

r-ec- : , ::t air, onte j f t h I s point 0f v iew 1: hI ::~ ra ( al l , h I :k ;t r a

and I [:cu-o [1972]; Di~ k :ora [197 ]).

Measuring efficiency by means of’ ercrririca i tes ts bar: the sane

lo~ l ciency a: checking correct ness empirically ; there is no guarantee

that the result is rec r ’;. mrc ble on new sets- of data. If air informed

choice is to be made coer wee r i two :,afcrlthm s for solvi ng the :aooe orob lem , :orcu

-‘ ore :y:teca a’ci C I :Ut ’ rc’ :~ ion ‘ 1 cit the algorithm s t cornilexity is cr eecied .

To toe co st u.’e:alI , thl : . i : , : oo rr:cition should U -c cro ’ i . I s ’ — i0: -~~ andunt : good

:Jc -r: i t hro: t en to- ror.’~c. good ever: if they are exl:res:e’o In J if ie rech

r ~rani n~ I :co1nia1-us- -~-r run on d.i. f t ’erer:t n:achirie:. Fuirtherni , re the

“ . -cr :mr e shoul d be both r - e: ’i :tic ouc h iou. cep t ib l e to theoretIcal s tu dy.

~n!::plCx t v r ~~~~ :Uy , ~~~, -si ta. mInd: : those which are

st as~ ~on,ie ’c enaent 1 ale size onci character is t ics of the in c ’ot d a t a t

- m d u i - s e w i u C . - .rc i coni c (ccc endero t upon the input data t . A ty1’iccJ

: tat io measure i s r uran. length. Irogram length in some sensu’ measures

nbc si:- .~ l i c i t y arc -i elegance of an algorithm (an algorithm with a short

c -c —r im and short correctness pro c i is s imple; an algorithm with a short

rogrcrrr and long correctness roof is elegant). This measure is m ost

a~prco rl ate if J r-ograrrolrig tin e is important or if the program is to he

run in fre -nuent l y.

11’0rn ami c c-or :m lexity measures provide information about the re: crrce

requirements of the algorithm as a function of the charac ter i s tics of

the input dsts i . i’y~’ic:ol clyt amoi c mea sures are running time and :alr-:gL-

ace. There mea;:Ure : are ~ ~ roort ote if the program is to h~ run ‘ c c : . .

Running t~~ e is usual~~ the most important factor re:t.rIc~-i c;g the - . 1 o f ’

problems which can be solved by computer; most of the problem: f-c t l i

examined in Section 5 require only linear space for their s :lution .

14

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11 W o o ’! , 1, r ~r.blorns with o i f loa r —t - c: c algorit.h’n :, so rage space :r:ay

i - : the l imit ing fe - c r . it ’rage r:c ace ha: Deen used as a r eassure in

pr- oUr of the c —m o utat i oncol ito tra ct :ss ility ~ 1 certain 1 roble:r o: (see

Section 2), but c ost efficiency :tci-ot es -onjicasize runn ing t ine .

Dynamic measure: require that we spec - i fy the in at data. One

s:ibility is to assume that tb d.a±a U cr a given problem size is the

w ‘r:t c oss ib le. A worst-ca: mea sur e ‘H’ running time or st rage sl ace

arc. a t uj icti crc of problem size m r dos ’ a er form an c co guarantee; the

I c- gras. will always recuire no more talc - r s~ aca than that specified

by the bound. A worst-case measure is in this sense not unlike a ir o o f

r -gram correctneso.

For s ~rre algorithm s a w -r s t case bound may be overly ~e:: i m ist l C;

f o r in s t a n c e , the simplex neth .o.d of linear progra~~ing (Dantzi g [ i” ~- 1 ),

w~~i c i i bar: an 0>: : -n e ci t . iii. w - rc t — case t i m e boun d (Klce e and 2 1 nty [ l2”(2]

s can s to run much ~‘a:~ or than ex-ponent l al on real-a r . Ut r c-bie ;c.s (Dantzig

3 1) . In such case: an ‘ average” case or “ ruc r ’- senl at vu ’ case ma:,

give a cn— or c real .I :t Ic to crc t . : ° r’ c e r t a I n r i l e : ’ . domain: , such arc s co - f icc,

and searching (Kn ot t ; [1 ~ ] ) , c ’oo rac,’ ’ case anall, : s is aL’cco:’ t : .iwcc, : ::i re

reali st ic tnan wor st -case -era. y :i a , and i n these area : much :irer :tc,s-case

analysis has’ been i cr i e . h r -wcv ’r. : rv er cr f ’-r ca , ‘ ‘ma cd c’s I s has i t s .i!’:iwt ’acr

It may be very hard to cho rc a a good r :bah . l it y mea su re . For icc r a n ’ - ’ .

assuming that d i f fe ren t art s of the cc l ci data are Inda~ u n i o r h I :

d l r c t . r i t o i u t - e d may make rhe ana ly s i s ea :i”r but. rc o ,~y to o an unreal i:’f ‘ c

as:ucco ; t i - c rc ; furthermore ev’~n a c - at .vely s i m i l e - r I c, - r o thin may ra~ ~ Ily

destroy the independence . Wi th average—ca se a na ly a l r c on -’ a ud : t l -- na l il;

run s the r i s k of t e l ng cu~~ ml sed by a very ra r e but very bad. set of i c c cot

1’i t a

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:00 c. c : r - ’ ’n: c r:co lt . :-a t~- , :‘ .~- : :rrr . crust be L O L C - .L U a c c r : c : n ’ ’ r ’ ::c. dei . :cc-

:o i s s i s : - t.~ ct n lcc ’ is tb .’ ran 5 ‘- ‘~ ‘ c u - . , . : n a c t s i r j e ( Co I: cocci -cobb - . [I ‘ - i ~~~~ . ‘SSc Oli

1 rc ou r a’u :t ro~cI-ion Ot ’ :s o.’o- r’o.c — o orl i u se cr .I 11 :1 i a I co - c o o cc . Ttio cc , c-y 01’ - ‘oc t.

a -: ,ach int : co::s’ I ’ i rc c r os : cr0! ,:; of :t .:rcfe cc . . , eniob nt -ic t O 1 . io an

.i rc t ’gor. lice rctuc’ccgc s Ills c u e sos -cU re : c:. ucc u t in ’slo’ f r ’ cc — coce the c:.s ’ : , s

t o i c , cell is .11: :1’- . Th .: r:ca ci: i~ ce a.i:o h :0: a c l x ’. - .: : 1 :c.I te iool-

c ’ r ,oI : r u c ’ .. each able mu Ic ld, an iotc - ’:’ r’ . ( I-o cr s o L o u : ’ .: .I- ’:- ’]xH, ’

r :uc::s ’.:c , we all- -a s’t- . rac,e cells ruoci, I” - ’rt . ’Oe r~’ t ic .l i r c~~ mu m~~r s. ic;

omo . , ‘H is’ n :’.ctrlr: u -cam t o-noi s ier tt - , e .: - ‘ss’o’ ’:-t s o c r c i : i ’ :r t o a Il c m

cell choos e ‘cct h’ s - i’ 15 ccc a regi :tu~’ . :-‘ ‘ r - - : 0 . - c - to -s r ’ -~~~~

‘ - t c r Ice c r c - c d -

c c a :c-:rc’.c,’: ccli ih ,-:e c Ads- ’: 1’ icc cc regi :t ’:’, r per: ’ c’: , - c o’.ri t ism c i :

-c e r c ’ . t : , i c rr tic ’: c o n t ents of t o ’ r- ’ui .I - : 0,: . sci .o o::ci ’: c ’ .’ t b 5 o I l - - co t s ~~1 ’ t a

o- -:g ster: . A :~ as’s :. os o I l e d : ‘ic . I ‘ - i ’ ’c yt t : ~~ ‘~5 i f l ’:’ the : t u ~ n~~ C:

crat ion; ’ to be c’ .ri’ I c i mit . I t s ’ ’ .LrsI ’L :il con :’~~mu ’: .1 r . f c o o

rer resor t : tis ’s _ rsc ut . ‘ , a:;.: the f i na l e - r o: ’ I ~yao-ac-i n cc ’ c:oc r : co r I i’ ’ :i I”. . ’ . c . O S

t ire ‘~~: ~s . . Tire detai ls of thi s- mach ci ’s :‘ . - oie l sm- c c,u’. c :o c c ‘o I Si t i icst

r o s s c - - i l ’ : n a.r-i rct .i - o r .’ ic rio t a ff e c t 0100si .brg tIc::’: ocr Il. 0 0’ - .0 aCe by more

~i .no. a c r ~ tant : ant -.-c - .

A ore . -c c . ccc ‘ - .c s’ rn-- ich icrc ,i s :eqric’cot ‘ ‘2 : t o carr . I - , ou t, - -ro e step ct

a c In: ’:’ . t,rci wool : has co .’Ec fl don e in the :‘c’u cot all nod c c”o lex .I tm of

s a r a l u i l ‘d r-: r ’ i ’ h c ’ .:, i.-o.t- I shal l ci dJ. ’~c:s ‘Hit , a r-lc here .

‘rh o r e : [ rc— acc cz : rn ao_’lc . ci ’ c’ ,~ deJ. r - ‘ml he: :‘. useful ft c, - r

r ’oadi s f 1 c:adly measuring t h e e :‘ :‘ic ienocy H ’ ‘irmi oco.’r c o o :‘.ig. r’.i tb:::.

but It has serious -ir w c - ’ c . ’- , 5 c r . , s~’ c- cc- uomt - t.u t c e : . Usi’c c ’ a

s i ng i’ : .‘t ’c r a .ge cc i ‘1. can h i d -in csrt .. .I 01’ c,; lc ’orsie ~r :t ’ - f e c . ci t I

J s~~iD’i c on a roe s cii a c ’ :;:: c, :aoch.:~~e I , . : , ~‘r’ ; out c 5: . ; :,i cI! I - i i : .2;

arallel by encoding :ev ’:roi i smai.l ‘., ‘ctr t ’ . ’r : ic . t s .ccc lar’g ’ ‘ c c ’ - . 7rs •:

‘—~~~ —

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‘ s c ar 1. 1 te l : s o ales: c ’ ’ cI : : rL : . .i c s g th a t the t b~o c-la i r I r a n ircalger

‘ ‘ci I ; 1001 oIl asci i to t l i ’ length of it-: b l c s c c r ,-; r ’. . r- ’.:en n ” .t l on

Ito ~~c r ’ . : O . cor d IJIL::an [p c y ~~}) , or by re’i o c r : . L c tt: ’1t ‘di i c c t - r cr : be‘~.

~ su i ted is’. absolute value by some c .cr stcmt t ic : . e: the s i ze of t h e li~ i,ot ri ot a.

ban d ic - :—a cce :~’ cc- sIr m e : ’ ‘u-c’ extr ’coelm ’ - a’ .’.r:’u l 2 -: co~~t l ’ s . 1 ’ . ( L , l;

I - r : ’ c”rc : ‘u i t h o ’.Il c c :~c a ico -o sser : . This ab i lt i to L u : ~c : ’:sl f o r O s : re: ’ ’..’ c c I c c ’

- s . f i .c lc : ien : t -n at co r r ’c, ’;s ( Knuth [ 1” j ) , r ert ’ o r:- :I c c g r c .o l ; - ’. .rc . ( iicUL ’:’. [ 1 ’ I -

c r h s ~ ’ ha: lc c ab l e s ( Knuth [ l”7 ’1J) , and the l i k e . II woo ’ in a ’I ’ :o i”..I r ’ . i r sg the

tf: : :r’ ‘ i c c -c t l i m i t s of ‘t hi s c:io :rbilit. y seems t be a har . i ;r

- ‘ I ’:c ’gocc ” : t -~5 % ] . K :1:0 -gc: r ~’v and li st e c r s k i I [ l i ’ 3 j , tAr -cl . [~ 3

‘ c’ iAi: :t. ’cge 1 i~~’x ] , :icc h Tar . i an [1’ c”~ } have c r - : - :ed ‘ .:ri c-hc l c . ’ ’ ci. I sis’ I s . w h i c h

- ‘ . - .c:es: to ’. - ‘ : c - r’ : I s U’; ‘::‘:: licit re ”: rcr .- r : e cl i . t in -h no as: c r o s s - r I 1.h :not I c

1: -~~sij bl Lc . I shall call o uch a :‘ chIne a J.iucl -:eJ ‘0 - r - ; :c , ac :r ’ cc,o. I’hese

- , ‘ . i , :c c ’ cs ao . :r r : c t ~i; o..ali the c r c ui I l lt i e s of 1 1 : 0 — : : ’ . ‘ - sing I ’ S r f i:~ ” .c

s corc h a: I i ’ s ‘so t Ucoc ’ Iol , .’i . — : c c ’ : : s l ccg fe ct o r es al ‘-‘ .f l ol ’:,_c, — , ‘ 1 ’ so o ” .crgoic c’: .

crc A J g ~~~ arc -Ut i L l , coo t the ’,’ ar c ::’.r to b’:’ :o’o- re ccj cm ’ :: ,orb ole t rccai.’m is

ir c e c u c’arc b ::,~~~‘.‘O O C 5 5 c c ccoctc n , :

Ai r ~c to.-r s ’ ’i~; s ic : . ’ . Ic :c ” ., ’iili e : 1,1. 1 1 . -S Thr ico c ’ c :: :cchri c c ’ . (~~.: r’i ccc [l’ ’~ —~~~

ha: ‘ cccl cc: ’cJ , in c orer , , t i:’: - r - - t ccii. . c : 1 ’ , . A T c c I i r c g :n r , c c h i : o .

c’:r r s ..i ..i ing :5 ’ a tai cc . Tb— ’. 1 : ’ . : ‘ - i s lv .1 .i~-’.-i into S u . . , c - c u c-c ’. Sc : cc

A 1 o h ] rig c c ’ . f a 5 1 Il i ’ cr u cc , ’coer c ’ .~~ ,-‘: :. cc I . ‘Ii:- . c ’ : c rc c i r I c ’’ : I ~ 5 .‘~~~ .‘ .

a f i ni t e a lt rri al mccc -r ,’ and a r ea.u w r i t e head ~~iich ccci s i : r e I n e

emi l “.i. a t I c ’ . ’ ’ . In erie ,. t o ’ : , the c :: c .c. c ~i I n c roar , r- c: ic c c~ tai ‘.‘ c e l l . aol e a

n - ’w s~~nb ol in the cell (erasing ~~ - c t was there j red ~: o~; l , no v’ t~~

‘.3 ‘wr it e head - - o ne cell :or- w ’ir d or ba ckwar . i c . - I . e ( :11 - , icn .d c-hang ’-

+ - ~~c n t ’cccc-d oco c . -:r cct ’ ,~~’ ’ . The : Ieeism ‘fl ~j~~: ~. 0 ’ . c ’ .c - ‘ t u -orb : 0 ’ ’:

d ’c1’ccnd:: :~ri Ly on the current i c i t ‘.‘.rna~ 0cc’. r’ ; s ta t ‘ and t t i ’ ’ ‘ . c :n t ’ r i 01’

the tat e cell b e i n g read ; t h I s dcci c ’ er r i o enc led U r - ‘ac : l c d - r c A

~~~~~~~~ “

•:

~~~~~~~~~~~~~~~~~~~~~~~~~~~

1~ - - —- -.-- ,—.- ‘

_ ‘— ..—‘,n,’j-w _.’~_ - —‘----_

~~~~

_-

~~~~~-

1:-r i e a~ d each t.cic to :‘rc :.l’ ol in a dec is . i ri t able ’ which t. ’ s rr::: ths ’~~~c ’ -~

01 ti le :cc c cdh i ci ’ . ’ .

Thr ic ig ro~ - ‘r ed hi.’ ::. c’.chircu model in lo t ’ , b e f o r e el. e-ctr . r , I c dig ital

5- : ft j t ’..:rs cx . : s t em ; he ’ so::r s ~ t t o -~ t sU’s ’ to model comjs utational r c u s S es in

the 0 0 : 1 1 ” ,:: • aol t i cout oi.’ .c ’:r ’ccce to c’u:y 00:0 . computer. ‘l riUUg :l Thrs:r ’.’’

01,01 is i:.:- ’..l:quatc 5 : 1 - 0 o ar -go a r c of cc - r oc re to c ’sscni:lexstl: r ’ :cesor : i ,, .I . t:

- ..:c’.pL.rcitt ; curd the fact that any r:.r co. .’.. - c’ . access ::cach:Ine cars i c

.~- i s a ‘I’ur~ c o rI n .aci i .I mc with - ‘rrl ~; a i olyco oucrial bl~ w— u ~ in roco c: .icsg t ime c’ . l-:e:

t i r e i oU’irlg machine oxtr ’.c c : . ’.’ly u:’:S ”Ho “or studying very .11 I c ‘ I cult c c ::: u t ” .I I :c . ’J

tasks . It is: also valuable f r :1 rI ; rig o r 1 i L-c :: : w hor e ta o ’:: are the

orno ’ : der i c ’.: , Or : for if l . ’ I ’ u c: ’: I r ’ . ‘ a: - ‘ s c-t l rr g (Knut h [L°75}1 .

Tn 1. wo o t wo-a soud -:5 who :1 s ’~i, is . : “t.cc cc or, :or.re cr 1 tical :s er ’.cc arc :

One . c . ’,int S in the rucs. ng t I::.e oo’.cw’r’s: o’ e: oil’,’ .~‘:‘ that c r c ictr i. .r c

For i n st a u r c e , ‘s ri : rt . .Is , c I co Os.i :‘,..c-cco l fl 10 : 1 c c : : it is: .rs ’:s ’uI ‘tc’ c ’sc ’. t

c -scsi ar t son : (or gen’:ral t - ~ rr c .ory dcccl 51 - ‘c i : ‘ . c:easuri c:g l i i -: c ‘ .: leo-c l I f ’ .’f a

~ u- si-cm by tb ’. d o t ic - U a - i - c 1 : I ‘cc Ire , : : ‘ it ( ‘ill.) , II Cr I I . arid IiiL:.cu:

[ i ’ ’ f } ) . In arithi: ’ :’ :t .,’ c non ‘o i s - ’ t ’ c - o’.l c r d - c ’ - .: , i t i s us d11 t o o~ucit

“ucI (hrc r ’ :t io ~‘s ‘‘ r a t . i -o . rc : tur d ‘ o a:c’ ’zca: 5 : . : ’ . ’ 10-s deci s i ro : art ’ .;,: ; I . e . , ‘Hs~~c

-- cc. : 0r at ] t ic or : -c ’: : -cr ’: I : c ’ i .c , ‘ ::a ’ cst Ot ’ t i r e i c r : cu t -dat a ( I r a sot c’alcco

t r l lccrr , s i r : - ’ . ) . In this -n . e :ccno ‘ ‘ ‘0 O t t ’ S t h c ’ C.cr~ t e x l t ’, U a c- ,:o- loc ’ , t1’ t u e

lo: rc~~ l s u t - s - - rtc i~~ht - l l ro ir - gr:ss (-b . , ‘ c or U - , ce~a - IL’:”, , ~1 ~~~ I

- - ‘. :, “ l :i tu a t c s c s : ctc.n: r ’ acce .. : ’ : : slay bc the cri t ical ‘ c - c .1 ‘cc, .

r thi c- o c r I c I r - ~t 1 us’: a : ’ r s t — ’ .’ :L :e ru n n I c r ~I t I nt: cm a c ’ c s r ’ r

cac:o ,I c s ’ : ’rc ;’ a c’ .’:as ure of aig 5 r ’ . t h c c . l c c a :cc lexic. - , . I b i s r: , e’ :c:OL” - is

an ti r ’’.aai . t i c O ’er a wi de rang’: of c c c c t i l c . c c t , - r I ad r h o ’:’ . : . i. s h ot_I l.

I ~~c - r ’~ cor i _ : fac st t ac t . r . in runn l rig t O r e , tic .. ‘ SUCh f l ; ’ I - r c . j ct S cot_c t. .‘i’: sec e d o

upon t i c - ’ exact c:. - d l , of cs. - . ’ . :c t scl I cc , l Ice ’, ’ are c i t e d hard l o C ‘:‘ I c t , , . arid

L. - - ________________

— — ~~~~~~~~~~~~ ---

~~~~~-‘-— - ‘‘

~~

- . -

they ten d, at least for large- Si sed. problems , to be washed out by

:ocoy’cco~ tot ic growth rates. To indicate functional relat ions :Lir :, I

shall use the ollowung rc ’ .t at .,on . I’.. f r’uios g are func ’t ’oo s -us cc

f ( c r ) is O ( g ( n ) ) “ mean s f ( n ) < cg(n) for all n , where c is -a

suitable r.o:itive constant~ and “ f (n l is I ’ . ( g ( n ) ) ‘ means f(ro ’ cg(n )

ts r all n , a-bore cc i rs a suitable positive cons tant .

9

- _~~~~~O~d _ I~~~~~~~ ” - ‘

‘ - —---.---- ‘~~~

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _

- .

‘ c . : I c c i ty ~f’ Tnt coot -rb so iroblems.

c i s c i re .s by ~ll Is’:ri, ~Imc2- ] end other t ’ o rocrali st :, c :.a therc .00 . 0 ; ‘ml .ans of tic-;

e’ .rr -lv tw er r f l e t t i -coci tur l , -’ bsl e’i to find a :‘ ors.cl system which w~’ ulou be ade ’tuate

for ext re:: I rig and ver~ i ly i n g all matlrerc .ati c:c .i t i-uI A : . These bar ‘- s wer e

5 ’ bed Lv lb S . - - [ I d l i . who i n h i s l ’ c.c s cu s I r s o - . ::i5. l . ’. he roes: I.he .i’ec’ c ie c ’ .c;n:t r co t o :

that ci - - ccc e t icoct of ‘sro ’..’U coul d be b cl i : subjec t to :::echaro’scui , v e r i f i c a t i on

and - :werfui en “ug h to 1- - . Oe all the~ re:::s of ’ el,s ’ . ’ . c . ’ - ‘ c.’.r l tt:’ .co : t c . Their

i c , t e ro : t in the :‘ ;wcdat l -ccci: of cc .atI ’ic” ,:~t . Ic s l i- ac - ,: t ed logic] ‘us: 1 . cc - t

the ic~::I. .I ’ - c c , “Ii:ct is mechianical ‘-: ‘ .uA c .1 cat] - cr ? ’ or e 1uivaient l’~’ . , ii; t

: ‘ur rii.g- .’ritbr:? ” . C h a r d ’ . [ lad- ] . Kiee’ntr [ l ’ 1 7 ’] , 1 oct [1 .‘ t ’ r ] . ‘Aac- . l :sg [1 ‘~~~- _ 7 ]

cord ‘j cJi~:r - ~~~r’oa’~ d’. ’.1 :‘. rcr ct_~ ‘i’: c : I r ;c i t i . arrs ’ of an aJLg ’’.r I t I , :: . . Those d’.d ’Ic ’ l f t i c r c

clO d all’,’ J i U : ’ - .- c- - ’c c t i.::t r o b .:~ equi’:a1ent~ in ‘I t re s ’-rr :e that

a r-ocb lem is solvable ac’cur ’.i I : ’ . ’ .’ t ’. ‘ Y e ..tef i c c l 11 . - c , f an c.c..J.g r I ‘ . 1 c c . , O i l e r s

11 . 5 solvable accs’rd, ng to all t ic ’ - -c t t ’ie r del ’ i n f t i . . r r : . This r- ’ccu , :l.cc ’ ::s

a t ’ the nut .’. ‘r . 0:’ an ai g.o.r~ thc”. is usually stated. as Church ’ s thcm ct 5: nor :,’

c.et g .- rit}’rn’r ( in the i r i l ’ oruc c ai cer i se ) car r be expressed as a Tu. r ircg :c:aci’,i:, ’,.- .

and an:,- ‘ru.r~ c : : c’cac hira.: ox~- res’ .:es an aig: ri 010.’..

‘n c- c a ‘ - - rccc-i. del ’]: . .. t ion 01’ an alg .riI-hm exi : te- .’t , it war 1° :sible

c . r cr.a thecsa’t .i c .I air : to :I,udv the . wec ’ of c omput at i on. Thric og pr ’cve ci t hat

n . algorithm ‘cx i soted for - .ieterioii r in g wt ,c ’I }re r a given Daring rc: ’schine wi.th a

g wen input w’i 11 ever halt. Other re :ear chers d,l c ocred a cr ’u c f t ’er ocU such

andecidable I .rublems ( . ‘ . .r ie c {i: . Y h ] 1 , w’rc lo h corrcso cial ic ci c o och er science

to the irneomi leteness results ’. of ’ ].b [’o’ i. and -other: in logic . Ic ’rhac : O u t

cc oo :t orr ’o to t h i s r°search on c rca crt : ::il ity is lua t . di.-’ evi c ’ 5 1’ 0 : r 0 ’ ,

bu 1I ~ Sing an ear l . ] -o r ~~ rk by 15r’.ri,j n Davis and , r u l i ia Rotc irics - c i . t h a t Hit : ‘. ‘rt ’ :

t. ’ -rc 4 ,h i:r ’:biem icc undecidable (Davd, .c , Ma t ij ” .::v cc. and R ob ins-s-c o [117

H u bert ’ : tenth r u t - n c is to determine whether a given polynomial e~ uat oi ’. ’ cr

has a solut i on in I

10/

_ _ _ _ - - .-. ._.

~~~~~~~, .. ~~~~~~~~~~~~~~~~~ Si 1.’...,, co. , ’ ____________________

- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .~~~~~~~~~~~~~ - -—-~~~~~~~~_ _ _

Two proof technique::, di agonalization and. sircouj atl . rc , ero ’ , he

computability theory. Diagonaliccati .or i is based on ‘ur .: I ‘cc, ’. :eIr ’- r ’ e: ’ e’r ’ - c ’ . c -

raradoxes; Cantor [1 ~L ] used .t to r ‘,“.: t coao th ’- :r c . or- . r’ ear nu” c c e r , . ~~~~~

I rcI .egc’rs and G~ del used it to or ove I i i : icccom’t letc ’rc ’:c’ . !“- . :s ~’ ’ . . - c . ’ c c i .

use it in the followi,ng way to devise an undecidable I c ’ c 1 - c - . t I~~ O:t ’.

We are i nrterc:ted in y e s—ri ’ . quect i sn: at cut t i re i c r t ‘:0 . 0: , :uc .- ic -is ‘ 1: to

even? ” r “ Is: n rime?” U-u t - i ‘ ‘ S C we iico.’e a l~ st s r c ’.- A 1 , . ~~~. . . .. Os ’

a? g. . r,. ’t.lcir s for answering such :c cece . n : ( U o’ -‘ccc:: D Y ’ Iii ’: :t

‘. Iefj r .i t I con. a: afl algor.I ’tbi cc c it 1; ‘:‘~:y I. r ’s]uce . ‘uc ’h a 15 s ’. . I r c . - ’~~.

C’ orc :.slde r the set 1; of I c i tegc.’r : su b , t h a t ri is no: - I - c : - - - . ’ ci ’

‘nd only 51 ’ colgor i o hoc: A0 answer: ‘ n o ” ( ‘r d -cc rio’ - -o ,:w --s- “ .~~ . all ) cc

I nil ut c c . ‘j’herr the cue:t I -o n ‘‘ - n at, d enser:: . ‘.— 1~ , c .1 - :sc ’.oc,cc d : -

since each algcr iths. in the 1,. . 1 A . , .1, . , ;.~., ... 1~..’ . s cc ’~s a n-n ’ c~~

-uo: w ec - ‘Xi ItO . .. e ’a ne ini ut ( A , is wr - rig n in : ut n ) ari d by ‘.Thurc i . ’ sc

t i c : s 1 1 cc ” - , .n ’. a l n r : all s- s::i tie - ‘... g - rI t his:, lulL cc used t i r e

s-none I n c a to ,ch . . w th e undecid.abcbl ]ty - 5 ’ the hall . . ng rr h’le::c 1’ ‘r Turing

:.. :i- ,c c i i ne s .

Aimu .laOI ’. r r is a cr eth d for tu rn ing one ~ r oble ccc .:r o r - c i ’-cc:- :olv i ng

met -h od m t. ’.- cur Slier. dn rc e we have eri e undecidable or wi crc F1 , we can

Irc’~e another problem 1 2 undecidable by showing I ’hat . i i ’ 1,., h ccs an

algorithm then this algorithm can be used to solve F1 . To cr cc . ’cccs ]J:b

this we provide an algorithm which c ’crcv”rts an innc: tance - ii’ ~ r ~- ‘ c

into one or more instances of pr oblem ~2

thus reducing I~~ to II

(or transforming F1

into I’,,, ). Similarly, to show that t w o -I ’cf i r i i t i ‘cc:

of an algorithm are equivalent, we chc .,w how to simulate an algor ith.rco

according to one definition by an c.cl~~’r thm acc.’rci ’ ng ‘.. ‘ t h e l i ner -

on ’ S nitiori .

11

I

- ~~~~~~~~~ ~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~ ‘,,__,_ ~~~~~~~ . . . —_, . -.--- ‘ -‘ ‘-‘-‘

‘. . _________________

.~~~~~~~~~~~~~~~~~~~

-- -~~~~~~~~

‘~~~~~ ~~~~~~~~~

‘ -~~~~~~~~~~~

-‘ -~~~~-~~~~~~~

y ,--— -~~~~

The a c’ . bo~ !c ‘ ‘cc l of ,.Ie dici’ cS i — i cir ~ - c c digItal ‘.o- -

.

rc’.:: ut tn r s : rc :ade u1:: ] ble

ccc ’ .- c c .: leccct .,c ,t:rt ., cc c ar d ‘.cc- :ocut . i on of o . c’ .tl.I cated .ad ..’rI lii,’:::, ari the

- c ’ . ’ ; . 00 :‘ e ‘- c~ ut ability L’eClL ’cle a c:.c’.to ‘or ct ’ more than rnatlrerciatical

I c r0 c c ” ‘:1 , . i: It” ..- O - i’ , this the. n’:, ignores .~uec: .1 .rc: - ~O I c : - 0U” ..:’ . : :i:e’, which

tj:c,iSc it: r. , cc- r t o i .so’r , t i c s,’ nor:’.’, ,j:IL’lt it’, t act? cc. Ccli c ’,’ ’. i’ bloc:.:

wlsici: -ft’S - us],”.’ icrn ’~’;’ aig. ‘r.S tluc .o. :e’.” Ot. ’ cmv- c c . . - ‘ - .1 cc, t ci’ “I I’ ” . . . i c r

I r ’ . : : - alice , C ‘c::l ocr’ the ‘- ‘0:1:’ .~ 1 cc c ie s- ’ t : , ; e” ’, g lve rc :‘. - .- ‘ - . - , :1 ri o in it

a c- .ac-,Lc-:uo’, c cc:. :: or of vol’ ‘.-~~~ , , cc Ow - : . c o t : . ‘ . .‘I rot . cc grc ’.s. i c el th r,

co’ertlo- .’: h o’.: cnil ’,’ 0 cut - cot: cci ’ ‘.“:rt..I c’ ‘c, cur cl:: ‘0:0 .1 c~i — t 1 ‘ j . e -Sigurltirc ::

a- this c’ 0 Led : ‘OXi S: ’, _ - . b ;’.”’’’’’r n. on’.’ In:’: ‘,‘ e t d..I ‘cc ‘- , ‘ ‘I’ ’:- . : a c’d- :t- ’.octi:J.l’,’

‘ ‘ccO - ’!’ ~le r .,0 hm : - I’ 0 : 5 :0 i lecco.

I cc : Ic. . ce, S ~~~. 1_.i. cu:c, s-a t- : tb~, 1:1. 1 0 c.ciIoe of th is ho er: c:.e:l cm.

I’a:le ~~~~~~ ~. ‘ b’cato, :‘, so cc g cic. -.c: co ~~~ sC~~l o . . c-olcl c ‘,‘::rl a,.

S ‘ci i : . CL’ ’ 1 :111’; sto w ‘ that. curl. t aro ‘ ‘ ‘.c t e r c t . .eod lrco ,; less acid lo.

.i rcrn ’,r :arst, a: or r’l..’r .. sl oe incr’’.so.- -:i ,.n large :r. Sol eri.’ tsr ’: cz:y’ccci t~ h o c

- of t hc .’ I m t cj s h d -cc , I . nat’os the cc. ‘ . c , c t c o : I . .c. ’ o’.:i. r. The t:.rble

cols’.: sh wc ‘,:sc ’.t runnIng Ioc’.e grow. cx’s 1 :.: ,. -,‘.‘:].y it ’ ‘-ir e time bound is

- -cc - r: ’ .:r r t ia i . Table ‘ .2 ‘:. :. cc .rstecc th’: ’- . c ci-: .1’ .c~cc i,:c: of : ‘to:: .. - . I.’,’ c,rs .’J e

i. cs a -~ci ’sen ‘ci :. ant -of ‘ lot - . c , . c r e - ’., I:o - lii’: nor wlt ‘f t l c : r c (~~‘~‘ t S r t . ’ creed

of tb -c maclurid by a large U- ’.,sh r’ : . ‘‘: ‘ . 1 ub ,-t’ur l.,iall’, icc ‘lu ” , c the

Sis.c0 05 ’ ; r u .crco :alvac’l” our le : : I - I c e “ t h . . unci gr ows’:: .. re slowly t i c ou c

r. ’: nt ;al .

Tcclclu , 3.5, ‘ ‘i, c 3.2 - ugge. I a rico ’ : c,r ’cit ct,r. t’ cb. . ,i , , t , bet ween ‘ . all g r -

(those with worr:t—ca:cc t’,i riO ic U.I i d , ’ i ’ - v c c ’. . . “ .i, in the :. . - t; ‘U t i r e i co~ us

ccc’id bad algorithms. c d,nono: I ‘. - ‘ 5 ] wcic no ar- ci t ly t h e U’ oct t . , c : re::: this

12

L .~~ ~~~~. ~~~~~~~~ ..~~~~~~1 ’ ’

— ’~~~’ ,~~~~~~~~~~~~

‘ —

distinctio n . I :hali cal 0 a dec . olabie rile:’, t c - : t . ’ t ‘.5 1 ’ U i ’ c ia : a

cr1 al — t i me ad go ri t ic’ ,: a,nd In t racta ble .‘ t l i e c ’w . l cc . Th ie - i i , I c t j , ri

between t r a c tah l - .c and intractab i’: ~“ rob I e:ccs i s ic s lo : - : , .c- . - : :L Di ’ 5 c , , r ac - Ic I t i e

‘ ‘t o , since any of the common ly used o uch 1 r,’ ’ del : cru c bce s~~~u ’ a t - ’d by

any ‘thor with ..n? y a i ‘ .lyn ’ccc c. ’h cii loss ins run:: ing I - ‘ . . • A: jab 1’:.’ 1 aria 2

.chow , it 1: ci f r ’easible t . execute cxc oncen ’, I a l—ti :cc e alg-;ritr.”:: ‘ci larg’.’

r ’.clc : - c ns . 5-Icc: ’ , ’ c m c c c , o l c c : ch . na? n- . ’.t : lern s-: are ‘ :t:hly : lvcib I: in exj r c ’n c c t i a l

t . imc ’ by exh au stive l.y cinecckl . cig ca:’.’; . but solving c ccii r’ : lecr,: I c c - , 1 al

I:’ . - :- ‘ecn’.c t ’ n’ - r’: :‘cir c ir gr ea ’ -a ’ In c ~~r t . ~,5 :s t In. -~~r go i c’.. ’ .c : rI S~

have time bounds wi le ir are :l~~,orcd a. - of’ small d egr’-c: ( ‘. (n ) c,r be’, C I ’ ) .

It i s a cna ,? or ta sk ‘5 ’ 0 ‘cc .: t’:xity t h~’n ’,~ t - cc identify ‘~,c ic hi ccci ’ c.cra . !‘ c . .t’c ’ ,.

are tractabl ’: and which cci”; intractable.

Hartmani s. Lewi s, and rtcarns tack the U r:4. , t - ’:c S war erd,~ hi I i rrg

natural intractable ~roL’1ems (Hartmani c, Lewis, a n t ‘- t e a r c c s [1l~’. 5]~

1-Iartmanis and Stearns [10 ,5]). By dia~ . -n a i I c -oi cci’ over all alg ‘ri 0.1cc-’,: wi t - h r

a given sj a’.ce bound S1(n) , t h e y were able to Oh c . n ‘.j r ‘t lecr , : - s c -ed

in space 32(n)

but not in :jaice 21(n) , for any sc ace 5’ ‘usc:c ’C

1(n) ce:.c

S2(n) satislying lim ini’ U 1( n ’ 2 0 ( n) = 0 ar: : :r :‘- ‘w I i r ’ .ci’ r.eo ’i s .n . c ” d

constraints. They proved a :Jcc:ilar but s’ z-rowh’,I w”aI- ’r’ r’ c - : ’ i l t c r

t ime complexity. These results ci m’j ly in art Sea’! cu- t ha t there are c’ 01cr.:

salvable in exponential space but not in polynomial :~ ac’.c , and r. ’c ’.lc ’c ’ :

solvable in ext onential time but not .I ni ~c1y’n - r o S a ”. t ’ : cc - ’ .

Ofrifortunately, the intractable rc blems jr ‘.cluce . l by di ‘ -,~c - :cc,li c’ crl .1 cc

are not natural ones. Meyer and Stc’ckrcc- ’yi’r {1”72J : r- ,’v-~’.b h i r ”

intractability of a natural jr ’S ”l en.. They chow--S that I - ic ” o r - 5 ] ’ ; ’ . 5 ’

determining whether two regular ex]’re:c.c I on: with :qurir - I ng ‘.c - c l ’ . b e I lie

13

L..~~ _1~~~~~ T~~~~~~~~~~~ -, ,~ - ~~~~~~~~~~~~~~~~~~~~~~~~~~ adT~~—’c-_ “- _~~~~~~

, ~~~~~~~~~~~~~~~~~~~~~~

- ‘- -

s ame sot l’e-ga l” cs c O orcc ’cctc . al ~‘iace (arid cer i se 000 c’nenti-.d t l c ’ .o)

for its : -luti err . A regular ‘.acpresccion is a U o~~c.’~ .c,- -o r:: I ruot~’.,i c ’ ro ’.roc

the cy’rcbcols , , 0 , 1 , U , • , ‘ ( , ) acc;rding to the f’ollo ’wlc”.g

rules. Each such U - rmul a don ‘to: a cot . of s t r l r c g . of :c- ’.i’ . - - - cool n r c , .

~.1 3 is a reg’rl-ci- expression denoting the set [U )

1 is a regular cxc r -os -o I-c n denc .:-ad. ng the set [1)

~‘, i s a reg’u,lar cxc r e sc iorc d,cc n ’.: t I n g the set win . cc s I n g].’.’ elemen t

is the empty o tr .1: , c .

5.2 If A and B are regular cx’: 005:1 ri : denoting ce’t co L ( A ~ and

L(B) , rc:ncectla” :ly, ther:

( A U B ) is a r’c-gui.cu’ ex’rres:ol on denoting the set L(A) L~L ( B )

(A.B’ is a regular ex’fressiccc n denoting the set

jxy xc L (A” and y € L(B’c ’j

A ’ i: a rio gui. - ’,r ‘:c’c’cres :l’ccn den ting the c ’ -l, c’ .ctcsl tlcc g

of the ’ o~~ ty string and all s t rI ngs U -- rccceo by o: ’r :c ” .t’;~icoch I

-c cc r m ore .t r l n c c o . in L(A)

and - :t -~oc~~~e~- . r ‘rdd, ’cci c’.si addit .i- , r iotS rule :

3.~ I I ’ A is a regular c :’m r -..’s s ? on , then A2 is a regular cx: r- sad rr

denoting th e c arce set as ( A .A ’l

To r ‘ , ‘ - that t i n - c c 1’oivalence r ~~~~~ for tw- . cueS: cxc n ’ s: I c . . I:

.cc tr ac t 101cc , ‘.I~”,- - ’r and U t oacc:’oycc r used s irr,u,l crt . i . ’ .n . Th1 :~i - t” .’I . -; : a

~ 0 Iyrio ’.rcci al — f cc , - .’ :.r,1,~-c r l h, !’.: ’- whi ‘I’.’. , ‘ c - - c a Turing maclu c , - . an ‘. : , c s ’ . ‘c ud

an exj o-nent l ’d . ::; cie€c c’. . ,uj ’c. c , w ‘ : 1 , conct ruc r . a regu.. - ’..r ox-cr c : : ? ‘n

1~4

—~~~ - ~~~~~~~~~- —

~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~ ~~~~~~ •

-~~~~~~~~~

- ..~~~~~

—~~~ ‘ , .

~~~-‘ ~~~~‘—--~~~~e=~~ .-~~~~,

, -

“c c ‘: ‘ cj t i i c ~~ t lrc c . : ‘ i - i f . .- r t . S on of the T~~’ing - . ‘ o ’ h d t c ’ : ci t c a l a - ‘ - cs : ‘of

‘ c ’Su: ‘ , s r ’ - :s? n i s oust : I-h at i t b , ’n . - I L , the :- c c .c s ’ b a:

S c ’ ‘ cc c l cii’,’ i f the Turing c cc achnl c -:e -.5 - ‘ o rs ’t acccr I I - i ’ .’ - I r r : r r l l.’ ’ ’ i i I n t he

°i ace S. - nr c ’S . 5~ loll wv that the ‘ -~u.I vat -c o-c r .oi,.l’~c - ‘ cc

- ‘ - c c r c :s i, ’ .r . : w I t h : cc cu at - i ng i, : ”.~: i ’ . c ’ .r - : ( t o wi th: i c c r ’- ’. t yc . o l d ic c c ~ .i, -o ,s

as .j~ 1, 0 : 5 — c : ‘ 1u ’ c t i - . n ccs:w ’:r abi ’c in -cxc ‘ t ac t . : I ’d c c mcc c by a 1 :c’i r:g :c , .” . ’cl r ]n ’. .

, ; ! r,c’e the !“ . : - c ; ’~~. h . . Low - I c , , ‘:,c- ’ .,rr , . :“o: ’,ilt ‘! ‘-:~ l I ’ . tL -ì t ~~~~~~~~~

- - c-c i s I . : c’rh ,!o ’ ] i can he c ’ ’.lt’- : . .h . r c s .T ac’: . ‘ ~ - Ud n t - s c . ~~; ,‘‘ ,

~~ , ‘

. cu I ’:’d- :ci ’ ’ - - o r -t ’ l -cico f:.r i’- ’g’,L. cir cxc c- e:s. I .n: c :u:I r ’~ :‘.r l r - ,’ ’ ’- c : ’ ; : ’ . . r c t l c ’J.

in , IS , -; ‘ c c l S olve ‘rear:, c- ’;’ r d c - c- i- s c uob - . r’ -.- : c i’ , h - -;- ; ‘ -‘c- c:

‘ :C - ’,~~ : ’ ’ - , :. i~~~~~it [ l; ’. O - c e ’ , - - - : tha I 5 : ’ _ ‘ - . ‘ I : ”.c’ ’:r’,’ erc ’ ’ - ‘. - ::I ’ t . c.

‘ rig t ho ~ . .‘a’o.’ ’.’cd-.’ciee r i - blori I c r r e , a l a r Cs , : :’ - , ,,. : ‘ ‘ 1] ru - c o i l s - ; :

c . r. c 1o 01 1 0 O c - ’ . ,.t . c’.’:c, -co ’os arc~~ , I cc’ [ 1 ‘ ] . d , ’~J ’ ;O ’ , ’ ,,

:‘c c c t i ’ :. I : , ‘s ic ’ I ’ c : t - - c h e r :i’oo’.r . l r ig the c -c looIv ” .i ’;cc- .’ ’ ’ i :1’: c r

c’ -:~oc c ’L oc.r ox: nc:: ci: has a I : . d r _ , . ’ ! , d r , : t” . ’. aa’ . : :a.c e : ’s r’, s . CI. ‘ i i ’ . cc ‘cici e rr : Sc,

5,-~~’ h . ] r - c i cc i f . “ ‘ ,: :O s c t he ‘‘a. ,! - c - I I’; i t ’ a U: :” : . ‘c S c f’ . - 5 , c ’ ’ S

- ‘.,c’ I t ’ !cccne t i :” ( t h e the -r ’; ‘i f ’ :0 .11 ‘ t r o t cru c - .5. ’-r : w I t h no’ 1st culy ‘: ‘ r ’ ’. ’ cc, . ct’i

c ’ ’ :‘,cs ros; ‘ :0 0 c c- . U r : ~re : -o, ’it ive yt ’n: t ’~ :’ “ • - ‘a LL’,t : - r : .

‘Oi’.i 5.~’:y’: r [1 ~~‘ - sO w- . c I t h r i t the ’, c r S :r - I ’ ‘ - . t ’ccr As . ’a d l a j c c’ ’-: . . cc-

- ‘ , ‘ c :c r i t ’ ci ”. : co s c- . “.~‘ a~ i- , Sg Icro . . c~ii i’HcL’c h , [ 1 - 5 ]

--w’:’i that t e s t ing t h s - .o circularity ‘ .f at . t r iL : t c gr: _ cs,..c’: (a - cc b

‘ ri s i n g in ; c ’ gc-ar:ming langu age semant ics ) requ .l r - - c c x c -:: ~ ‘ S I ’ d f

The idea I n all th:~’:’: proofs 1 5 the sane : one .0 ‘w: I. w I i ccf fi c ’ ’ ‘ . ‘1~;

c~- r iv erS any co c c I u tat I ’ . I i with a jart i cal - cr sj ace r I ‘ ‘ . x ~1~’ -

a,cs Sn. 4 anr - - v U tb .- gI ven c c ’ S I . . and ‘ c u e ~ ‘ ‘ aJ.: S t S c ’ l0 : c’ ’ c c c i , . 1, -c: ’ :.

5’ .t , ’;cc rr ’,;: r e c u it s to assert the cx? : t ’ c ’rcc ’ ’o ‘ . 1’ an ~n : l r , - t c ’ t - ~ l,’! e r i’ :5- ’ ! c: a-cl_ I in

15

___ - “.

.‘-‘~~~~~~~ -— ~~~~~~~~~~~~~

I

. : ‘ L ] ~~”.~1o:: ~~~~~~ o ’u

:L~, “

i: ”

~~~~~~~~

’

:~~~~~~

’

.: I.; ,i : r :” .c” rs i’; i r :1’: ’ : , c’oi’ :c i,

0 Ii ’: :‘o cid c: ,oco’ ,‘.c ’r,caO’le :~ ‘t. ‘ ‘ I c’ . . -:~~‘c’ a I occluded Icc I’ ], ’ ’ l ist ’ o~’ rc rc c-c ,

‘ ‘ . ‘. : ‘ c r o c - , ’ c ’o i; : i ’ . s ’ lc-r c ’ . c: . ‘ i ’ i c e c o ’ i ’ ;ob le: ’ : i c c , ,’s’~ L i l t ,’ c ’ t lc’~;i ng r o ’ ci ’;i”l ’:. If

coo l’ ’. a s o -Ole::: is S crc rc cd ‘ci a 1:0 :— cl . ’ - . :U e:5 .l :s , ‘ -c, : t , ,e ci ci,’. ’ : ‘

lice-c c t i r - ;r o - Is a iyico:cU c ol.— ;o c,g t c r ~ ~‘, - c ’ cii ’ t ice ci . , “.c’ c c . “ cc-l ’ : , , . ‘ - -

-

su c c o s e cc~ c’c:” : i , o ” o . e c i : - .- ‘ , : c’o ,l, :’ ’.u :::~Li-: :01 5 1 ’ , ’.::: - ” .. f , l l s’. : C.

a giv’ ;c : c~c- ’ : :: U cu:c cc i c : a c tabie c - I U 0. o-~ci”. iceo’! ’ ic fi :~. . c c: , c,’ cc

I_ ye: . -c c : . eric. o ceco - ’ ‘ b y ‘;c-oicI I .l t I : c e s’S- - U , ,.’, cot r,t ’i .b c i : c - ,-:i c c - ~

To c cj i c’, c : ’e ,I , c , ‘ I ‘cc 05 ’ :o ly’c: r c .l_- J — . ;‘,-:: ‘.~ti: ‘!~~‘ ‘) ‘. c- ’ -.; ‘ cc I x- .j ’ccc

cc c — c . ’ - : ” - l ’ . : c : , ’c , ’ i _ ,:c:- : ,:. c ’, :s’ .cc — c ’ - t - - o ’ c v l r u l _ t i ,: : ‘ ‘ , , c i : I r c ~’ - ‘coy. -‘.‘. vac u ia:

:.L c c , . : 502. I c c : 10.: c~~u ’e0: ; : ,I c c . ro o d -c- . .a ‘ ‘ -‘c . ’, ’.. i. cd:’ ’, ‘ U . do r o t c - c t . I i : ’ ’

c~,aci:.: c .: accccc : , . ’ a rd’:;:, ,l c ’ , c c .d, :‘ c.S . ’ i - ’ ’ -:’c.I .’I: s,::.c , , , . : c o- ; :cc’ ’

cdc I~~h ; cau cc’5 t h s - .’ : c ’.cc ° ’ . Isne o ’ ‘.:v . - c ’ . f e c ; ,.d : 5 , -s e n “ ye: ” . ii ’. ,s , :’o ’ . t i , - f l o e

( ‘ c i ’ c~, . c c e l c ” ’ : ’.rir ei iv; ‘~~,e c-c ’cc:,’ . ,’le U , :c.c ce ’:t cur . l : , : c r i sc l i e - f i ’ . I : c s u ,

ar’. .:arct of ’ ti:c ,c ( -c ’ se as”.’ ci, ‘, - ‘ .c 1;,’ arc acce: t l i sg a corn: u f :’s’. - . II1L foui c ’,ci:, :’

nc c — U - c t ’ c- al c .t I ’ . ’ :d.g :‘ ‘ c . :sc: . ls.v’.o’ c . i re “ - c’±c’,uc: stait e set cc’ i ’le:r . I c .

c u d f l ” .’;t c ”ir s t, di’:, : :r sub; . e t . ‘I ’ k a ’ . ’r col ..”::. - c-c c , clues ’S : all

‘~ I r , .’t ’ ’ i ’ - cc ‘;‘ .- c ’ ’ I:: -,; , c ’u r c s ,S ,~~Lcoc c,cc 1 , . ‘.‘ciceo c i t ti ,~ i - .-; ‘ of’ f c c ; \‘ert ,1 c’o :

‘ci’-: - , t ’ - ‘ -c: . Id , y .s ’.~c 1. ’. . - t!r’c clot:: ‘.cI ’ ~, ‘ c : — : . - n’clclec :ic :~~.

:‘ .r, ’ r~~~c’:i:’~,I, cally in , 1;’co ’ c’ ..I cd. 1 , 1 c c ’ and let ~~y c ,I~;’, , )t~ Li .c c ’ ! , , , so: ’

l _ r c - : — c ’ c . i r bloc’ : : ,J a oib ’ ,. - ’ : . r . — : - - c , ’ t ‘ n i s t I c : a l _ y S r i j lyri c - l c d . t ’’:c . The

. rc a- w ’ :i I . . - ‘cc , ’w’ r is . “ 10’ ’ f i r e r ’ . r r : , tori ’c Ij c r r i’;’ - we l ch - ‘.rc in

- S it , cc .‘ ‘5 : , - .

~~~~~~ [IL ,~ ’l ] ‘0 - .d I .e-cot T~D ~c c c t c l ’ o :n c :‘‘. ‘rta ’.n ‘‘ li -c- ’ . : - ~~~~~‘‘ H’ c c : . .

‘. ‘t ~ , ] . - c - 5 71 ’) - ’.: c l - c t. ” I - c l ’ - - , . A H’ l ie- c :’. I is ‘f~,) _ -’ ‘~~

iot c 1:-

1’

- ._~~~. . . - . — —~

__ . 1 __________________ _______

ppp_ -, .-‘--- - . -‘- - - ‘ -‘ - -

~- . - ‘- ‘ -.--‘--‘ _.‘ - “.~~~~~~~~~~~~ -_---- ‘--~~- - _ - ,

‘

it ;catj .:”Ics fa’ - : nc- c c i i I c ’s:

:° .‘~ I is in ~~~~~~~~~

3.5 is ’ ~ , is I: ~~ t l c c o t l Q, is reducible o c S i c c i ’ .11cc “do d. S I::, ’ ’ .

To cay tha t Q is reducible cc. . F in p - - lytc :cac ”.. l i t ’ - c’ , ec cr : c hat

thers is :-~ (de t ’ ’:~- ,.l ‘ .1 c t i c ) i’-o lIc’cc , ’.rcd. ’J . - O I . ::oc ’ cjc~.~r , ~ 5 s ’ . ‘,‘~ , i ‘I ,, g ’ c’ ::, ‘cc

In st ance of ’ 5 0 b ’S’::’, , ,‘~ , wIl l :‘..rr vert it ci cc l, -OO i I cs :t ‘ci ’ .’ - c : cc h — I c c : , I

cash: that t h e cc :wc ’r ‘f-c t1c~’ , n stance ‘U Q, is ’ “ yc ’ ’ ’ ’ . 1’ ‘c rab ‘ccl’,’ i l 1,1cc

‘urs a ’er fe 10’ .’ i c c c t cn r s ce of F is “ye: ” . if Q ,1: c- . c’ s . : hole c: :- in

t . lo c s ’ ’u c ’ .t F ha: a i cll,- ’c :’. c’sl. a l—t i : c .e cal ’ n I l . ’ , t i , - ’ : , cc’. U ‘es’

Th:s: if ar ’s’; “ ~~ — ‘.0 c , : ic-to : i” : 1cm ho’.: a c”lvnoc:’c,i - d — t ,i:oe a lig n’? c i , . 2 = c’

Cook’ s coc ci_ I cs i”. , alt w i : t ’.c , is cw that ‘die ::co .1 - 1: aOl I,1t’; 1 1 ’ - c c ’ . ? ‘c’

~ re~ ,c s it i ‘r : . ’d sccdc’ulu : .15 “ 2 —c c ”rc; lote . 51cc’ : aI I : : ’i ’ i ’ l ici ty I i ’ : bolero is 0

oi ,c t-enc’r ine a~r c ’ f i c’:z ’ a gi-’.”:cc logical l’ - rocui. -c 15 t ru e ‘cr c lca.t ccc

c~s:’S gnmotct ‘of’ c_h,: value. “true ’’ ‘ ‘,zc . S “t alc’ ” t o t S r ’ - ‘c’ ac’i ‘Ic’S.-;:. ‘1 i~ ea sy

to - ic. ;’w tic if tb c r c - S lo:c. sat.i :I’it .’s 3. 0 . Cook c i ’ “s’. S ~~~. 5 5d-~’ gIvi ng a

:].‘fli ‘oiaj.— t .1 :- d.g r ,i th cc ’c O ’er s’en:t ,ru ’.:S,I n e , f r ’.ccc , cc g I v e : . c i . c , — :‘.-i’:r:”oi ’nist I s ’

Turing machine , a given irc~ at , ar cS a given .-lyt c c:,ial t , ir - ,e h - c c :, a i’ogIccd

1 ’ - .r~-c sLa r ush f ina l lice c -;rcc,-,ili. a is :at i sci’dl_ c ,b le if and c:l~; a: ’ ‘ i nc Ucsa’i c :g,

ccc ,cichi.ne ac: cc 1.: f loe in~ Ut w i th ,i n the ti: - .o t ;- . ’ucisi .

If erie iu-c - -w: a :cngle : r O l l _ c c ’s, F to be T ’Y — c’ :c:’ , : 1’: ’, ’ , ‘ci - ’ cocc i c roci-”. . -

another i r - a - I ’ c m Q 702 - c o rn~ lete by sh owing that Q, is in ~c Y arc , t thoU

is’ reducible icr io1~m omi al t ime to Q ro:-erty ‘s .5 then c ’,ccllowc :‘r ‘ccc the

transitivity - 5 ’ polynomial-time reducibili ty. bar’s [ 1— ’~’s’ ) us”., ’cl t h i s

idea Icc; exh,ibil a number of natural ~~j -c-crc ’s] let c’ c r - b c ] . , ’T : : .

continued th is: w . ’rk , and the number s.f knowr ~~~~~~ -cnc:c~ lie-I .e Fr ’cS i ’-c ’ ,: 1:

- - —

17

~I’1 c . ~~~~~~~~~~~~~~~~~~~ - -~~~ J

- .,.~~~~ ~~~~~~~~~---w~~~~~~~

ii x I : t i n , }cw ,d ~” :’r: (s e e c”~r i c r c t ’ ,ccce- fe r n . I t c s i , ac -cd :lr ar: , .i cc [ 1 1 7 ’ J ;

. cic’c’; . U . i r r , c - ’ . c c . cu r d t cc ci’ , ! ’ [ 15” I ; Ocir s”;, : bL ’ : :- ’ :i , cc: : J ai’~~arc [l~~c’~~i :

ban- c [ 1 . 1 5 ] ; scaimi [10 1’!]; Sethi [l ’:t ’ I ; and UlLr cccr [ 1 - i d o l ) . In a d u i t l s ’r r

to S i ce oa t is l ’ic ,cOI l .ity r . Lie::: and t . h i _ ’ - ’ , ’s”: I c :. c ts: - cc :cble set r i _ i -s , thc

c ’ ‘ ‘L ’s cl cc c rotcj , ’.:rcc s’- c_ c ’ s - I’: — - ‘ c - c c LoS e .

?Iut:gc’ coh ~s’.oco:o~~ ic ic: c c (Ccc ~k [do 5 1 J ) . Gi” .’er i o~~ ‘ c c c 0 : a r r . i U , is

1 isc’o, : - o i c .I c to a sub g r aj b i 05 ’ Ii , ‘S

: 0 , 1: c i ‘O ral ( dcccv [ lo72D . O;’,’ oc , cc o:’a~5 . ‘5 , c c c i t s ci’ ’~ ’t icc : ‘c c

c lor d wI th 0. col cc: so that cr ta :sccta,- e c ’ r ’ U “.‘ ‘ . ‘r’ t ,I c - - ’: i ’ . r i”.’s’ 0 c c ’ ;

5 . 0 0 0 1”:’ TicS : : i-J.’Iem ir . ‘/? 2 — c -c:~i l e t e even I f ’ Sc = ~~- c - c c - ,: C is

d arner (Uars’l ; , 3 - : : , , c , , end Ut c Scc’o ’ ”,.jei~ [ l it” . ] ) , i , : c s c ! ’ , : u : it 1. 1.1 cc:

c r c::, dc c cl “ c c i . b iake :r ’ cc r ococo ’ ci ’ the U sir c co : 1, :n’ cuc r , ’ e . :Oc .r1 ’ e (Ac el

ii:cicc:r: [1 7] t ir as f - Sc o r e 5 : a , -~~,‘ri “ .1 c a — l I :cre cj gccrl tOo ;. cc : ‘r cony

lcinccc’ cy’ac :i wol ti : U cr cccl -r .: .

iicuado ton cy’ - ’r ’ : ( d o n [l’j d 2 ] ) . - d c i ’ -.- :: c.i gr c~~bc C dot : it - c r i t : c i c r :r cycle

ci , , , c l ’ . asses c _ rd ue~, ‘:c” ,or a vert -,;x .-x a c t l iy - ‘ r -’.c ’. ’0 flnd s ~ r - -b-is’:: 1: a

cc c c l al ca:e of the tra’,”.clling :uJ ’occ :cai r i r - ‘bo le- cc , ( c e - s - c c ’ s , ‘on ~~ . i t

i: ~‘ 2 — c - rca i-.:ts’ ’.”v’;n i i ’ U i s c lunar (Urc r eI ’ .’, , “ l’s , c c , arcs’s Sa c - i a n [1 T ‘

“ i: , ’ ci ‘~‘Iarc [ 1 7 1 1) . -‘;:iv’ ’:i -’o c’J of r cc c c , c,j , ’ :’ c fl~ .ri cci-’.,

:u:. c o c ’c5 ’ , 5

1’.e: ccc’ , ’ ’ 5,0 - cei oS ’ S i n - ’ c cc i - .: - , : - c ’ : ciLco c t . ‘ . -x: ’ .ct -S.’,’

~ . sar’sar cuh gra~ h (Liu acid S. ’ldocr:icbr ’’r [1 ,‘ r ] ) . ‘ ‘lici’ :cc cc ‘n c r] Si -.5 . s ic :

i t Con t -as cc Os 0 l an -c c ’ 5 1 5 g1’ c~ c Sr a— tO at l ois t k ‘ccdg’. ’c?

A major ciT c- ri c r i , ’ ‘t ’ ‘o::,~ lc:-c .’.ty t .b1 , :c ’. r ’.’ 1., to c o t ’ r” : ‘ n ’. ‘,~i , - s i c ’ : ’.’ “ 9

A r i o t ‘sr o ci, -c c : reach t ’.’. thi : r bloc: : cc - c c I cc b e 1 o t c’y c c . i cig si - cd ’ c , ’., - ‘ . a

‘:xlio S at a rob leco , in 719 but not in 9 . :O W ” , - cc , r ‘ ‘ S c ’: ’. ’. ~‘ c ,, y : ‘ . c cSc et ’ .

0111, and So1~~iay [ l ’ f ( ,’ . ] :~~~g ’ . ’ :S : 5 5 c c ’ . ’ di’ ’.’. ’ c c ’ ri . , :at i c c is ic- i S r i 1

_____ I ____ —~~~~~~~ ‘ ~~~~~~~~~~~~~~~~~~~~~~

- ‘-- - -v-fl—

re s ‘l”.’icng t b c , ’ 9 = ‘ 19 ? ~u - : ti - c r . c .~c_’ , , c c without a iroof that 2 = ‘ 1 2,

it is s: t csll :‘ru~I c_ f uJ , t o add new natural problems to the list of

~~~~~~ -conn ie-I c- one :; the large amount of time spent by bright pec’Fle

fruitless:ly ccar’c}iclng :‘cr ~ulyrco:sic1 :d.-time algorithms for ‘S7p -c ’ccr :~,lete

c roblerr ss is strong e~~ dence that tbc~ 7~p -com-c, lete o rcblec :c : ca’e in fac t

intractable.

19

- ~,— - - - -~~~~~ -~~~~~~~~~~~~~~~~~~~~~~ I ~~~~~~~~~~~ — — -_-

~~---------- —~~~~ :.,c _ ,__~:. ~~~~~~~~~~~~~~~~~~~~~~~~~~ ., — — . ‘

— . ._,~

,.- — —‘ —

0 . Techc . i s’.’ ,: :‘c r S ‘.5 AJ.g ‘n ithcc c s.

Although many Scar er tacit combinat ;ni al i rob l ’src c: seem to be I n t ra c t a b l e ,

many others ha”s’ e good alg-snith.c ’r’.s. A :rccall n c_so,hei’ o:’ d a t a ciarac ; u !at ’I on

t ectnr r ?. , cue sc l’ cc rm the bas S : c ’or these a igc c r c t ~c-’.c. This section ‘;xar ” .i . nc- :

these techn i que:. w 5 r i c c c i ‘.rc :ut l i o s - ” i n Table 0 . , . .

[ Table 0 .1

data St ru :t, c. i r ’ : sc .

Any al g or I t h m (good r bad ’ re ~u: r - .c ‘one r ‘:, 1” : data . c r aster ’; . -

r eco r” .os’:nt tOo ‘2? emsont s of t i r e r :’h iccci I be :o~ ci’ ,-’ . c an-U tic-s i n : c ’’ -~ t I - c ,

:sm,1 ’ot’sd duri ng the solution r ecess . A data structure 1 s a c.oco~ o: :tc-

ob ,’eet o- :m~ u r e A 31’ eiecc ,o c;t s rc- . at e.cb I n :c ec i l ’i e c n w e i c . A s o . c c i c ’.- - o w~ t1,

tO’s data cctru ’ctur ’c I: a c ’c’t of ‘ler ’at’S :ns U r ‘c : :c c lc c_ il ~‘.t In g 15 ~ ele:: , c- n t c .

c ’ .oc - a ge.-J iso : L c c n o n t a t . . n . 1 ’ a g i c - o c ’ . data s t r s - . ’ t ’ c r o ani ’t c -:r,cc1, I n:

is kn own , . r,-; can regard the crat - ens as rj ccclt i ’,’e: a’ - ’ : , mo . o:c snt lon g

any a~g ’ r i t cc c : . wli , ch use s ’ the -lat e :t r s co : c r y . The e S ’f ’ l c l ’ . c c c ’ ,’ :1 ’ S h e

c ’ .tg c’ I t ’cciro w l . 1 d’ci’end a la rg e ext cc c . cc: cc tbs ’o ’l c ” c ’ s i - : : ’ ’ , ’rctc , t i r, oc ’ l Os - ,

u’i ccl ’orl y ing oc et a s t r - :- - O sr ’ - .

Th’:r ’ ’ Cr-: t w data sc r-si. - t : . c r’o . 1: ci ais le -h al - tSl ’:r. c,r ’ : ba , c c l :

o . I cc l c ’ - .c t r c c - l , i r ’,:, .. ,‘ c as-c-cc?,’ ~?: a :, ‘S l e ct I : c c ‘of :1 rag’.

s e t . : cc ::.c - - c c ’ s .‘ ci. - c e - i t v c -t y . ‘l~ c c ‘ora L , r i o sir- .- a: c ’c a’s o’oc a c _ h an

‘cr r cc~’ . g ,v c r , S b ’ ’ :i rnt, ’or of a c ’s ‘rage :~~- ,,, r. - ’ can e j l i c - .- ’c- , S c e a c: ’ . 0:0

ir , t n c c _ c r ’ . -’ ,’ c c i i (.i ’’ , ’. r ‘, rig t ,io ’ ” ,crn ” : :s S c’ s Jc ’O cc re tr Ieve the - ‘ ,ic ’c” ’r t

‘;a, u- : r ’ ‘.rc , S sn a’ r ” .~ - - cell . The- a ‘a r’y ‘ cc ’ a ran ,s “ acce:, ’ ”oa c h l ne cccl

c. :’. - 1 , g c S al c :‘,‘s cj t ’ sc ’ , .1: re -, ‘o r - ’.’, . ne- can “. 4 , : ci rc’a, ,’ , t~ c’ :: r’-;,

cc - c t - r , , rr ca ’ c’ .c’- ’ , . t n , . r , , ass ’S , :. .i i. , d c o , , r i c , a n aI ri,r r a ’;, ( b r i c i t i : [ l ” ” i ’ .

20

_____________ I __~~ .I ,..,,T t T~~ -_-- -

~~~~ - - — - - ,

-‘--,-- ,-——--- ‘ - -

A I ic n l ’ : , ,’d ci :“, i ’ . C I or ’ -: ‘.- -n s i , ’ t ,: ~ S ’ c~ cci l l , ’ .’ ‘ ci c ’ 1 - 0 1 ’ , ii s’ ‘ - 5’ - c ’ s:

d l c i ’ l ’.itJ 1:, : a :, il , cio, r’ , r ’ l t ’ ’ c s . ‘.r i , ’ i i a I L S: arc i d ’ cC ’ :’l’.’ l r c g ,ac~’ . ‘1:10

c - i _ I l cc ’s do: ‘ s ‘ - ‘. . ‘ ‘ ‘ca l . i S ’ ’c’ . : a r - :‘ c a P ci , . c ’ s ’ -

h. ’ : ::: :‘. cnrl r . 0 :5 c : ’ ’ ~~~’ ‘ :. iia~ a ‘.~~~~~~~~‘ cc c o t s ’ . !’. 5 - i ’ ’ , . h ’ : : ’ , o ” : c ’ . - ’ ’.c

- n ’ s c , , : ’ r i r c t ’o r , ’ ‘ - ‘ ‘ ‘ ‘ c c . ‘fwo , cc ’, r a t i , cn s c c ’ .re ‘ s o~-~~l h S e . r c :~ 1 : 1 , 1

::,c ’ c : : t ’ cr ” : 5in’ vctn ci L : ’ . I f l 5 s ! ’ t~ a res’ ”.’ i. ~~~ can e t b s e r :1 c’ - “.

cc I S cc’ i n ~1i’ - cc - - e l .cr ‘.‘ ‘.~~“.:‘ ‘ . ‘ .-‘,‘ ‘- the :-ccr r ’. ’cc l m l ’c ’ ’ c ’s - c , coo i t c - :r c

11 ’ : ’ c - c- : - : . P’i~~’:r-~ I iI ,u c 4 r , t ’ c s a l i n k e ,i cc , r u - t c. ’- - ’ . 5.1: ’ Sea:

‘ ‘ cc c’.,,,; - - t’i r’ ’’~ ‘ - t i ’ —: icc ’ s c : , ’.’: C a : aS I s ’ tc - s I cii’s ears I i c i ? a ’S - - - 5 5 ’ ’,’ ‘si - c d, ‘ - -I ‘ cc

cc , ‘ral l , c f l . rcO -: ~~~~~~~~ i o c l S -j r - al ,~w , s c 5 cc l i n k e d :d cc : - c c . c c r ’ . - i cc ’, ’. c ’ .

so- , ‘- ; , t c c’ :’. ’.’- .- , s - - S r i - c c _ c s ’ . ri ch t e s t ,i c . o U -r e r c a l 1’ . Tii’~ cm’. ‘.- ‘, - a

dck’ ’ .t rc’sercn~:- ’c c .cci ’ci c i, n - r ‘ S ri l c f l h l c ’ : S . ’ ’ i”,i, ’c ’ c l ’e , ‘uc,l ,’’ - ’- :’ ’

C ar d 00 n’ccciarol’ , ’a a. on ‘n r :c ’, c c - ‘ci li cnl ’ce ’.i : 1 c c - c a r ’s : .

[Figure ~. . 1 1

it is eci s’,’ t ’. S c m l c ’ , c ~ nt an ’rsç s’ and li r i l - c tn ’.’l rust i s - - s s -c’. t c , a i l

_ c t s ’ ss~~’ d c ’ s - S i’’ - ’ r ic - c i ’a i l’s:qui r’.’ ‘coc at c-u . m t i : - . L O r e ’ s , td ’c~ ‘ : c i c ’- .. can : ‘ ,

I :c; J len . ’o ’cct ” i as ~o7c’ ectj ;’n: cf c’cx’r dj ”.”s’ (see Fi~~cre .1) : t i cS . :cc :&ke - I I , S —

cc . ce :sing Ca - , ’- ,’ i n iar c 1’s ’sicc€’ : : such as i”dbl’I~~T 5 ah i ‘. ‘ i ’ . A in it 3: .c : cc: ,

cc’s : j i c c i t l i s t - c r c-cs :ing c ’a - i i i t y. f t c ’ . c ’ . o m n o to he imc .-::i1 Ic t.~ i r sn ’ i ‘ “ c c c ’

‘ cc, e r r ” ., , ’ “.: q - .i :nk ’ . ; :c 1 i ’ucture i n such a way that :1 c-al ’- :00

mu se ‘ .‘nrc t, ani t I : m ’.c , though I kn ow ut ’ n ‘ s.c - c t r ’ thi s’. In ’ .’:

Using array s and l inked structures , onc e can ircc~ I - ’: c c , : comic: ’ ; ,t ‘U - ’

la t a t ,r~ . ’ ’ cli ’ ,”S. . I shall corn s I der her e f . ve c l a s s e s ~‘f S-h a s t r u c t u re s :

l i s t s : , un )rd ’sred c c l s , - “ rd” :red sets , g ra ~’ -1ncc , and S c’ - -

A li st is m~ ::qucc nd ’e of ’ ‘ - t o r r e n t s . The f i r d o e,l ’ s : : c ’. ’ n : t ‘‘s ’ a 1, t 1,

i t-s head ; S c ’ ,’ i_ oct element is its tail . Sim~ its -~ ‘ ‘c - ci t I - c i : — i i a li st

includ e scan n ”.ng t h e 1,1st to retrieve its o1, ’.’c :c’,’nI : ‘I n ‘ r i o ’s ’, i d , l . c g cc

-‘ .‘—- ‘ - ~...~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - .—~

-

- - -~--- ‘ ~~~~~~~~~~ ‘ ‘~~~ - ‘ — ~ - -~

--~

:c: the new head of the l ist (making the old head the c e c . -nd

‘.- ‘s - om . :rs t) ; a~i c icng asic “rIsc , , .cro t as the- new t .a I 1, oi ’c’Il t ct ing arr cl r e t r I ev in g

the in’o ad of a l is t , and delet ing and retriev ing the tai l of a l i s t .

Lis t s crc a-tsich .cc ty a :‘ c w u f these on erati .o- ns are i cc:: lb t , e ha ’,”: ccc ‘.0

cram’::. A ct ,’. ‘I-c is a 11:1. att ic a o l o l t i . c c, and si’, ’, e t L n al l. w:”.s -rh. y at ‘,he

: 0 , 0 , - I . A c c c_ c is a l is t a t t i c ad-U I 5 ? , cr a il s -a sS on ly aS t e e tad S -ir s S

ieI .ct ’. , n aL s-a d on ly at tIn e head. A deccue (c ieu is t e—en d ’,oo .4u’: ’se ) is a

l ist sin ~ ci ’ c , , ~ cc.iiti~ n . r dele t ion is ‘o:si’ole at e i ther sol id. ‘one can

Icc: , y ’:y ’r rt :‘. I -S lOe either as a circul ar array (adsir e:ses are :.-sc o st~:o

co c A-o le the s ize of the a r ray ) or a,: a s I n gly linked stc”o’.:l ’,:re ( i f c ’ c l e ’s l O f l

c r - c c : the tal l is r r , o t n e c es s a r y ) . See Fi~~ re ii~~”~ In e i t h e r cc ’ .c ’,c , all

ic-ora l s ‘no ‘.ox’.:’.o~ t scanning rcc ’.co ire ccc ist ar’rt t ime . The ar -r - SI’.’ O c t ,r t o : e r s t a t t no

u ses no :5a, ’ cc :‘ , :r ot r I n g ~. . c I c r t e r c out r- .’A u . r c . tr i a t arc ar:. .- ur ’.t ch’ ot cc” .~c--

equal t o h , cccmc, xlmarm c size :‘ the 1 1st be : ercc ,ancerct .y all. coOs - i to th e I t - ’

[Figure 1.

Other imr .~rtan t 1.1st ,siceration : include concatenating t w . l i s t s

( c ’,akiri g the head -of the second list the element S’ ou lewt c~g the tai l~ of ’

the c ’i r :t ) , insert ing an element before —or after an element w’ci ’-se i ’c a c_ i ’ :c

in the list is kn ., -wn , and del e t ing an eJ.omoc ent wO ’ se locat t -n in the lis’t

is kn wn . The :- ’ .o~ e ra to c-n c require a linked structure I c r t ’ ,c:’ ir et ’ f ’lc? ’:n :

l ccc~ i so -c’:ntati ‘n. A singly linked structure is sufficient l’.-r c c l ; ’ s~~’o c c - i ’ s L , nrr

and t’ ;r i n s er t i on af ter cjn , ’t Sr ’ , r element . Insertion bel ’crs- another s-tem ’ , c ,’r ’ . 1

arc-I d e i ct i or r’: j~u I r so a doub ly linked structure . See Figure , • 3 • An

alternate woi y t h -j.n d.t~ - S o l e S . , . r i is t - : r , :vide each ele ’cc .s ,’cr t aol : ’. a f l ’ ,~

which is set I, “ t rue” i f the element i s to be ‘I e l e - t ’ . ’c . The e t ’.,’cc .er s ’t I:

or o,t e cp l s ic it y dc acted until the ci ’s xt s -an t r i r ough the l i s t .

[Figur e 0 .3 ]

22

-

~~~ . .—

~~~ - .

- c - . -’

Ifhco list °i torcition: hardest I Im I lement are , n r s e r t . cl ii; ar c j lsc:c:ent

at the k-th o- c crc it io n in a list , retr elving the element at the Sc - th position

in a li st, -S r deleting the element at the k-th position in a l ic t . It is.

possible t: toonl-,-:ms’nt these ,cj- eratlons t:: run in o(log n~ ti c- c, where r,

is the size ch ’ the I i : ’. . by using AVL trees ( Knuth [l7’~~}) - r h-~’ t rees

(Aho , F’Ioo’cr c ’S , and ‘JlL’ccan [i971s]), which are rathe r c.ocml? cc , i ed i , ?nksr -I

s t ructures. Recently ~,i iba: , tIc fr s- t ~ ;i ’.t , Fla::, and Robert : []c T7’l ] i’r ” .ve

:‘om,cnd a Wa’; t scars’,’ - ut these erc,ioi -m c irr cc( i .g k~ tic:ce.

An un-o rsterso-ci set is: a coh_c ’.ction of di :t ,snct ‘-c lecccen l .’ -atth flu iru . ’.:sej,

relat l sn sh ii. Basic set cr -or ati on : are addi ng an element t a cot ,

deleting an element from a set , and t e s t i n g whether an e - l ecc .en m, is tsr a

set. “Inc way to r’~”: r’.cs so:it a set is: ic~ a si;ngly linked l i s t . AAdit .an

requdres constant t im’.’ but testing and del ’ t i . n r oqu . i :’e O (n~ .?me , ai r - - c ’ -

ci is the size 01’ the set . Altern atively, if the elem ent, ‘01’ the set are

values which can be coc’.-c’io ared and sorted , one can represent the s-ct by an

AVL tree or a i;~ ’~ tree in such a way that all three or oratIons r-; ~uire

O(log n) time (Knuth [ 1073]; Aho, ic ’mr c r . ,’ft , arid tJllman {l” .-t’O]).

Another way to represent a set iso by a bit vector (Alio, ‘S 1 s ’r — c h. and

‘sJl].man [l9”14]), which is an array with one storage cell U r e ach ~os sible

‘.1 “mont . A storage cell has two possible values : ‘.rc: . I csd . cad i rn f hat

the s o t contains the element , and false , indicating thai it J.,-es ::”t. Al_i

t hree operations require constant time using this represen ic :nti-:cn. Bit vect~’r

representation is only feasible if the number of pso s-sible ‘.clements is .‘mcs - ,i i .

If the number of possible elements is large, one can nthndc the beh avior

o f a bit vector by using a hash table (1~nuth [1973]). A hash table consi s t s ’

of a moderately sized array and a hashing function which map s each r’ossible

23

~~~~~~~ --~~~~~~~~~~

- 1’~~~ ‘ L - .

- - L ’ . ’ ::: - : c l in to an c a l m , ’ ca stle ’, :. If an , ‘lLe: ccs’ c i t is resent , IS i e ’ . lemerot

( ‘ . ‘n’ ci I- :’ . . intu’r to i t ) is st rc-ost at (or near ) the address :t sccl: ’ied by the

hashing function. .‘i , rcce t a ” .- or color ’ - : elements may hash to ’ the caine address ,

:onrc- ccc’charrt ,’m must be ~‘-n’ovided for resolving such c- ,ollci :i cns . Hash tables

sa’e used extensive ly :ir o sl’-: :, i 11cr: , and c - any Ian ers: have been a’ritts’r’. about

th so:cc (see F~iuth [1~~~ ], I-Ion-rI : [i S P ] ) . Wi th a h:i:i’i las le , additi urs,

cci - ‘t.i - c i , can .! testing r” . :i~~~c’ -: 0(n) time in the w -rst c:scc but i ’.iy

c;cnct ‘cot tims’ on the “ .‘.‘ ‘ -o ” ~~s

Act,i±tt c -c r oi . set cc s’rcit i m s are ’o: es-ui if two ‘.;‘r score set: cxi st . These

include the cii i It ty to forc : a set whi ch is t h e urr ior . , .i ni, ‘or ’ , coO t I o , , ‘I’

di c ’ c ” ,”rs ’nc e of two cot-:. For most re r -resentat i .°rr c url.i orr, ,lr ’cl-or’:es’l, i cc.

:5151 rich L ’fer cr s u - cc re , ’osire t ime ro~ cr0 I :rralL I ~‘- the s- icc: at ’ tSr ’: s loe: ;. I

s e ts . Hcnwe voo r , I: the ‘,snl’.’erce of ~1,ccc s-’nts is sccc’ssiJ.l ensu’.~h ’ . sic that a b I t

‘c’ se.cto r can f i t. ci,r’.t’.’- a ‘ca c’m c c cotccr a rdc and tI re c’cc :ccn uS. s’ r o : .‘ -::s’s S t

o’-o .s t - r operatch - m c , tS ,’:r , mod m c , I r i t ’ . ’ r c s ’ . c t , I cc , and di:’t ” .cre-r,ce - rc’cjucii’e c . c r s t ccc i.

tircss’ .

An ordered set is a s-ollect on of clement:, eaclo wi th an ass o ,I ‘.ted

rc’.L’cner.rc value . ‘iv- - ion : ‘.“rtan t cc oral . - as sri ‘.-r’.iertc’cs set: are ccc’s ., cc i ; t i s ’ ...

“t , em’octs in i r . ’r ’.:ac :ln ~,’ cr- cr and se-i - -cd c m i ; the element wi th 1-s-tin lsc’.rg’.-’ct

‘,“h , ’.ic. A va r i c t - ’i S ’ S . woa~’s exist to sor t n d o m e -n t : in O(n log n) t scc : ’

(~~~uth [197v. ] ) ; t o ’ b ir ’. :a~ ’ c’.:’.c~ arcc s m : arc the -only a: er os . I -c-m s used to

marnipu.lat- - the va lor -cs- Si r -r n ‘ . I (n log n) time i - c required in b .,, -th the averu~”.’

and the w orc t , case to sort (Knuth [1 7 ’ S ] ) . Selecting th e 1-c- I-i- largest

element r equ ’i ru : - ( n ) time (Bl’t,mi, ‘. 1. ‘yd, S rcstt , Pi\’sc :t, and Tarjcuo [l~ ‘on ]

‘.:ichrsS’.a~ ’.c, S a t e rs e- mr , and Id1 j e-nger [1’ cc ‘.~ ] ) •A priority queue is: an - ,rdec ’ed c -S. on wht cli the “ 11.. at cii; ‘i’ c c i - - “.r c:

are allowed: adding an element t . ’. the ‘~u c u ’ . - , retr oivci : i g L}:’.’ c”.ir i imum-valu e

-I

L. -

- - ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

-;‘ler,s-.’rct in the queue , and deletin g 5cc element w i. - e l- .> c a t t c .’:, is cc. am ’.

t ’r- .c c : c th~ qu-..ci.rc. By using blnu-::J al tree: ( Vu.iller c’Lri ( l , r , ’ j , S r a:. [ i- i ” , - ] a ,

lef t ic ’t trees (Kn uth [it ] ) , or 2— 3 t rees (Am a , Hoj cr :’. :’t , co ’.o ‘OiLman

[I s , ’ ! J ) O fl ’ c can .Lsm-lement p r t cr ity queue cm e’rat , i :,c s - t hat t i ccI re

0(log so I 5 ,. :: ’: , where n is the sisoe 0t ’ the queu e . Tic’:, “ I ‘cc lcoc ’ u nt c aO i “on :

als o allow one to combine two queues ~x i t u a larger i;u’.”ce ( ce : tr ‘ci rig the

smalle r queues ) in 0(log n) time.

TI ’ the values of the elements in an - - r dere ’.i cci ar e I :ote~ ”r . of

cm ‘le rat e sict oc e , then the ordered set operations can be so c-cOle ” . ! un . ‘ .‘:ing

a k - s - a ss radi x :. -i’ t , , one can sort so integers in the range I t :

in 0(Ian+n) time (Knuth [l c ’I 73 ] ) . Feter van and’; Boas has devi sed a

method for to , , leroent ing i- ri crity queues with integer values in the rang’.’

1 to n soo that the onue ue ‘on erat i on : require 0(log log n) It s i e

(van :‘hsde Boa:, 1-lan:, and llt ,jkstra [ l-1”~~] ) .

A gra~ph is a set of vertices and a set of edges, each edg e a :ih r

of vertices. (~ne way to represent a graph is by a twc-dim ensional array A

c alled air adjacency matrix. The value of A(i , j ) is one I :’ ( i , j ) is ccc

edge of the gr aph ; oth erw ise the value of A(i , j ) i s - sce r- ’ . An ‘aJte’rr:crt ’.c

way to represent a graph is by an adjacency structure , whi ch .0: a1’. :a rr ” .:.’

of lists: , one for each vertex. The list for vertex i c’.ird cidri .‘ vert ex

if and only if (i , j ) is an edg e of the graph , f cc-so Figur e L~~1 •

[Fi.gure l~. ’~- ]

The adjacency matrix representation saves 5.0-ace i t ’ the grap h is i ’m:”

( i . e . , most jco ssible edges are present); it also allows cm rs - i, t e s t th e

presence of a given edge in constant time . However , Anderaa and kos’enb-urg

conjectur ed (Rosenberg [1973]) and Rivest and Vu.illem.in [1075] j -roved t h a t0

testing any non-trivial monoton i c—! grap h property requ ir e: c2(n” )

*7—‘ A graph propert y is non-trivial if for ar ty ii the property is true i’-:rsome graph of ti ve~~ TEes and false for some other grap h o~ fl vertices .A graph property is monotone if adding edges to a graph does not changethe property from true to false.

25

- — ~~~~~~~~~~~~ ~~~~~~~~~~~ ‘ - ‘ ....

~~~~~~ ~~~~~ _ _ _ _ _ _ _ _ _ _ _ _ _ _

‘~ ~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

o S i d L e ’ s of 0 :1’: ci- 1 , 5 crc- ‘ O i o O ’ , cor al r x in i i i - ’ a rst so use , w h c - r ’ e xn .0 5 01 - ce r r u m b ic r

0 ’ ‘c- - i - t i s’ .’: In t h e - ‘c’ :’.: r . i~y using ‘in adjacency structur e , - Ic - c can search

a grur h in O ( n 4 m n ) t in . - , wlo ’re ::: i s the nurcber of edge: in the gr :cc ‘cc;

4h us re m r e. ’ -r, t as-- I on by ‘cc adj acency st ructure ii: preferable t’cr co cci-se gr cu 1-cc .

A t ree is a gru ~ lc wi th s -ut c y cl e : . , ince a t r ee is a f r : r n }c it cain be

‘oO ir e s ’e r1~ s - ,5 by -so , :td , icic - .’rcc y c oms - o t s r - . A si- rc c -c o on ‘te t way’ I. rec r s’sem :t

a tree is I . - .ch u e a so ot 0 ’ r t h e 0 , 0’ - - , - - -cc :’ : ~ti,c the ar -n i oS each v e r t e x

wi th res pect to- this m’ u- c . , and ct r ’ d c ’ . s i t o ’ m o o S - : on i r c -c o :sr r ay

(Figure ~~~~~ Thi s r en r ’.’ce r i t : , t i - , , ’n i. cj :”s b , s - ’ ci: L a g a: ¶ i i e -~~~r ’ - ’ t 5 t , ’.

be sOx b red from leav e’: - so ‘- -t , wi . H, i s ‘ 5 - c . l b - c u. ’e ,I : , : i’ tc h. cc: ,c

‘0 nivolving t rees .

[Fi gure h .5]

- ‘ cr: ion .

An im~c”t ’ci~t and very general a].g’cri thscct c techn t que .0: recurs-I ~o c .

hecu.r sc i ofl is a method of solving a ~ r - d lem by reduc sng i ’. ‘ . ..‘ one r c or e

c’ stc t - r u b L em :. The sub o r .,-hi ’coc : are re-clue -on . n the same way . h’v’.’~r l 03.1 ly

ice - subprobl ern : become s:ccaj ..l enough that they can be :. lye .1, I r oe t iy . ‘She ’

solutions t , - bhe smaller cuLl r ., hol e :: . . ‘sr t inec s ‘.c-oc ,.t .naJ , t . g ive c iu t ‘

t: the bigger :uh prohlte:r. s. until the c - iu t i ,n t’ t i c e ’ ’r - , 5 ’ a c c , r ’. t i

‘ -cc i u t e r i . As a simple exami-le ~f a recan-sionoc a lgori thm , c o t ’.: Ie ’~~ 5, 51 ’

: ‘ j l ,l ‘-w an g d efin i t i o n .f the n- tb Fibonacc i nu mber:

1~.i F ( n ) : if (n = 1) or ( n = 2 ) then 1 i - i r e F ( r i - l~ 4 r ( r n - 2 )

Using r ecursion , one can oI’tem r ‘ tat e cdi ; r . tin -c. ‘c u - c l . so s o ’ s I S s :

than would be possible without recu .rs i I I . M any o~’ ~~~~ c - . - . 1 - -s -

Inc tuding Algol, PL/l , and ~i o o k , allow rccur sciv e 1, r occur ” .

2~

~~~~~ -

~~~~~ “~~~‘~“

~~~~~~ _ _ _ _ _ _ _ _ _ _

‘—-- ‘V

whi sob -call! 5- h- .” c : ,ce l yes ’) . In ci lan~~~oi 1 ’e w i th ‘- c r 0 S I t : fac t ii t-~, sueS ’. as ’

F-.ob ’l ’l-L.Il , 0 c c - - can 1’ :: ’ , 1’ c c - c o d a recur sive :algcrol t’ Sl rr. b1.- u si : ’. , -n ctack to

cI -r e Lii ’ generated c”: rt ’n r -1-1 c c ’ ,: (Ah o, Hopcroc ’t , and ilLs-cu [1 i~’b ] ~~.‘~-

Dynamic a r ’ - g r ’ -u”x , ,- a y (c ’.- lhn ct n [ 1- 0 51] ) can be vieawc - - c “ .s- a c s es -I el kind

:1’ re ’s’ ,lr:i- ’nl in wS c~i s’ic ,nne k e - ec sc t rc ’ .c ” . ’, on: the gen ’cn” t ccl c c , : ~~~

0- S I ‘ - cc l: and

never SOlci-cc i_ S ic - ‘ccc I c’.cb icm:: twice . As di i exonin c le- no ’ the or’ n - so wh , I ‘ c i -oar ,

‘be s:ived in thi s . i c c ’ . , c ‘mc c i icr t h e ‘. ‘orc~ utat S on of c_ Soc c r — i _ S o Fib s - c o o t ,

c s inc :her. A re -c-un - : S ‘cc i r - sce iaar- bci:- c - -i so 0 . J re ;u~I re: t1:c,-c so ‘cc 1, nal

to i_ Sic sine of F (n o t o so - ccc: ut so F(n) ; such a ~ r ’cce-luI- s e n - c ’ -cc .. :

F(:ifl—i) c- rio utal,ion , cc ’ F(i) for ~‘ac:o i in the r - n : , ’ - so - ccc 1 to on

A Sc ot t or way- t c rcj ute F ( r ~~ i~: to cnccc i’ute each F( i~ lu - S - m cc c c cc

each value nO ’ t . The m o , t eff icient way I . ’ i cci- loo cc ec ot a !v’cc ar :mi c r e- rc e ’o: .ing

-~~go-rit 1-im I: 0. ’- set . on -:t i_ al Ic 0:’ solut ,i cmi: to all , on! , o so I - ,‘:‘:, - ‘ nc to ft II

In the tat -b e fi’ur:c ~cc :Ll’. c t 5 ’ - ’o larg sct ‘. “.j i O r - b ’Leoc c. Gometi , c: iic .’ ” ne c ar .

-I i s’card t1i- ’ soIl - 5 0 1 ‘. cc ns for ,:mccal. l sub socob l ems as: the corn iutat i - -cc :

and re—use I- l i - c so ace I c r larger suho r -bl , eons - . One can evaluate F ( n ) i s,

ho time wit S ’. two s tor a g e la-nati ons h I; using t,h t 5 1 . S i -’ ’ . , (hI’ ‘ . O U r ,

u s i ng a ci -c ed -c ’- ’-rmcn exo r- .’:c ton t s r F(n ) result: in an even :‘n st fn r

o sis-~~’r t :5 ’. 5 c c . )

Ioyri csc ,l c progra mcunu ng has - been ~ c’’ d with gre -nt - : 0 0 C c : : cr ’. a c cc-:: - ‘ so of

combinatorial r - ‘t Iers :, in- .c . -cc .ng sin r ’t ‘ :c c ath r ’.’oi , ,’c, ec’ ,~ (F1o-1/ ’i [ I ‘.- 2 5 ,

context-free lan guage j a r s i n g (Younger [3. 1- 7 ] , icIa r i ’..oy [1°’7 - ” ] S , e l’S I’

correction in context-free languages (Ah o and le t erc cci [ 1”T ’,I’ } - , and

construction of - ‘ . t imucn binary search trees (Knuth [1 t’l ,) , It ad [ ic c : ] ) .

Graj -h .‘Oear ch lng.

~‘~o:S graph rob lem: require for thei r solul i - o n a :1:5 coo - ca t I so cs-e S 5~

of e~cp1oring a graph . A search i s - an examination cf t , h’ 1’~ edger of a gr oan St

using the : ‘n Ll. nw’hr cg r ’ocedure .

27

_ _ _ _ I -,__ ~~~~

- - -

_-w-— — ~~~ ~~~~~~~~~ - ‘~

—--—- ,- - - _— — — -

i_ c: 1 ( S a l . b ’ i~ Z c t t i n : ‘“ u’s- all ic-ige : and n-e r t ices - : the gra-rh cl ew

- p , . ‘r’ ‘ ,‘~ -

2 (ch - .- : c ~’ a c:” w :ta ~~ . rsg ver tex) : If cc new vertex e x i s t s , halt .

(‘The -on , ri- gra~,h In-a: : ‘:‘ e - l O ex~I ore-I . ,t-he~~ ’l cc , ctr- ~c:e a cc ~~w

c’-~rt ’,:X ‘cc’.i c :c: ’.rk c. ‘., - i t ( - c-on - I ‘c ’-’ - S )

‘ en, ’. ( - ‘so’: : S o r - c~ . - n .g’ ‘ : IS ’ r, o 0 , - c a edges ieau ccw’sy c r - coo :11 “. rc_ice :,

g to . + -son, 2. ( AIJ of the +cr-io So recnchat le frorc c the cu .rr-:r, H

-.- “rtex h-i : Le er s ec: 1 r -~ o . - 5 , - n’ w : e , choose a new -c ,~~e lea-c l ~~

‘caal’ :d- ’ - m an .11 y e - :’ ‘ ‘ . ‘- ‘a rc o d e ed ge ~ .o . If the - ‘th oo r er ’..Io, - ., r c c ,

..1 the edge so oc ’ :i+ . cco ’er e. I t c . : . ~ ‘ “c . ea ’5 s-t en, 3.

- ‘ - s- is - so ]_ l c l t y c ,5~’j S’ all v er t i c e s i n the g’- oac io to o~’ :caneci’ce’d

‘ci”: o- ’ cs- -ho t S - I ’ - ‘ so coos the . ‘ ,I c-st . - a r”. v-or t~.- ,-: ,, e le c± ci: in ste : 2. Thor : the

ccc ~~ .‘ .o - n- - c c . The r e t of the spascnc org t r ee is Use

c. ant ‘;‘cr c _ ’c:’s . The e-dgc~ of the spret’J ’..rig t ree ax’e use e~~~e’~ at , , en lead i _ O

c ’c ’ cw ve’ott,oes when cxc s-- c in step 3 . The oro p er t ies of the soranning tre e

end ‘rc o r : the -r .~t e’r ’ ‘ c us-cd t ’c select the st~~~~ing vertex in c’tep 2 -‘~t’c

i_ me -- - s~~ s to -cX 3 lore In ctep 5 . For :j me simple graph 1:rcc to lLe.rns, such ci:

c ’ -.rc : ~rsg - ‘ - :5,0 ’.- ’ ‘- l , C l L c ) ~~~ cc erc t s (los--cr , :-I ’t :e t .s t Tarj -an [l~~ 3 c ] ) . 0-i-”.-’ o r b - s o cc: ’

‘x ’ . t r a l , : ur , is :ati:i ’ asc t my. However , c ’ .r harder graph ‘ic r ‘-tIe -cc :: the

“ c-ct i:rc~it i or’. .rd’-r is ‘c ru sc,i c’j .

iii a de1i_ h - f ir st s-c-arcs, the so. .cge selected in step 5 is an edge -°ut ,

of the last explored ver ’,oo’Y. aci th can d,i - ’iate edges. If a det t l - c - c ’I rst search

is perfor med on arm ur: ’.c, l O-” ’ c - c gr ’ej~h , the generated spanning tree ha: Zo o’

r’c~.ert-j that ill nun -t r ’ - c - - - cr - ’ s cec nr ieet . ‘so-o rt l ‘.ces related in the S i

(Tarj an [l’~ t ’ - I ) . , ‘ e i - ” S gur’- ~~~~ i f :uch a search i s 1er f rlc s’ i on a

directed graph ar,S tI . - ‘,- ‘- r c. , i n- c , : are : : a ” t c ~c r ’asi O’rorc. 1 t e t o as- they ‘sr-

marked - Id , then no fl~~r i~~t c’ edge leads: from a vertex U. a vertex which i s

-

28

-~~~~~~ 1’ .—

—-_-

~~~~~~~- -

bo th ii ’ 5 I c e y - cr ’ s ’ - : ’ - o’ ’ ,: c ’ . n i - : c o o n’ - l c + t ecd in t h e , ’ : c ’ . : o - i c g S s o - - - ’ ( - t - c a , : , [1 - “1]

he ’ s ’ 5:1 gn.u -~ - 0 .7 . A o’.-: ‘. 5c ~ 1l a’:t ,‘ea1’ C~ r can ice LIOn l ’ .cc :: - :c :c, “ .S ‘cc cc c ’

‘ ‘. -m’ aI I S : c c , exos 1 ,1 c it . :1 cisc !: S . H -re ’ t h e ’ old n-ci t - so-

511 g n s c ’ - I .~

In c c c ’ r ’ ’ + S c : — I ’irr ’ O .‘ c,’’ n ’ .’l c , 55 ’ i, Igi ’ s’ IL- ‘. ‘ ‘ ‘ c l ’ s ,‘ ‘ ‘ ‘. ‘ - c o_i , c m , ’- ’

souL of ti c’.,’ :‘ ,I r , - t. ccs’sc , IL ’n - - ’ - c n” . - c I oc:- s ’ +°~ h

c cir t ,i t iconsc S n ’ . - v s : r t i ’_ ’ e ’ : I n ’ . - v o ’ ’ c ’ :~ ’ - o c d i r c g c : - 5 . n I d : ‘ s -_ i s ’ ’ - O r - ‘is ,

L I c e c t . ’sr’t os’ - n ’o cc-s . in ar: us-c a: 5 ’ - ’ . ’ ’ S ‘r u i s -.c ’ _I ld L’d f’, .0 m sr ceC ~‘ ‘ cc’ ’ ’ - -

I - ic e c oins- c li -v -c l r in l w — - c ’ .I . h - ’ - - o : : - - - “.‘e l : : I n a , 15 i” . . ‘ Z c : ’ . l 1:. r ’ l ‘ ,.

1cc- tI c ’ t ’ n ’ -o ’ c c o ILo ’\’ ci S a lc ”..’- , - l 0 ’ . ’ 1 ’Ji - ‘so O S - _ I c - 1, - r o n - o c t c.”i’ ’ I . , Z o c Fl “ c r ” ’

0, L n ’ - - c + c i U r — c ’,Ir ’ c ’t cean ’c- i , j ’:c ,c : 5 ’ ’. I~ :. c b- - c , - c , : Odi 0, - i rs ’ . ’ on t u e - - s c_ so _ t soc c Isc I

[ 0 - 1 ‘. ‘occ’-

Both - , l’.’ c t , h c — I i n ’ _ ’ c etic i - r a - ’ , c o l c — o ’’ r r ’l , cc:’tr - - ’Si , c i : ’ n r c - H ’ ; i~o n J ’ “ ‘ - cc ’ -

u_ c r ig ou r s ’o_d , ’ ac- ’!icl, - ‘5 so : - ‘ c’ I s i c o ut’ s’ b ,c-~~ ‘ , n ’ ’ ci n’ c ’ ’ ’ . o ( r n o ”, l ‘. 1:’,

eiq-l.cc’e c _ Ic c r — n’ . - r S - ‘ -c , :‘ , — i - - :g . ’ ‘ c’ - : : . , “.,5~~5 5 ’.1~,5~ cc , , , - , c in ,’ oc:5’:’ i c - -n s o - so c

s ’ e :cia’h , me: t S c S . , , - - “.‘ - r-el - it I: ’ ‘n’:, I m i c c i o i , U r o t - tape-I - ‘.~l .~ ‘si :e :dl”i - (h : iu c~l : ~~~~~~~

lex ,i ’ . - n:’ ’ :~h,5 ’’ . ‘. ‘ : ‘ .,c ’c l i ( s ’- t ic [1 ‘ 5 ] : ho’ - - . ‘ . : i so 0 ’.r. c ur-c m l ,cr ’ cls- r ~~. “no d

.‘lro r t - - : 1 - i _ I c - c - S cc -arc h ’s ( 1n1 ik : tr - i [ 1 ’ . - J , ‘ - ho c. ‘ - ‘ - i i [1 ° “ ] ) . c c c ”

u: ’l ’: s l .

Oj I ,dcii zati cnn 5-I -I h in d :.

A large c l a s s’. of n — - s te-c’,, ’ rc ’- - c i , i ’ - . ’ ‘ l i i ’ ’ c o c , -i x i : : , I c c c c t i - ’n c : a ‘‘ s c S i cc

ii -’ Lined on a c’Ora~ h with we I guS e’d c - O f - ’:. [ 0 i s cu r - _ I 1’; so_-c I-Ic ’ I 1:’’ , ’ ’

t . } io ’~ e r -ob ltom r ’ ci: l i n e - c o _ n’ n - l i s t - -go -n- :‘- - ,- ‘l’ ’ cO - c ’ , , ng r u t ’I , n c c. ( 1 ° - u s i c i g I i -- ‘~~ 1.

Nemhauser and Garfinkel [1, ° ‘dj ‘I , S c o t t , c ’ S S ‘ . - r :e lgnr i tho’.. t I r c e t ’ . ‘. ‘, o n , ” r’ ci , — c s r’s - s-n’-

-~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

-

~~~~~~~~~

-

~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~ - —

- _~~~~~~~~~~~~~~~~~~~~~~~~~

_ ‘ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

- ‘.“

I I n t o , -: ’ . r or S c i t ‘ce -t o’ : r’ - gn - ’in c sc :d c c g c :,ethc ’cJc ar e cj v a , i lab lo- for their s ~iut ion .

Thes-e’ ‘Jg cr l t ints: use’ S w t o - - Icc I quo : , ~~~~~~~~~~~~ and au~~nentatio n. The most

general setting ton’ tht ’ .csn e S- chni ques ic c in matroid th eo rt’.’ (Lawler [19’7i_ .}),

O ‘ - n t on e can ccnd ’.’rc f ‘ m i and cc ’ s - : ly the t i - - s -h i s I que s - t o grap h rotclemr wi, I S i - cit

0 0 i rc~’ ~ib ‘ i t cccci t re ,r ,1,.

h - -n , ider t S o t c J I ’ 1,1’ ,” , of n, ’ic , - LI :,g, in c c ce- I, wt t [s we-s-I ~h t e - -,c i,, ’.c cr :t ’ , a

‘o, cc.x ioc’c’” — -,o’ . h c ~. s~’_i O s ’et - - ‘.5 ,i , c~,’ccs g s o - r ’ t c e j n s add,i t ,I -onnaj, so , ’n c t n - c c : a t : . The

c oil so 0 - 41 c ” ‘ ‘ , ‘ s- c e - t h y - I o , I c,cs °, : ~- usefu l I r s 5 olv ng thi s- so - - c - I ‘ c c , .

t, t i , , c ’ - , ’ -co , ’c nO , ’ I v w - - I ’ . f c c . . : : c5 , . I r ’ . ,- ‘ - .‘ ‘ - - I - - : n s i c - n ’ , O c c ’ a v l ’ . o c c t t , ,

I l gh c - ‘ a , t csscti.U, r sg c~ ci , - I , • ‘ ‘ - ‘ l’:o:, c-nt — 1 ‘ ‘ ‘Lc ’cc : - r r i . When i:-~-cJc _J riJJig an

c l- c- i ’. , , a - I - i it. t o i _ I c e c c s c ’ : ’ - O i f ‘ s o ’ ’ - - ’. ~‘lt s _ a ,-f the :~ t s - e t sat isf ies

lie ’ . : , , ’ 0 01 0 i, ” s s ’ ’~ .’,~’ - cc ‘ i o ’ -~~ t - so’~ et -c ’c ,~rs c s-w i ’, . ‘FIre r’ ’ :uJ,t -”j rt :

‘ - -r ’t-ainl’,’ c c i 0 i , :O ’ ,i o ’ :’ ’ i c - - c c c ’ I~~~, 0 , ’. . ‘~‘ ‘ ‘‘ ‘q : 0 ’ : n - “ .O e so n c - O ,I t ‘ - ‘ I c , ’ , the

s’.d, :- ’ t -h’ . 11 5’ -- - 0 ’ 5’.’o_ X l oct10 ’ . ; u _ c s - - l I 1’: a - - I girt ‘.1’’.’ so -14,-on to wt c I c i , thi s

t i _ hi - cd ~ , c c~j 11 t i - I ’ .’ is c’ ,.S o . i i : o ’ u- :: -urn . r s g ‘. r ec r ot 1 - - oc ( Kru s’kal [l-~5’ I ,

} r im [ 1 - c - I H’ L ’ij k : trc i [ 1 ° - H . Yao [I , r ’~ J , h is e r , h t o r c and Tur ,l an [ 1 . T c f l .‘~z,-cs i f the go- -c- , Iy c o o ’ S 0 : - 0 doe: r i o t nc ’ i a -c e -

~ t im ad, -oniu t ,i n r c , , it rcc o ’

produc e solution s whl - ’S ar- so le-ce t, -o ~-t~ o’caJ, (Garey and +‘ohiis ‘01 [ li_ ’.’l’ ] ) ,

and it S, s u sually e’a, I\ t o . c ’ , n, i - :me-r: t and fast.

In si t ua t i on: w h o - c c i_ lie- gr - - ’ed ,y s - c - - S i : - - ,I doesn ’t work , a method of

j , - ‘rat l ‘. 0: ir o :J so cv ’ - o: c ’ ’o : ’ : - ‘ o’ . ’c S I-os ’’: .i n i- : . The idea is to sta rt wi th army

:ulut n ’-tm t - the ‘.corn s: S - r c ~ I a ’ , and look t ’scr a way to au~ nent t hu weight of

c_ l i e s- iut ,i on by making ic e- oil ch ’cr g ’- . . The new s-solut Ion is tI ’ .e-n ini ~ r- -ved

S r i the same ‘ roc ~’, arid 1- l i ’ ’ n” .’e’~ s’: i s ce - i r S inued un t i l no ir~ r -v’so- :t’. H is

J . c ::i S-ic . hinder -il l -ro b c -I o t t , ’ so - nc -lit! O r ’ . : such a locally maximal lii i .5 - ‘ .r c

j sc also glob ally ‘:. :o_xcd’:: , ic ’ , . 0-Oven if the s l in t , : 011 . 5 ’ not guaranteed S c 1

r , - ,x ,I oo’cmc , + , Si , - c - c ~~cc o ’ rs ’ 0’ . ’. - - - i t method coi a,y be cm good heuristic ; I ’ r !n:t ‘ c r ~~- ’.’- ,

~

~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~ ~~~~~~ ~~~~~-~~

---- --

5 ,in [10 O c ] 11:1: c’co- i ’ lj ed 5 . 5 wi t S c gcocc ’. i soc ,. , ’ ci t : t’ - I , S r ’ - n’o ’ .’, o - t i I r c f :‘si’’. _ ’ooc - in

-~ c ’ s ’ - t i c s - c . The trc n\ ” -I J_ Ing : a , I - ” - _ I s I n ’ r l o l,cocc is- I o n f ir s], a : i s rS- ’,, ’ I -oc_ c,cloc ’

t hin’ . cc ’.tI s oil], ‘c’’’ . rt- ‘ I ce~’ -it ’ a fl’ s: i s a I I I ’ . 1,_i, :5 cure -c , , .rc t l ,e ‘.-dg’ s .

i i c u :cI ’t t r n : cycle on’ ‘ ‘ O - o , , ,“~ ‘cc -i - al so” . : ’ ’ ‘ . 1’ S i c ’ - ‘ n ’- ’.’; - LlI :”.g :of_ es ’os ,s-u : :n ’ :

I .

i ’ .~~t ‘5: r i a t i n~ 0 - 5 t h

born e n’ , .ble:cc: require more :n~ h i :tic:it ’._-~ , c’ 5t”. cc ,a ns I ’ n ’ ccl at i , .o c i s_s -cr I:

I ’ o::cc lie “.-j i_ t S r the’ cl’,c~ ion’s dtt .~~ :t n’uso’l co - , d,, :cos - s:e- S - - ‘ on ’ ‘, ‘ ion thi c c - c e-t I , -c - c ’s .

Tl’~i’oe ac.Lvocncod teoh ’sr i ques have been ievi:ed h r -I ’ ”nl.in ~; with , 0 E on ’ ee-’ d i v e r se

pr ob ’Leo cos- wh ich re- cu , n’ s - ,.LI ,’ rc os - :. I t o iii -ao -’.t, I : ‘ cq ‘if .~oit :‘.. Thec’..’ t ’-cl r no , I ccc: ‘.rc

-s-t I , c,O ’ . ~~~j - l ” .,~~~ si a. c - c c ’, , n - ion I’s c c ’, c mec c :ccrc t , ‘c_in n linear arl- ccgeo: ‘son ’.’..

l ath c’s-Ic,’ : r - c s c , c i o ’n .H a o , c th ’ . s i, cc! ’ c ’.cc J,v .I rrg the c ’ ilcr ’si o c g, . s ’ s , . - , --

I’ cn r cciden’ a uncivc ’sr s - ,,c ‘if - I, - , c - ‘ ~ , O , , ccc ’. I t I - -nec -I I n I t cialbIc,’ ,lflto singlet on’s

ce t s. A:sc .cciatcd wi_ ti ’. e t c h ‘ - 1cccc ’ .~’c c c I: a ‘c-J o” . c’ e ic’i sb S . s-c c able S.c

-s’arr’j out ‘S-ir e 1 ’ -l ,’i , r wlng on ‘‘r oot -cc . - rc t he s ’ S

thl’iion : C ornb i c’s- - I c , ,’ s- ’ d c s - o r ’ ’.: cc s- I~ongle c s c t . ],c:S l’ ”.’ r c n t5 , cs -s-id

‘T i d aL ’ . ’ : 0 - t o - i l : ”; t S r ’ ’ ‘-c -c l_ac: -c~S’ ‘~‘,l “l’.,’c’c’.cchc c in cc giver. :ei Is- i a

c o n s i s t ‘ co s t a- c j , ’ .

1- Ov aluate: h’ ’. n .j “c - ~5c~c value a c : s o c J - t e - i wi _ Sc or given ‘ leone-nit.

A situation co f thi s kind ec cc uc’ c in the cc “ cc ci 1:5 ,1 ‘c . oct F,dI ’TIIA b “. ‘-7 . ’ ~1, c -sa cS

r - IQ T IVAI ’.sc S fl E s-t oil ements - ( c ”;.Ji cr cunci Fischer [l’.’.o ’In ] -u n-I in several . -tOt er

combinatorial problems (1 cm r , ’,a n [l ’.’t ’ c b f l . The s-c t un ,i c ~n so-t i e-- t cc Inc

discussed in i”-’c c t- I ‘r c 5 is- the :imi-ie:I such r i - I ’ ’ : : . . -1 :tl ’l . ” .n ’ -n d Fischer

[ ltc ib J proposed an a1gor !t ’ii o ~. fo r thi s n r - - h ’ , . ec” u : ing I n - - : :0: ’ a d : i t s-c

- —_ “ -‘

-‘—-----‘ -‘— F ~~~~~~~~~~~ ~~~~~~~~~ -,— ‘ ‘—_-- - “ -—-‘- _—.-_ —

_~~~~~~

— —

s i_ r i : - , - : . :cl’ ’c. S-Ic [in-n :- ,’ and 0-horn s’ c ’ .c r i !’r -at ed the set . us-’ .I o n, J I ’ S e-os, aSs - c , S rjic .g,

- so -c : , :-u t e ccc l n icocuoc : s- .j ccnning trees and ~ roj’-ose~1 an I soc reve l co ce c ich: si ‘ic ing

c-ath cocc i iress . : .n on trees (‘.~~o, l-Iopcroft, and Ullinan [1 1’ . ] ) . T he i r r cetI ’c c . ’d ,

whiclr is verb ,’ ,c im’l le t o program but very hard to ccn aic_ s :e , gene -ro il i ’coe c’. t- ,’ s-n

c ’.u s-cob er of o t i c ’ . ’n ’ problems (Tar j an [1’ ‘1’5b}) . I shall cU scu s’s t h c _ s ooc ~ 1h’o u

and it: rc ’nc :’o’ c’:ah ’ Ii e running t i s - c in , ‘ , ‘s c t i s or’i 5.

Pa: cctho ’r n -ob len oc icc ’.’ c O v i n g disjoint :et: is the i’ s - l i_ low” or g . hu n -n ec’ e

t i le vertices cf a gra~ O i are initially ‘ c ari-itioned into several subsets.

‘s , , ‘,-,‘ I c i to c ’ci,rid th e coarsest i arti . t ,i sc n which is a refin ement “- ‘ s the gIven

- rn- cS .1 — c c curd whIch is-- 1-res eo ’n-- e-- ’s under catj acency, In the sen se that If tow- ’

‘;s’ri ,L ces ‘so ari d w are contained in the same sub:et of the cc ,r t i t l c c c r ,

cN- cO: c _ i r e s e t : A( ” .’) {x (v,x ’) i s- an edge) and A(w) = (‘co’, x ) is s-u’. e-~ige l

I o , S, - - r ’ , oct -n x-’5et i , ,’ the c anous- :‘occ-’,i_ er of times with each sub s--ct of tine partit i on .

TOil s adj acencc :’.’ — c r e - se -rn -on ag s-rI 15 ‘1 on ir ’ easi ly -c so- n os~ utabbt e in O(nm ) time .

5 5 c r ’ ft [lyIlJ ~e’n-’ised a os-c - r-c soncphi:t.Ic ated algorithm which run s in

O(m log n) i c c oorec . Gries [ i”d’I~~~] give : a n ice de scription -s-f this algorcitoinc’.

S anti, t err ret ’locc -crnerot cI sc u:e:’u,], in solving the st ate minimi zation problem

‘or finite automata (Harrison [l ’ . s5 J ) and in t e s t i n g graph s for is-c ’. rnn r iinl corn

( ‘1’ m e l II, an’,i Got’. 1 - ‘.~~‘.‘ [1’ T

A third r U t - s - n c so - c :L:r r:g - a good cot t on updating meth o d is 5 0 c c - linear

arrangement l re -bc I -c : c : Given ci set oS n elements and a c . ct ,t ’.cct ,l “ to el

subsets of the ‘ ,-lernt-n t .cc , Can S - i ce elements be arranged in a line 5:0 that

each sub set -occur: -~~ - ‘ o c t . 1 gu -cusc ly ? Thi tc

~ r cb l em arise: in biochemicl-ry

(Benzer [ 1°’,. ’ ] ) and in arch aeolo~~r (Kendall [11 ’ . ‘c ] ) Booth and Lueker

[1 r :- } hav e devised a s- , + ,’ t h i o ) , t of solvi ng this problem in 0 ( n f r n ’ S “c c - ,

where m is- the total sir: ’: ’ of the sub sc’. ’t: , using a data struc t ure they

call a S - S

—~~~~~~ ~~~~~~~~~~~~~~~~~ ~~‘

~~~~~~~~~~~~~~~~~~~~~ --“.- ‘ -

- -‘_‘ ‘

-- ‘~~~~~

-‘ ,— ‘ -. - —— -— “~-r ‘ “ ““ ‘ “ ‘ “ ‘ . ‘ - “ ‘ ““ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ “ '“ ‘ “ ‘ ‘ __

~•luI!IIIII~

,~‘c ’ ’ s - i r 0 -b ce ~~~~~~~~~~~~

i ’ I i c r e - arc’ tw - rnet ,ho ,,i. - ,os - ’ ::-ln-’J . ng fl ’cs-Ic a r,’ L - le ’ , , , - s c ’ : - : . ; - I n - I or ’ . c c - c

shrinking, which are re lc it e-c to the algebrai c cc ,c r ’ .c ’ .cn n - : -f ,“i L. o ’.J,g~:i:ra and

is ’ -coroncc ’.orj’ h i s-c:. brie way i_ cc- solve cen ’i ,a,i or tIral Ii nor l ’crc , o I,: t c .- - ce -e ons- ~. ‘- c - ,- n - h o,.

graph into several :ubgras i on , s- solve the r. cb le’c, on the cc ic- ~n~’ : o , , , and

cocc i_- inc the :cclut l ons to gin-” ’ t i r e - :- loit i ~cn ‘s’ n the c -nc t , ir e gram Si. 115 mo:’°

instances wS ’. ,,-n ” c thi s t , c . o i l r s ,l que is , . ‘- : ,sI , the :us--gn ’ cc i is , arc’ C. ’c _ cc o’o,c c s t ,

c oni gn - ce b r , ) s -at - i c :’y i n g some c n ’c n r - s o t i v l ty r ’clc t ,i c , . I c ’c ‘ - r o e-n - t ,, .-

1’,- the t t ’ ,’:c nr . I . 1:15’ . )n c ’ .O. I O: n ’ . a - _c ’ . e f ’ f l c i ’ . c c r c t ‘soct: ; t o - j - ,- t ’ ’r ~’ l n n c the

“ c” n e - n i t : . ,~oc ” .d alg- n c _ i_ t i : : : , . ‘s-xi s:t f . .cr a ‘sar i ct’s of “ . com’::’ l vi, 5,’; ~

r .c hle -n cr :

I:c,r j an {ltyc: J , :i ss ~ en - : 5 ‘ _n c . L ‘i’arjarr [ l973a], Sic’ s- cr c t and i’a’s’, 0 anl [ I r s-’.o c J ,

b ” .caui t [I ~‘L ] , ‘i an’. i - u , [ l ° ’ I ’ . a L Tar j aYi [ l ~~0’ 5 c J ) .

‘. t i c - ’:’ ’,-e cc; S . s-u ‘cc : 0- _ c er :5s in r, - co ,Lecc o s’ ic t _ i n r i n n k ‘cct s -f the gr a~ ls

‘0 cc, :cIngi” ” - r ” - c-c , solve the c r’, :blej o : - or the - shrunken grap h by a’~’~ ly-I ng

‘ is - ’ i-lea r e -c ’s-or’: ‘ ‘-“i:- , and fr- s-rn this - c ’ Ici tion compute the :.“Luti ‘nc -c , the

or.l,c I oral grc i5 I,. The :ici’ ,I cr5’ : ,c n i g Crab I on so s-nr c cc “ccci : ‘t on t oil-Jon ’.’. a On . c ’ s , - : -mc O i l s

image of the ~n’-cs: in . -Jo.’nerally the J- :.~~t of the graph t o n be s’iiruiil’s is a

-c lon c l’- or a union of cycle:.

3

—‘ — — _ ‘V ‘ -“ ‘—“ - - ——- “v_~-

— . Ton Ti ’cr - ,c t :‘.‘~~~~‘ ‘ :‘ - 0 ‘i_ c .ccccs ’

There are i ’ .ci c , -.ir~- .c: of so , -mb ina t . c r i s-il, n - -roblencr s ’ ’, ’ , r wir I cci ’ . g cod algorithconc

‘ire ccc: we. Thi s ccci , i — cc o ’:-: - c ’so c ,i nes ten such n , rcb l eooc : . I have s-elected the

leo:,: -c c S- i s- t a : is s-f t i e - i so 1 0cc ’. n ’s -se- ° : , t i re - r:mge- s-f t - ,cchr , i le e: l i c e ’;

c,ij~ - , l c cc~,’ l ’ s - ’ s .l ,_ , c c : c ’ ’t’s wi c cl c - - . ‘s-lie- .l I c ’i I s - n’;- ’. c - I _- c _c, ’. t i l e ‘ c ’ f , oc : , ’ tl’c’:

cc be r e’s Ve - sOL !i S ort ,i’s i,’ of’ n’cilcico’:c with good nig: r ’ ‘c}c c ’ . . ‘1 an le 5.1

1,5 : tc i _ S i t I o n ” .’.l lec- ,: :_c’, ’ .S t l i s c t e , ’So , o ’ . b c s - c- : : used. in t i r e best - ‘~~ .;‘ : 1 i_ b r:,: t i n t S r - c ’ . .

FiI ’.c ’.i’’ ’ 5.1 ‘ - i c ’ - a ’ : ic - so so. ‘, o .ccc oe - n ’ct : in : Is - cS ’S cc tc ’Ic c ’o cc,cS cIeve d 1’’ ’; ’cr n ’. bs-7,’ ‘ - -cc’

th ese - r ot- I- ‘ cc-c .

[Table 5 .1J

[F i ~~ u’c 5. 1]

hi : c r e t e i- cur ler ‘i r - c o n -n ” .

- i_ y e -n : - ‘ s c r o—s ic I s - c ero s’iononl_ s-’t ’ soS , :, ’n’ ( ‘ . s , :s1 , . . ., a 1) , the dl s-crete Fourier

crcuc: t ”. so:: r -s 1 - - : , 15 t o :~~npu te the ‘sec t, r (h ,, , 0c 1 , . •~ b 1) given by

n-i - ,5’ , ~5 a.’~

~‘ . ,Li

-_’ .ji c- ,~ are the (c c:sn’ Itcx ’) c - s - S o r o t s os-’

11= 0

‘ n, - ’ . ‘i’i’ls -c n r o l - ,” ” -,,n - ,,, :ns- i, rc c I o ”'. r ’c o ’ . , , L c r - s o c - : s - .r o ’ . . Ann aino r ’t Oi , ’ c . ’ ’ , -r tine -

ti c c r e t e is- cu’,ic’n’ t r s - - o c : o so’s- 1: nc - c a l cls ’ a sob c - o .ct ine inc ‘sar I- -Us ” ar ’ thc- :ot .ic

l ’ s ’ I I s - ’ - ,:. - : ‘ ‘ .s - i i l n g ~ol~~’cco . c:~~~I ‘.“ c .~ ’, u a t i ,n n cuc , i i . n :t ’c ’s - v - l c d I nns-

-ar , d .1o .t cgscr cccl ‘ s , l,vr s-:o ,i a ’. ‘ ,i~L t i n 1 b c ’ c .,, ’ c , I r ”. (~~ uth [it ’.’ 1, “1’ . h ’ : :so ‘t .

aj ’ . I T ’ i , _ ;:scu r [ l ” ~’~- ] , F-c -is - ct ’ r~ ‘un ‘- Sc _ t r i m [ l o n ’5 ] i .

t is :t r c i ’c g i r t f -sow - ’:. i t o cottipu te the discrete F ourier t r :cn s-l’ -’- rclc in

O(r”.’ ) t ime , t o lc ’c_ - cnn -c i ’c,c’, cc l; ~~~~ 5 1 ,‘n ularized an 0(n log n’s -tics-cs-

rn - f l o e - I , ~~soj, j , c J tb - ‘as ’c _ S” u’.u ’ier t r c cn c c ’- -n cr . They were not the fi rst to use

5 5 c c met ,I c . ’d , w l r l s o h - rl~~1nated at l east as early as: Runge and K~onc .1g [1-il-U]

The t ’as’ t F c_ : ’r r - er trcu i s - f : n” uses re-can’. I ‘ix ton cut ,L- ’~a c the amount of

31~

— _~~~~~~~~~ , ~~

•‘

~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .— -

— .“‘ -~~~~~~~~~~~~ r ” ,_ ~, ,,.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -

:-,r’s’s ut .’.ti, c c . c ’.t ’cen :I ly Winsogr’da [ 19 75, 1 il” I rn’s- o, u-c a c’ ,c’t,i’.oj c ’ :r

‘.c cocil u t ing the d i s c r e t - j- ’oun . I no r tn -- ins ’s , - ‘s--coo u .. l rsg - - id ,’ ‘‘- ( r i ’ s cs - ’-c,~t i ’ s il -cor n - I cons .

This s-ctc’t.ho:,c oc oc,y be sun ‘cr1 n’ t , the c ’ ci:t F-c- ’ori ‘-r roc’s, 0 r’oc , , , r s r ” .ct.ice ,

alt .S r - agO: Wi n , ’:g rad has- n . - ’t cj ncj ’,oncec -i the vers-J .1 n’s-icr,’ b O c , : ’ t l o c r e ot ’ Oils-

a,lg on c_ i_ ui : .

0-t atn i x 0-Sc c,I t i n i c l c cc r t i -c , .

bl ve -ni S cc-so n ~ n cs- ci tr ic—c s- , the :‘ .o ,t n - c i x os- ui_t oi l ILiso at i cc ro ’ui ec :c is I ’

is-at ne c_ O ,e . i r o:o: ct rct s-s c - i ’ ,:d,u s o t . flit - -ct cc_r olan d Sr i gh se-Sic-al c- ,el S ’r s’co .i os-c ’

ccci t n i x ‘s-ui_ f l li ’.cc c’iti , c -n ’s re coil res 0(n’’ ) i _ c l c: ,e- . b t r acsc,’rn [I~ ‘.‘ oiI ’VI c ’ tid on

wc’c,,”, to mu,,cct , c c l v two 2~~ 2 matr ices with only s even c : Ou i i _ i r i l c a t , : , and

c_ sc -c d lo is- t i c I n c a n”ccc’~irs’ ye ocratr ix mult i t’l~ c.cat ion al_gcr~I tOs coc i- cc cub sIr’.’.; only

1o~ ,,, 70(0: c i : : , ’ . f l ’ . ir’ -sumc rising rc-:c_Jt has acted as- a stimuli: f,’ n’

cs-c ub r- cce -u’ ccS r in l ,5s-: c . coo ,c le -’xity of algebraic I. ro lc ieo:o: . No - ne i_-or . a- .

wi_s e t h- s - n - ,°;t r cr :s-en ’ s algorithm is- improvable. Stn’a,:sen ’ s-- ;c,lg’.’.nithrcc has

b-ocr: used to .‘ c:::c :ii e tncucc-itive cloc:ures- of gras-Or: ( -buur r - [los -i] , Ficc icer

‘u- s Meyer [1-cl]) and t- ’ do c -s-nt- ext-free language ~ a1’scing (Vali’rrr t ~lJ c5 aP

• 2.5.1 -in 0(n ) t . oon c: .

A pr ocUie c: : related to os -- i t r lx multi : i lca tion is” the s’lcn-rt e:t :

n - c i_ hr- . Si’s- e rr a dI rected graph with -sitive edg e di :t”unc -: , the s i n g le

scarce sohorte s-ct i ath problem is to find the minimu ms -, l i s t - c i r c e - ‘.1’ s-c. ci g iver ’.

Vertex t - - - ev cr c- I her ‘c, n- -- s-on The all pai r s shortest la t i c n’ s- Sd -c ” is- t -

lin t the or-,ctnni mu n’ , distance between all pairs of ‘sc-r I ice: . D i J i - c :± n - I L i5~ . I

Jts-vi sceci ar t algorithm f-s-n the single :‘ource ‘i_ noble s-c whi ch re-~nb re ,c e i S O , ’ . r

0(n 2 ) time or 0(m log n) ‘ti: s-e depending upon the i s-cc l ’ cccc ” ns l . - ‘ ‘ . . ~~~ . - no ’-

n i s - th -s- c s ’ococbcn’ of vertices and in the nuns-her ~c S ’ ni_ge- : in the g r e - So

(Johns -con [1- -r I , ’ ] ) . Floyd {l I ” i ] gave a way of solving the all air. :n S b ”

~i 5

-

~

icr ” ---— - ..,

~~~~~~ . ‘Si ,,,,. _ _ — ~—i- ’:t,_, ,, ~~~~~~~~-_

icc —s - (o: ‘ - ,i” ’ . F~”:’ , S c ’ occ ’s c [i, ’ i l - ,‘O n on w - c d s - S c : t o ‘ S c - : a_il n, .-a lr : ~‘ ,ool ’s-’ :’. -.c ’in-;

b’, - ’ ’I” .’t- -.c us i ng -,.‘ .( ni ’ ’~~” ) c- s - or ‘ i r I s - en s - : ,‘.,rj d c m ; 0(or ( l — c ~’. log on ‘ i . cg o n ) -

t~~:de- total. Avis-, Ri-ceo- b , curd is-c c [l~~’ ’ ] ‘s r c s - - e i ,~~“ . t cos - i- -c_ s t ,(cn2 i ~: n -

c- m i cu-i :,s -ns c’.re- r - ’ ,c c ,cJ n- r , ’ 5 i,on ‘t On ic cs-c r - ’~ os -s- c Ic-cc s -c’IL ’,’e Os- , -.: cibi l ~ n o’: i_ i - - c ohn -

ibi s: J, . cwn -c_ u c~.i ‘is- ‘O re -1 ’ 1,0 :- i’~’is- l’o,c c cs-c : ‘sc _ -i’ a ‘, : ‘ ‘- ,c c c , s - ’ . c .- I e ‘c cco :’.c Icc o ’ . c , ’.c f ’ o_l

‘ o- :‘bauatioii c cO o a S b c_ ri c a ’ - Is-asic .

us -c A is an cc -, ccc c: ‘ 5 0 - , d -,s- U is- an 0 0 -‘ 1 c_as- ’ ’ ’ is- s - c ’ c c o: s ’Oc - ” . irt,

c-s i s - c’. n - ~ 1 ‘, - c c i : r ’ ”.’ ”.’ cc n’i ’li’l ’ ’s - . ;~n c- ~1 cs-n cs-i. . ’:, cc cool’s--: lb .~~~st- ’ r u 0 ’

c s-_c ’-, ’ - - n , : Ax = L . A :t ’ c: c -ctac ’l con e-t ic-c d 0 ’ - n ’ ccc i lcg tS cc L c’ ~~c S:nuc : .s ‘is

c’l ,boJrr ::t L,’.: . , F ’r ’ . l ’t O:c - c sn r d t .lnl - .’r ~i ’ .- ’ i~~, i e c _ c ” c ’ , c , ‘ i ’. I U . :“ . r ’ t . . c _ ic ”

i, , r- ’ ’.~’ j - c m - - s ’ o ’ 1. 0 , 1, ‘i~~~”~~~~ - . - d ’ s ’ s - ,-’.’ s-i cr ’s- rI - L’J • ‘it s-

I l i a c I : , l as - n c _ r i c- .oc , ” :lc_, o’ (i. ., L icac Iso c - c -- ct- c’ - ’ , , ‘. ,, be “ . :- v-: s - I . - ,

J cig~’o : c .J) and U Is al ’s -n - t n i s - u n~ .si_ar (i. •~~ 10 has- Os-u oc i_ :— . - -

be-low the -L s- ,~on i :u ) . Th or Ac-c = ‘cc 1: solved in t,~, - :1- sc , by solv i ng

= t’ , ‘.c ccj, ie’l ‘ci’ ’ ro t . -o l’;_. 0 , , ’ , c c c c ,tx i nng ‘ix = y • c _cUr ,ccd :. c__ cic~ . -1 , :i ’ .

Beca_c:’s- L arc - , S ‘0 have cm ecial c. , so s-cc , c r nt :- ,1’, .0 o:g ‘ u ’ s 1 acok: . . .lv,i . rn g cc , .n ’ e

v— ny c f f i c i ’,’n t : i _ Or ’ s - , b ‘ -‘s - oct ‘ ,so ’. “ s‘ 5- iu s s- l ax: -:lI c :c ,b : : ’ ’ . - j , ’.,’nc is t u e f i r _ c t

‘ ‘c c ’ , decomposing A m t i’.b

‘th e c te c ” c,irco: ,- c i t , i,orc no ’ A o n’ c o d e - c s ’ . by , c - ‘ ‘ ‘ c . ’ 01’ ~‘

, c’s- ±_c, s’ c c t i n f l . A r c-

, J erc c , t I on c c c c or : cL s I n c of c c , c - I l ~r ’.~ a o ::c _ n_ L t - ci I-c - ‘ 0 ’ one n’ - r’: of A - son o f? , , so cc a

of A • n - f I. toe mu l t i 5 Ic: is ’ c c i c o - r d ~c ’ oc r r - ’- ’ . ’ t b y , t h e - c- odd o ’ t e - , i fc~ ’ ’ -’,’ i . l hccv n

a zen ’.- j~, cc i ’e- Vi oiic r ’,’ c i . - r : — : : - -r-: ’ - ‘ c l i i _ ‘ . . By sol ,/ s - t c o c o : t i ccal iy :15 m l y i n g

ccc c,oh row oi er’:cti c s - i t . , - - r u co in n - tao s - n c ’ . on gI c c ‘c,l c c ’ . : r t - r ,l x .-\ intc ~coi

ui ’; ‘-r t . rsc c , rigul ’cr rc:rc,tm x ~1 tb ,, I’LW cd ~ ‘ c c l , oti s : : c r 0 os - s - - ’ - S , i e t ’Ic r c a L ’c’c” i ’

tr iangul ar m- - J r , , c U .‘u ( ’Sc is- at. 1,01 = A

L .~~ ~~~~~~——‘

- -

~~~~~~~.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~ .. ~~~~~~~~~~

~~~~~~~ ‘~~~~~~~~~~ ‘ ~~~~~~~~~~ -“

If A is .r . I g,?cr: clb :, ’ a -,t e -n ice (o n -cocc i_by n s - o ,— : . n cc , o c r b o - ., t i i e c , Ut]

-iec -n: i: os’itj on requl resc 0(n 2 ) s-~ ‘ t e - t c and 0( n ) s- Ic :,- .. ‘ _ c , c : n - , n r s - cal’,-’ I org

and h a- .’sk :nlving neqcolre 0(n4) s- ,. c .. lO: rn - c c~, ’ large s~ ’ cL e °’ cf eauat i om : ,

U c ~’t - ’-,’er , tb - c c’ ,cclni . x A i s c’~ “ .s- ’s ’ cc . i-or a si ars-e cc , cr r,I c-o , t i c - c ‘c, , cocc ai r c .c

s t o ”r cigcc s - - rac e - required by °Iausss’i:cn elimination depend is-c a coo cm b i_ ccf cc- ,:

way on -“n t O o ’s - s-es -’ — c _, ‘ n c — s -ce - icc ’- c”s-ructul-,: -s-f the ccc atr s- c-s . in 5 S ’ .’~~~~’ . C . 5 i ’ . ’, a

0’ s-OW os- eras- ., - s - n c ccci’,’ int o’ -duce new n ri —o s - s - s - - s (call ed n--I i c — b i n ) inc t o ,ds ’ in - i s -n , :

so , c,aliy :- s- • c t is- de-s’lr’rblcc to nearrange s-,he- o,:,::trix A by me an s

c: ia ’ - ’.c oson c: c olcsc :’c ’, -, ‘ ns - cc ’_ ccat i oi ’~s s-c’- t ioa ’. tO ’. ’: ‘s i l l— in care n ’c _ o-sr ’ . i r ’ .g t i : : ’ -’. of

‘In e Is-s_ I or c r c _ I or arc’ t ’e.iuce-,cc.

For tO - - s - c c _ n : se - lIc ,t s ac e-cud , to’s- re-s rs- :eoc’c s-ii’s- : - i’ - — n , ‘ c ,~ :er- ’ struoot’u.rc

of A by a ‘r e - S . 0 . TO”.- ” grouc h c sontaln s ’. :-n ’s n’ v r’O. ’:c-: 1’ r n’ s-s -cOn ron ’,-,’ an’s - c

- _ ‘ - I , .os-,rr o f A , s - r ’ . ’s-ric edge (i , j ) : ‘ s-’~~ ’:rc1r o s co — :’ ncr -,s-c tc cT,~ (Ui) in A .

1:’ A iso c:,’icrc’.” r . i s ’ , 0 sic on i ’ s - e s - I ” s’ if A ic cc’,c v’.:’. - - s - r .I s o , 0 is

,Li r e - o ’, ’,’ci. Ph’: ~~“c: Sc 0 r-cc n’ - s - ’ .eoic _ , A s-is-i all os-atri ,ce ~~ f’- sos-ce- c by

, i:- , ”.cl t a,rctc ’ ucic c ‘s en c ’ cnu t .,i n g r ’ .cc c u r - i - ; . clu,r- cr r co c o o ’ A . I’,’ cLc r . . t i_-a. o ’~g, n - Os -” . ? r ’ O n l ’ t c u c

s-- f 0 ~t mcrs-,’ be ~ .s- s Icc b ’s - t ’.” cIa,: ci n” - r o-c r c c , s - versl .‘.n of A such that

ai’_ cs:icic e-ls-cs-~~0 i c t S . : r ’. Is : ‘,. : ‘ :‘ b - c I , o , n . ( I t is nece :s- -~nc to I-u - -s- n O S - c t ‘IS . ’ . ’

o ~-so::utat.l - s - os’ do n -ct , c O c c i _ r c a S ho .’ on u s -c ’s-rI co stability c’ i c -c c ’lcis-c l n rat ,b on

r -“c __c. I cOt-o h if ~ i~-n’ - c tO n s- i s - S U e ic- cr c ’ : see s-~ rc s- ’i S o ’ . - ‘o’n,i -tolt ’ so [l’c5’7

o i - s -wars - -n [ l ’r , -~0 J ,

i ar t c ’r I l ” 1] ~~ c - s-c of the first to :‘.cg~oe-: t U s- cc c_ cs ” s - ’. ’n n i c o ’ . c s - s - s-f S O ’ . .:

‘ - , c ‘ ; .. The i-i ’s -a ha: S - - : o ,c n ’ . cxl c - n r c ?vo ’b y Jevcl ’.’-c ’ e r t . F n ’ g ’s-nen ’ai c ’ : ~‘ . s-

c c r r c ’s ’nn c ng t h e r id a t ion ’. cud : n - c - t w o - - n i Ilaus- Sian c’lioc .l t i - c c i on and gnu’ So ‘ c c ’ on- .

c’s-nc P c - s - c - [l”r’:~ Ha~rary [1- r .’lI : Rose [l’~1’~-) : P s-s -c, is-n, , ‘cc , ‘c I O - : I

[ IL - -c ’ - ) : Icc_c S ’:’ [ i r ” J o and Is - c s - c e and ‘i’ -’ tr , b ccn [107”].

[Fi gur e 5.2]

2 ’ ’

-

~~~~~~~~~~~~~~~~~ — : ~~~. . ‘

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~

‘F — - — ‘~~‘ ‘ ‘ ~~~~~~~ar - ”fl . - ‘

A: an ~‘x:c cs-c lc’ ,,,c f ’ the ::1 i’ -ccc , c o : c c ’ o r S i”ossib le by t aking advant c~~o.- . 1

- c ansI 5’s- , c c o c os -t ier the graph i n Figure 5 . 1° . l och a -c ~ k ~,y ’- : fr -n O r

ar ises ’ in iOn ’ - cc io ”r er ’se - , - , o l ’i t ’c ’ . crc of d i f n ’erential equations. Orcc ns-o.xa’

- ‘S e - r is c e c Oa-.:s’ . 1 - u’. t ’i .i c’ci or’ c ’ s . i s- ni requires O (n~ ) ~“s ace and 0(n ’ ’ ) tnmo.’ 0 ’ .

cue-h a cc ,c.lt ‘s- t on , if no = k2

. f l it - band~~ 0th sso heo c ’ . ’ .o c i ’ , ‘s - a r c e cc i i c-dn ’ , t i on

ic- ”.O U , co ’s- i t s ’ s ’1 ’ ore - c to o(n~~2 ) and the time f -

- 0(r.’) (Cuthill ‘ i c - S

‘- . ‘ S?t c o.’ [1 ’- - ‘ 1 , ‘ I t - w a n _ c o n [l’ ’7~ 1 ). Ge -ni - ft c [ls-7s- I ‘0:, : - du v e-i’cco _ c c c c -cs- er, is-et ’s ‘s- n ’

‘- , ‘. ‘ ‘S i -’c , c:Jlcd nc- - c o c c i d i s ccet ,I __-ci , t u n c i clc r e ’. iu I re s- - ( c c log o r ) so ole -c and

~~( i’ ‘2 ) t I c : . - .- . . 1 - ‘ t I- - cc , - c c c ’ t ’ i n , ccx l -00 cc ’ [ i ’ d ’ s - ] c S : - ’w c d t h at , ‘I ~~t t i s c I o

a ‘ - r : ~~t an t c ’ a c c t O o ’ , o c c c c c e . ,r d.’ i s s -cct , r cc n c, n’ , c on ui r cc s ’ ‘ I - i n ’ least n ’ .i li~~I 0 c o “s-H

- -o s -no -ut iri g i i c c c , ’ 00 ’ c_n ’,’ ‘n - -len ’ I rig c s - i ce -c ’ : ” . I ’: s-’ Gausc scci cc_n eb b s-i cccl I ‘oc - Sc. k k

go ’ . -on s-’rct: 5,:

Nested di scce-ct ,i ‘ . 0 0 : _ i s ” a c’ c c cO’ .ls-’: .i’ ; ’ - “ let i c - c c wh h -sh uc e - s’ Lic e ‘ ‘ - c ’s- I c :c , t cc

I , -: ‘ l) ~ (, k’ .11 L?rid gran -i c’.’ n ’ . . 1:5- c “s- 1’ or I-c ~ k g r - . i gr’ H o , s -c s- i tOre

I -k+ l -vertex bound ar y betwee n ‘I - li ens - (Figur e 5.2). 0-1 . -s- I; “c cc’ cc r,:,trlce:

which a n i s e in p ractice do not have co c ’ .On a n ice - , ‘t n ’ n c - c ‘ , o ’ - , d c i - r oc ’

ci:k whether n ’:i - ” c ’n d,si s-re-ct ci cs-n O r s - i c any i e - t t ur:c_I]. g ’cc nen ’ .-cl . I :cc c J c ,~ ~~~~~~ 1’,’

Lipton . R o s e c _ n - I Ta:’j :ui (Tan , i:_n [ -, ‘ ,-“ ‘ b I d i s cos - c e - c ’- ’ c a ,-r cc ’, t o e:sO .~ - c ’ . I : i o . ’:t . ’

- .1_i s- c e - c’ S ,i - cc i i ’ a r lc , i tn - in ’s - ’ lc’cnan gr am Sc: ” sod ’s- that lOc o c : t — ‘rag - ’ ~‘ o one ’ ’ 0 : . 0 ,i I i

r ) ( n l o g n ) and the r c_ uuc In g t i m e - :t,ill o(~~~~ ) . ,~ cc lc fo al ’s 5 , 5 a r i se t o o

- — dm~n~ ,i ‘ ‘ t s , ’L’l, finite - ‘ 1 ’ s-c ’ s - n t j-rolc J coors (I t a r t ior and flcn’- ’y [~ rc ~

- 1J. c:n j aj . P 1 ‘w Ar i aJ . j s i ’ s’.

2y:S . ’- c’.. of . 1 1 n ’ . - -ar e - ; ’ c c i i i c o n s c-rn sot in con t ext s c ’t l c o ’r than linear

- ‘ t , ’ - - o - i ” i . i - - n . I c : l : c c c c , o , t i c ’ ’ c l c o n t e s c i , 0 - cc th r cc h lc ’c’ . can be I ’ . -r cc . ”.c ic~t ’ - , s c c s c o c .

t e rn of c - i l i a c I , - ‘ on : , cs-I Lb c :,j ,n incd c :iti s-n O ti s- hat’s- ng s- t’.iI t — a cifldl ‘ i- s 0 0-i )fl

r ’~ l ac ing mul t i ~- liccit .1 ‘ n ( S c s c ’ k ln ’ .o u :c e mid Canré [mr 1’’3 ] ) • Another sic t-uat l o r

L _— -- 4 -- .

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ — . -~~~~~~~~.

‘.~O c ’i’ , ’ c y o c t c s - ocrc c t ’ l inear equatl ons occur is in the gU ci-ci . ‘1 - ‘on c u s a ly c i s -

‘I ’ - - os - uto r 1’- ‘gr ’ csioi :. lu ’s - i - o s - c -, for i n stan ce, that w” wi sOn c . ccc O i l y a

c -ccc i ut-~cr i’ - ’granc so that it does not recompute an ~‘s-~’~ no::: i or c. u , i c ,s-:c U sc :

‘,‘alls i c ’ OS ’ on” . of the variables in the cxi ression has c }i : u l g c c - i .

The l ’irs -’t c it-i ’ in the analysioc is to re~ re - ,’ n - n i t th e- i- - f r - - c . by a

5 ’) , ’. -in ~ rai h. -d ,cln ‘c_- ocr ’t .e - x ~i o n the flow gr :ci O r r e-’j i- c - :o , ’s - t , , a c c , , : - - 0 , - s--c 00 ’

t h e _ ‘ —cngr rrm (a :eL of n- n grocucc :t at cco c secnt c having a sc iosgl e c c io ’. o n - , ~~~~

m d c_n I

a s ’ .i i c gl o.c ccx ,i t~ j o c ” c i n t ) . ‘aeOn edge in the I’low gra) ii !, o, ’; n o c e - o r ° : ci n c u , :0

‘.‘s-i ’ c, s-n t rol I rons: t ofle basic block t - o - another. The n ’-r o c b’), ec c , s - i ’ - i-H

:‘ - - r each basic block, the set of avai lable e~~~re ss: i ’.I cn s ( t i n s - c e - win - c oh don

no-c L 0 : - c S to i - c - i-cc -rio ut. eo,i ’i can then b e t’onuui.ated as: a :yst cc:: c t ’ linear

0 u ’ . O i - s - n i : wi th - ‘ no. - “.‘ - sr l able f o r each ba sic block. The vc c_ r .i able is cc bit

c ccl i’ , w , i t i c ‘ n c- hit c -rc ” :~~. s-ncfl ng b’.. each program con Ic -nc - s -s--i -o s , and

c - ” , s o . o t t - ’ - i ’ l l ,’ - - - I, or t i c o s - c it I - as : r e - i i s -c _ c c : adtht i c - r i and c o oc _ nl t i~ U cation in

the sys - t c ’m ot ’ c - I r i S i on s . The s-j c a r c i t y structur e cs-f the oc : cr t - r i x c -r i- es-) conri so

S o the !‘Il cw gra s- Sr r ej ’nesco: ting the r - ’grs - u s- c. For l l i r t i ce - n’ i.e- i- a ”. iso of t O i l s ’

u cs-rn ero l ,s- ti j et ic” o - , scc ’t c Kildall [ 1973], l Orc lna t ’ i ’ cc r [ i l ’

~], and Allen and Cocke

[ 1 ( ( c - ] .

-inc cart use standard ‘Gauss-ian c i .0 cs-S n H .100 S o c -i c c c l s - c ’ ’: I,, ’. c - 0 : 0 ’ s s - c t t ’

‘ v s -n labI L e no-c1 re-s’s-- b - 0 c c , but it is n , ’I , eo r d i c e - l u _i. to S cc_ k - ’ r i ci~i t s s-iJnc - -f

so ca-s. . 0 ty. The fiüw graphs of many computer rogrcrn:: have a “~ cc lab

n ’ : - -r t , v called reducibilit y, which mean s S o c essence th: rt es--cry cyc ’ i c c ha:

a single entry o s -m t from the starting block of the 1’. J , ‘ gn iu : o . Aflen [1 - I f - I l]

and Cocke [.1 - T I] ] fi n’s-i formulated this- n c -i .1 s-o r c ’c f reduci bili t y , re - c c ’ s - nt re - I

an O(nm ) — t i m e algorithm I - test for n c - W i n bi l . i t- ’,’ , and used thij c i - c - c t in

an O (nrn ) —time algor ithm t ’, cr glob al ‘l ow analysis: OS ’ r c c o i n , i c ’ t i l e gi’a~ h s .

- i~l)

- —‘ - — —~-‘——— ——“ --f• — - . __ _, _ . — --‘- .-_

— _*, -“-S. ’— . . .,,-~. - — - ‘, - -.s---. — ~~~~~~~~ —~~—~~~_

~~j

- _~~

_ —~~~~~r

“— ~~~~~~~~~~~~~~~~

,~ ~~~~~~~~~~~~

— — -~~~~~~~

“ ‘ - - - ‘ ‘ -—~~~~~~~~~~~~~~~~~~

- — ‘— — .

~~~~~~~

—-. - - -

Si ’ . i c - i c - f t mid ~~inan [1’ 1c 11 I db ,.e---, ov er ,cci cur c’ c (n : . log n ) —time test r

i’ed ue .Lb , i .01 ty, cs-u .S cli ~~~~ ian [1013 ] combined with cle-s-’ t ’r’ u se-- ci ’ t ’ — 3 t r e e_ c

c , - “.‘o os-j r Q (’ c , ,i.og n ) — t cLcne meth od ‘ ‘o n ’ g U t c c l i’io.°w s-ri ca Ly : c . lsn ’. I c c ’ ’ s - o .f

[1 0 - 5 ] (U s~ c , ’s-’ el’ t ” ,d a r a t I o - n di “ -r u - cl c : :c . ho . co ,I 0 0 0 0 ’ g h bail o ’Il , cn: u c ’ , , c - c - c :.

wl ni ch 1: 1(1cc log n) — 1 ,1 coc o s by °c re s ul t - - I’ A,ho and UlLoan [l - - T7H .

and W ’ - g , -can c [ lr1 ,ctc ] di :— ccovered i s - on to us’s- ‘s a t in c -os - c m n - c ’ s - c’ i c - r i I . got y- o’ ’.

- e - c -, - t Sr ’ s - so O ( s - r 1, -g n ) -time ‘dl . f - s - - i 1 5 c c . Taiyi an [ 1 - t .. - } gay-c - ,”. c ’( on ~( . n ) I

— c ,L o :oo. ’ :i.g - s - - u Sc ” , f n ‘. :t ins-g o’ ,’coc I Li ,S o’; , cu r-i bat-cr ,i , ’ c ,;

t h n Gr c d i s - uco—\ - l - ; so - ‘Jg r .S 110” ! 1 - - run in - ‘- (or i ( c : , , n ) ) t ic:ccc - (Ta r t s -cd [1 0 ’ ’~~~~

’ I ’ ,

‘ I t - i ” ’ t ( n ’ : , n ) i s - a : ‘c o n o . ’ : . ’ o ’ o c ’ i . i n c en s e -1 ’ - ‘o n , c ’ c ’ r ’ coc u cc c ’ s ‘w n c : t c oi n ‘I - O t i

- c ’ . : - “ Os - c 0-Sa ”.’.ching on S t ri c cgc .

Gui. i -s ’ c -’ :-: ‘cs-ic y s-ni ’ s- ’ tr - ’ s-Ic r 0 rig: -,‘ . c s c ~ ~-:ed -os- f cOO c li ’ - c _ ct , 1 - s - c ’, ci,

0 ’s-s-c ’s-, a I’i n ,I t - . a’s1 Sco ,’.L~ - c , cmi 1-inc rn .I. s Oc ‘I,- 0 - c - s t - In s ii c ’.cn ’n ’ x - ce -u s- ’ -o: a

cont iguo us sub- i n - l o n g c-I ’ y . I t ’ r o i s the length of on curd n is- t h e

1 - - cog 1 Ic c’ t ’ y , t i c - c a a :tra ’i gict o ’ - n one - s - - c_ n s-i n - c thm occ o ’ .l,ves this yrc ’hle oc i n ’ .

-J (nm ’c t is-~~

c,. 5~ iut }c . Ico ,c r ’ O ’ I : , mid l ’ s-’ ct t ,t [1- I ’l l cic”ci sed an c ( n c ’ m) —tim e

‘ Ig ’ r O l l s - c c S - n s- S t o r n ‘ t t - c b e - l t i g . Tin- - i l - ‘ o. l g - r l t h m s o ’ ‘c’~. .- I c i in c ‘ s I t e - s o c on

- ‘r c - r c t Sn g ci - i o i toc :1 I ’UC i, l i O ” ’ c ’ - - ’ n’ s - - o c t i c c ,- ’ a c c - -gr : .c_” t o i’ t ’ C ’ . ! f O l :ne S h o e - s o d s c

I i i ts - c i f ’ , ci’ . i thi n . c i n - c s- c cci. t.l:~ - Lning :z c ’ts c ’sr:rc t , er —h ’ ’ —c,- i cc r i ’ : o ’ : . ‘ 0 ’ “so ’ - , ’ 1’ 5 1

0 t - ‘ - i s - ’ s t e j s 5 ’ t i n - so - -fr-c u :,. Boyer ‘ur l ‘i001’ dO [l~~( 5] I r e-I ‘ ‘ 5 C - 0 ls-c (“s- Oc

t e t S - - i ’ a i go c t ’ S S ho ’ c , wi c , c 1 c . aJ.. t h ‘ugh i t m acc an O(n l m ~ run sr s- , i n n g S ‘ “ - O o o 0c c :

w c o r o c t ‘case ( ‘r l i i u t . O c , Monr i so , mid l o O t [ i c ’ ’ , ] ) , I’ c - q ’ n.i r ’ c - : - - o i l - 1’ - .c ( n ( 1 ‘g ~- ‘

Inc on tb -- c;-’ r o n ~- :’s- , wher e ci i s the cn _ L O c c - r i ’ ’ ” :1 c c ’ .

A g’’n - r - a i _ iz a t ion of h- c i t t i -ni c oc - s - ’t n ’ i , ,i r cg r ‘ I t ” I s t , t, ‘ ‘ .0 n c - : Ui , i - ‘ c f ’ :

- mis-on cont igi n , , , c c f l c c ’ I . r c i o s g i t ’ I W o ’ s - S ‘r i n g s x and y . The I c c c 1. - n o ,

ii!

- ---_ .--_—~~~~~~~ TU ~~~~~~~~ ~~~ ‘ ~~~ .- . —‘-

,~ - ‘ - ~~~~~ .- - -

cscon t c ’Oi i r ng algoni~thm nc OoiLl t - - c rc -- i above do i c - c t :~~o.- cc . 5 ~ cs-n 0 lY ~ U s . cc r - i - i ’ s - c o d .

Karp , Miller , and Roscnh s-ng [ Iico)71? (U- . cr’ i b e - -’.,i cii i o( \ o c :~ I c ”. log(ncc ’ n ) ) —t is-s-c-

ci.gor ,stbm c ’s-or longest common sub strings ’ . lie - m en [1 T ,’- ’ J ‘,5J c c ‘cen t s-cS ‘cc

-s-c l ‘.1 s-- .l bbs -cc o. n . icig tree-c I n c a :s-cW wac t i , I c h s d . t bs-’e, t O n , i i ’. s- I c : : ifl o] (cr s - c : c )

t ime- . u.uc l n- - : . gh r t - [ 1 - 7 ] Ins - i s c r c - y i ’.L ’_cct a :‘t c u t ,i 0 ’bc : ’.s- ,l ‘no s-ton i c ’ I - .-an d _ c s - , -i ’~~c t i , . -ys

- ‘ t ’ Us-i s ciTg~- n ’ I t i t o s - .

I’ s - n ’ - - i s 1 C s-y~-c’sn c n t : .

The t r c t rn g no ml ’ -n ’,n- nt s : r i d s-cc: is to I t ct -c n ” c c o l i o n - t h e - c . r o ’,gI’,’ connected

Os- . ccy~’o orients - oi’ a given directed gra : Ii wI s-h i n cent i c e s - and 0cc edges.

Thi s i . nc-h le - s- : -s -c -u n _ c in f ind ing tic ’ i ’s- - o s - ’ . ’ .’ c i , -: c-1 cm- c s of a n o s - o —c y cc ’ c ’.en n i c

osc s-itrix (Fors~H S n ’ : s -n d Yoie-r [ icc ‘7

] ), I ns - 1.5 c o - s t cog -_ cr g , ul ;‘ :ub ’ .c icc 0 ’, c r c

and tras rzc i erit s t a t e s oft a Ma rko v el s - on i r, (Fox and Landy [1, ” .- 0 : ] ) , an d. I n c

:‘,I rd d.sin g the - so s o , cct ivi ty .- “ t sc of a set of ermutcn d i cc,, ( tue -Kay and hn-gn’:r ’:r

[1 i7 14 J ) . lang- -li t and W e - s t - s -s--h-erg [1’o c ’i. j gave s-in O(n ~’) - t t c n c cilg’.’srs- t SL - co.

tCunro [i ’l l] de-cer ibed cs-n improved :cltgorlthm od.th a r ’unning t i m e c f

O(r , log n + cs-i l- . Tar jan [IL’ “7 1] s-’ c s:c ’iit , c’.’i, an oi’ (n s- ’~- ooc ’r — 5 . 1 : : . cc ‘ ‘ . . c t ‘ rj ’ i - O n

which us- ,c-. c - c t i c — c i m t . stcarel c -ci i ci t ’os-W c i cccl bc ,c ‘.ini. a c ’s-rue i - cr’s’’. ’: .1 - c- s -c Ove

this : irs -hI” : : . .

Pla n ar it y ‘ 1 ’. :S _c.~~~~

Let G be -a grap h . The ~‘lan anity testing ‘ n oble -cc . is to , dc,c t - n-r ccc i c c e-

whether cI can he th awn I— on ci plane s-c U s - c -cl mc- two e Igc .’c - ‘n’ c- :c .

Kura,k owskc l [ l c c ’~O~ r os - - i - i- cI an elegant mathemat iccc,,l ch arocet - ‘ r , i os -u t ion cl ’

j lanai’ gras-bc , showing th at a gra ph ‘I i s - planar U’ and - ‘r ’r1 y i t ’ I’S d os-es

not contai.n - -n cc of the on -- f r - l i hs ’ Ic - ~s-i go~ 5 . “° c- cs- a go ’Ii ’:-l ’:i] - s’ s - n i c o s t 0 ’s- ’ ’ c d I . .

Unfortun ately , Kuratow ocki ‘ s.c co ’r’ , I S c- ri - s - r n ‘ - - ccc’s: I c be us- ”l c ’ s : : 5 ’ a c r :cc ’t I cal

test for planar ity. l ion . ] c _ n o t - i ’ cnn - ’! I art t ’r [ 1) - 1] r ‘ c c c , , - S an oils - t o r i thm

Li

~~-~~~~s - E - - ’ ‘-“-~ - - ~~~~~~~~~~ —

c - i c ’ cs- I n S e c I s ’ har n an i ty by tryin g to c -r cctruc s-t a planar repre sent ation

“ o s-lie ~-l r- i~ i c . They gave no t_ cne bound for the algorithm, and their

c’ - ’n tat i , ’n c’ - s - cn t a ln _ c cnn error : the r ot - os-ed -i lgor lthm ma~ run f e - r e v s - n.

;, - i - i : t e in [1’ ” .- ~- J C ‘rre ctlb , I’ ‘ro s-su,lateci tin S.: algorithm, and Ghirey ] i°- s - ’ . ]

‘ci ’,” c u r 0(n ) - t io c . n ’. implementation c-f it . Hopcroft s-ins-i Iar :an [1y72]

e- c - - i o ,loi c ” . ’J — ‘s e - c ’ s o i c — ’io’:c c-cs-f cc’. -cc _c a:cr.o-:-n i cice Jat os , ‘. ,- -cs c ’ ’ ,y’ ,

in an 0(ni I -s-g n ) — t .l cs-:e- b oss- lens -co st s - s - i -cc , “. i i : Ich i wa _ c las---’. r sl coco 1.1 fled ass- . :

t ’ . - 0( on ’. (FIo ~ er:t ’t and Tanj s-cn [ 1 ’ .’~L ] ) .

[Fi gure 5.1]

I- c o cc i - c l , I ’lvn ’in , -or nd C-.’J ’crb-cj u ’.’cn [ l9c-7 ] i resented ‘.nn-~ti ,-:r g ood ‘ J g c n - i s - i . c - .,

w J ’ O n c a n t s-r v urn g an --cci lIe-it t icore b’.. ”.nis-J. Their (U gc -n it h r r s can eas ily S.’cc’

Lei r~cn U t to ~ac,cc in 0(n’~ t i c s - n c . Booth and Luek er [ ii ’ - ‘] sin wed S. on

to cs -c - Sc e-c e I r i ~~~~ tree data st~~ cture in c_n 0 ( n ) -time it ’ .ss-c-r,t ’s - t i - -n c-f

4 ,t n cs - alLgonitrnr .

t-Sa .ximimi Network Flow.

Let G be a directe d gras- h with two ci cis-t ingu is-he-c i vert Ices , a

source s and a sink t . For each edge e in G , ii’t c ( - ) be a

non-negativc -’ necI-volued n o s-r i an oit y. A il .s-w f on .1 is- a no s - n -n ns - -g ’o t c v c c

value f ( e ) on each edge such that , o’or all vert ices- v exce - ”. ’. s- - sir s-i

‘. in ’ . total fl - ow .r i edges entering v is- equal to the t e - t a l S’low lea’s-l ag so

‘5 1cc’ value of the flow is the total flow leaving s (whi ’..o ic i-c c us-il. to -

tire :. - ‘s-t ,ai. fi-s-w entering t ) . The majd mimi network f_ ow o r i’, ’c€c c’ , is’ t o

‘ S ” t ’ . c r r r c l f l c ’ a f low f (-c ’.’ ) of maximum value satisfying :‘(e ~ -‘.z c (e ’n t ’- so

coil. - -c i gec c e

Classic work by Ford and ?u.lkers-on [l°7kc2] produced an cs - leg - s - c l a lgor ithm

wh i ch au~ ’nentc: flow along p aths. Unfortunately, in the wcs-rs c t case t i c e - ’i r

1~2

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ -- - . --

I thm requ ir e-sc ~s- xj --

- i n s - - t n t, S al L i os -rn’. , S i no c ’ ,cgl r it ‘so or ’ .”.: c c - ce - , . - ii c c c ’ , I ’ ,’ a-c ’ s-i

in j -nactice . I sis-s- lids :sic - i Karo [ 1972], by using bi” :’..it ,i,- fi r : t search

to guide the ~:eles-’t,b - c c ’ . c - i ’ cc~ Oc r c ’ . o o t ‘S I c c ’ 5 0 ’ ~. :, s- r o o o . i cn ce ’. i mi 0 ( c c c o s - ’ ) — t c c , - ’

s-’an,i ation t ’ the S corJ—i ’lclk ’ .’r: . ‘o n algoril-lic : . m dc”: e r n I e - o n ’ s 11 , , S o :i n ’s S c [ l — T ’( ’.b ]

use d U r’e .q o I ’S , Sr — 1 . 1 0’: cn ca rr ’}i c las icci ] rocs -ve ’.i a ni a ti rn ~: ooce t hn ’ -ds 1- - : lCOci CV OO mi

O(n ”m~ t i c oce n- - a s s - , I . The b est s-nlg . ’ri ’t Iuc- , s— c - c ’s-sm 0 ’ c s - s - ’ c ’ .~ ‘s’ so t h I s : s - n - i d l e - o s - is

Iii’,’ to Kars-cin’. -v [ i’ c’c ’ i c 1, tino .1 cs- is- s - n - : c - i ’ .’ ’ so ‘nig ) r , c t s - io o - c - obt :nin an

.t(n ) time boujid.

Sn’ . ’ . h Match icr f .

If I is an urncil n’ - cc t , e ct groin So . the- gras- h “-at-eisiscg vi- ‘t i es- c I s . : I. c c l a d

a oc-’.ax,i :cs-.so’, o s - .s-cr i en ’ of ‘s-ige -c’ in 0 , os-c O w ’ ., - h a v in g a c s-nc cc c -c o d : l o t - . l.nch

a s- c ,’. S, of edge-c: l s’.O a ro ,ax ,i ccouio. s-catching . ,-‘cs 10::: cs - ’f c to c t- . r c el ‘s-I ‘. ‘ c n . n o , - O ’ ti ’. l s-

o r S ie-nos- is i t_ c r e s t - i -I - - I 1 _ c r ’ . to to l j- o , s - -: l c-- go’s-ji m: . A gn’:’ .:S s Is I - i S cin ’t- ,l t e I : ’

it : -,ert .1 es-c s can be I ‘ crS I S . .o ’ ’ r i c : J ,i c n c ,. t o n c ’ . ’tc so ‘ Sho rt flo ~‘ o.ig ’’ c’:fl r n s -’ ’ .c ’ Ss-

tw- - v-cr’S c-ce - : in the s- as-os-c: se t .

The i’ s - c cj r ’t, o te gr :1: is matching s - - I Its - ’::: can I cc t s - ’ c u o : : ’ - c s - - t o ‘,is - :c

1 i n c s - - - ’n , r —t ,c r’s-e ci ’. g ri thm m t . ’ a on e- t one -r I - c f l- , -rc s - - r e-is-len : i n ti c S o . ’ O c -cl,1 n ’ s -o s - c c

c c ‘o r S t b - c c iT- ’ - -r ’n t ’ (Ford and Fulkerc ’~,o’s [ i- s-s - I l l ] ) ; fe-o r such :c : i ’ c l , ’ ” , i S i s - ’

Ford—F ’u.l,’.oc ‘r: c s-n cc, .! gc r c t hin has an r ) ( r in os l S . c::e bound. U .s-Os-os [ 1 55 ccc ’ .’’.!

n ” o s r s - i t s 01’ .‘lg- -mc nlry to cs- Sc-L aIn e- s c : _ c ’ o i s - I a lTh,’ ‘t i re c s - cs - cc cc, i’ ’ - , ncs -sI ,ied, the-

Olu ng ar l - i n :‘: ‘-t i : - -t . Il - s - n cu’,,- .c ’i- and Karli [ i s - ,- ’’.’- ] s-s- se c . s-o’e- :’.111, s — ’ s ’’,r :s- - ‘ c - r e S s

- - - ‘ - 1 ’: - ‘-‘cc-I s- icc] r - - ’:’ .c . n u~ d,a ’t - ’ .n g c o - c I S c - Ic ’ t - ach e-soc, is-s ‘( c”. ccc ’s c c c - :

The ir alg. - r I S-ic : i s es:s:c .’nt ,i , os . ly t I c s - ’ c ’ c,c,c’ ,e as’ i l l o c . I ‘., c - s- ( i-lv e- rn and ’ , -cin’ , S ‘to o

[ i n , ’ 5 ] ) .

Berge [ l’)5’ - ] and li -rn - r o n and Rab .i r ’ [ 1 1 5 ’ . ’ . ] r - so - - -1 5 I rc i l an ‘ c n s t ~ - o ’ c’ , c i c o g

path method can solve the max imum m o n t e - h i cig r , - i s - l c ’o - , c s -n ’ . non- i - i c ‘c ,r ’t S t e g i ’ c c c i n .

- ~~~~~~~~~~~~~~ - ‘-~~~~~ ~~~~~~ , ‘ - ‘ -—

— -‘-.

Sc n,- 9, a g e-s -i ‘~Lg r I t :50, 0 - , ‘ , t , - t 5’ 11 on Li - - ‘ ‘ s i c , ,, r i’ . - sr cl t :. : . Js-o u n s -’J. [1-c 5 ]

.1s t - c -s-vole c , t ’ n c 5c i n i g i tus s-is-c - ‘..~s- , c b o r g c a ’ S i , , I n ’ . ’’, t - - g i v c ’ ’ . c - -J y s - c - ” o c i ’~l~

He s-’ls-I nc’d ‘cc 0(n ’ - - c - c ’ c- -cocci , l b ‘ -~~}c it I s - as - t i ,

t , “ ,c t - s - c - --i - s- :. o . c . ‘ -~~~~~

‘ n ’i thm t c - ,c in j ( r s 2 m 0 t , so ,-,’ . 5, - owl - -i- [1 ’.7 - j

s- si s n i c c ‘on [ 1 1 ” . ] ,r s - . s - - - o - ’ c s - i . - : . ’~~ , c ’s’. ’ 0(r s-o ) — : ,~ . cs-c- ‘ o ,, c or ,, t c n c : ,c

‘Is-Li ’ ,, i ] , , ’ 5 - i c c - o : . I nn . 1, e ‘rcc l t n t - i 0,1 . l i t - c c : ~‘ I ‘ I - c c ; ‘so c ‘is-cd E - ’.,r’o cci ’s S-h is - ’

. co ’ . t cr - t n - c . - ’ - ‘ - - s of 5c c ,! on t ’ . -n ‘~~~n , ‘ s - c , , ( n n l 2 m lug n~ - t .~~c c ’ ’ ’ r tg s - t i m.

‘ - ‘ s - UnIt c - c .

Let ~~~~~~ Sc ’ ’ fl ,U. ° ’ s - in ’t ‘ ‘ ‘ 0 , , es-d o ‘.- n c t c ~, . c , ccg s-i si ngle

- 1.- n o n e - n t . 71cc .1 - - oc i cct set cr n l o c r . ’ — 1 - ‘ - 0 ’ i c c s-c c arry -ut a . e ~dcX c ee DO

0] s- rat ,5 u r n s ’ - . 10 ’ ScI ’ . ’.’ 5 - _ c ,,l, on t , ‘ - W c ‘I’m cs- : on t 1,e s - ct , ,

n - ’ t ’ - s - s -’ , 1c ’ .~- t ’nc ’ - os- s -u ’. ’. ’ . I the set c -r,ta i. o s i o n g o le - os - c e -nt x

ar _co : (~~. B ) : add all . ‘ ‘c - - cs - ’ . crs -l . . - S ’ s-ct B to sot A (des ’ s-’ Is- I s- f

:s-t B ) .

The- ojec ’:’.tiorno c ‘ c,r s-’ t . c:,’ ’.-- ’.,s-’i’ .ec! -‘ co t , ‘sn—linie o ths-t iso , each ,i o r s tn c ,set ,I

mu st be so - -ccc i lc.’.t ed i c e - fo re the next fl o s s - c is known . Assum e ion e - o ’ .n ’,”.’ -.’ o , I -s -r ice

Lb jc , tic - ’. c’ mu ‘nc-cs - c .-f cs - I -:r’jt ,i “as.’ c , cn t : s-j is-s exactly o r — c s-n . ’ - -or ‘o en-at is-s-n c

(so that ‘-j:’t- ,-r the last uni on all c.c i ’ -o:, c ’ r i t s ’ are in ‘nc.- set ) arid ocr ~‘- n

I s - c t ’ .”.rrrj xeci f ind -s-j erat i. c-c: (if a ~,. n , soc -n e elementcc are never 1’,s- und ’s .

al .I . c:r ar cs-I Fischer [ i c c 1 . 3 rn’s c s - s - ed an algori Lions - ’, S ’ s-n ti’s- i s i r , s-bic ccco in

wh ich each se t - is- r e - c re - s . ‘t i t , c ” .l by a tree. Each vertex s - i ’ the tree re~ r e -ce -n it :

‘ f l ’: o l e -me - oc t • Tic ’ ’ r , .~- S of th i ~,c t i- o c r cont ai ns the a s-ins-ce - ‘I ’ the set,, ari d each

tre e vs --rS.ex hoi s - a ; -- ir i t- - r t - - it -s , ot in the tree. Ice Figure 5.l’..

A f’S n-S - c - ri -- ‘I - ‘ o os - ’nt x is n - i ’ s ric ed by s :Lonrt ing at the vt --r i ccc rei resenting

x and 1’ 1.] -w in g arc-n i p .1 nO r: until reaching the root of the

l h I s - so - s - S c o c t s- ci s c S - S c : ‘ . 5 c e _ c ’ - • A wi ,s-i ’ s - e t c A and B It ,

1414

_ _ _ _ _ _ _ _ .~~~~~~~~~~~.

___________________________ ~~~~—-‘ - - -

emS ’ .r’cc ’ .-d by making the m ’ -o ’S s.f the A t ree- the ou’s -’ nn t of the r oot

- s- f the B tree .

[~~~gure 5 . 14 ]

Thi s- alg s-’ii-ha requires 0(nm ) time in the we-s -- ct cc ”.:’. , since an

‘s-os - f - so t s-mat ’. ’ .:- - m oe -I-sm” - ‘s-f unions can build up a tree cc -or: 1st ‘ cog no ’ a single

It ‘-cog p ath . — I s -IL ls--so and Fi . s-- ei c e r- o:r’s-di f ied the union so cc - - i ci”: is- i th cs-

l i~ wtrs -s-’ on ce ’ : Ii’ B contains ocr - -re - -cJ ’ . c c ’cen t s than A - then the so

c s-I ’ the B l c-a-c is made the I t -a r ennt - s - f the root of the A to ” .c ’ co , ins-s t O n e

nrc ocs -re A i_ c cocoved t. the old r - cco ’ o , as-’ B . lee Figur e 5 . 5 . This w’.c gln i i - _ co

union heur i st i c , ,i- os-1 ’ r ’ -~-~ s the algcs-rith .’.o. considerab ly c lSail-s-r and Fischer

~r~s-vect an ‘~ (nn ‘1 ~-g n ’S t , ,S ns-e- bound.

[Figure 5 .5]

0-ic! n-- i ’ v and 0-Sc cs-0’ - cc (II. , Hopcr ’ - IL , cur d UlL l.c:cans- [I. - - ‘1~] o- t i l l e d the

t ic - I i- ros -ce-.Iur t - by allis-ag, a Is-cr’s-s-r I s-ti ,c calied ~

c ’.t’n s-o ’s-ircc ro: ss- si :‘ n: a fte r a

f i n - i on elecco,’nt x , all vertices - -- n the pat h f r ‘0 cc X t the r -s-I are

made cs - SoS -u - et c f the i’ . - - S . 5’s-ce Figure 5. - . . Th:i s. in e - nc’ o:l c-es the ‘S . , ooc e of

a find by a constant fact- ’ so h u~ oo’.ay save-’ time on later ‘i rs - I: .

[ Figs-cnn: 5 . -

The s-et union algorith m with cc c i t -tn s-oors -i . re c.’I os-n is very c’s-c- ’,,- t cc r o ’s-g ro-is -,

but “.‘er 0-’ i- card to ru nal y s- ’- . Fischer [1’ T ’~~~~ ~~~~

-ve d an ~(~~1/2) a: c-cr houn d

and an i,i (m log n) lower bound on the w ‘so c - i - -case running t i c s - n e of the alg ’.rit io c’:.

wi th path c ’c. ’aj r e-c - s-s- i -n but without w c c ig l n t eo i uni on. S ~ot ens_ca [1972] 15:0 r , ’v c- s-c

the upj . er boun d ts-’- 0(m log n) and this - i c determined ‘Sb ’.: running ticc oe 5 - ’

within a e- , oncs c t . a r c ’ S . e-ac’t -so I’ ‘ r the case when m is -- O (n ‘° • 5-11th S - d I r cc s-ct 1 ” .

- ‘ ‘r o’s re - s-s-ion and weighted un i on , the -alg or I thm is- even inar ’.’i ’s - - 5, c cus s-ilty ’,

Fischer [i i r o ’2] proved an O(m log log n) upper bound on the running tic: , ” . .

145

_ _ _ _ _ _ _ _ _ _ I ~~~~~~~~~~ ~~~~~~~~~~~~ ~~~~~~~~~~~~~ -- ~~~~ - .

~~~~~~ ‘ “~~~~~~~~~~~

—_- —~~

I !;.’n ’ - ’ c ’t - u - c ‘.n J ’ . o -,s-r c ’ . [ i ’ ’~~ l i :- : c c s - - s-’ -:~n S b ~ u c c - ’r s-c -s-cs - s - i t - ( c_ c L~ g n c

I, t innn ’ .c cs-

wine - cs -’ log n = m i r i [i~~~l ’g ‘. g ... 1-g n~~~1j . 5 ai ’ ,ia c o [S . c- c I a ] i co , i r .ovc n n

c-h e s - o: . -.cr Sc -s - s - i s - n s- ’ . 0 ( 0 ’ , i ( c c c . n ) ) , w O n- cs - ”. - -~~c - , n ) is- s-c,

c ’s-mcI i s-is-al i c -v - cr : - - ~t ’ A-: cs- - c oc os - s -- s - or ’ s : c n - , - s-- s-s-S ( A ’ zs - ec ’:co ,io , - , ( 1 ‘.08 ‘S s-i--

as c l - IL on:.

F o r i .~ ° s-i ‘. 0 t h e - 0’s-i cc -c t Loon A ( i . 5 ) ccc S’.c o ’ s n , - - : icy

~~~~. - ) A ( l . 0 ) = 0

= o’~ t ’ o - -

-

= A ( L - ” ,2 ~ :‘

r I 1

,0~~j , j b = ‘ . l o i — I. , A ( i - j ’ — l i ~ O ’ ’i’ Si ~‘ I . 1 ‘~~‘ 2

( , . ) ‘ ( - - , : s - ’ m ioc [ I A ( i , Ion/n S ~cig r c } .~~~~~

Th’, : z ,o ’,c:_ cn ‘- (to ’ . s - ( c o c , n n 1~ is a rather c- ‘c~ ‘sL’a~ ’-o n c - c 01cr such -ci s i.cmjct e

‘t s -,goo ’ithr , . ,, ) r c - ’ ’’a~,’ ona ° si ~’ s- ,s- . i oi: K ci ’ S , ‘ S c - - c - j t j s- 0 cc’,: s o s - -a s - los - . Is - m i s - c s - [l’ ’.”’ - : ’.j

:5c c-wi-I S S c a t b - c r - - oc r - c~ - r si _ c ’j s- s-: in : t az ”.s -- ’: - s - t o , - s’:t cs-c s ’ - n ’ s o r -cc- S -c -- c

wc , l - I- , r e c l o - - - ~( co-, x ( c c ,, r’s - S t i coc- ,’. ~s- , - - - s ss- t, ’:- -’s b y 1 0 - , j a t l - c c’ 000J r’:’s ’ n c

a ’t g c-lo s- boo . ir s- 5 ’ s -ct ‘ s - c : . i . n , k-s - j c’ ,cos -~ o” , :: cc c i c t c n c c - n c - c i i i’ . - ; c , (o c c c ( ’ ’ . o c b )

t Jj cc -~ O n t I n : rs- ’ n ’ : t s- is- t solve the set un _ cu ~ s-’ ’I’1e-c”. (Tar ’ ccin’s [ li ’ D.I’ r c~~: ,1s -ck ’. ‘ s ’ -s .c. ’ - c u , n ’ t , i s-n 15 . t c I , , .-n ’ - ’- r :c in the n - c ’S . - - ’

-

For -my so- ‘ at rc clO: c i c ’ . ’ so x . L XJ - I - ‘ in S - - : t~L - ‘. gre-cc S - -: - t o O - ‘g- - so n t ‘irg- ’ n’than x

14 ’

F_.— ~ -_—-c—~~~ .,~ -~~

— Tl~~l’ ,,,. ,,_~~,, , ~———~~~~~~~‘ -

—

s- • Future Dir e ’ .’t,i cons - .

The field or ’ combinatorial alg orIthm s is tos- ‘s-as ’s- t s- ‘. ‘.- ‘:-~r S n -ci

‘s i ngl e- m a s e r - ‘ so even in a single ho ok. I ha’s-c t r l - - -1 s-~ i Sa t cci.

sonic -f th e- - nos a , i -so r e - s c u l l s and u,o s-d -cr l y i rng s ole-as- I s - n l b S s 1 c d l , but I t _ c - n’ -

ar’- c e - n t a i r n l y many I coc i c ’ so s- ant resu lts I have had t -cccnr i. t . TSr m_c ,cc, o c o ’c’,uch

on cc co ’s - .c cc ,tc- ci cs -a t ‘ r c a i alg - r I t , s - s - cc-cs’ is -a ’ Sc ’ .c’ccn Ion s- , rcr u chn r ’’c :, a ior : t he co-

in s - h i s cc ‘nc t ud i rig ce-ct ’ cc I on’s-oI l I l k -c t s - s~~ gest five- areas I -so c ’ s-cO o_cr’,

r s - - - ‘0 ’ i c . ‘i -’ - ,’. ,’i: Sri wi c ’. ch rc~ at i vely l i t tle work has been i- c ’ . - , - Lo ut ‘ S n

a s , , - ::, i ’s-s-c c’- -w a s ,cc as-’ s-o - t ’s - n s - t i a .Ly great .

5) =

Ac-sw-sn ag n’c- ’ n i s c q:i o t i s -u o n _ c r s-i be a c’s-cs-, s-- s.’ r ’ cc ii s - ’ S° s- cISc ‘S oc cc

‘S-r io - rj . A c t h u o .igis- ctca c’sy c c .” j le hav e at 5- -: rs-c~ ted t - :,s-l-s-- ’.cc t0 r -I~~ cc rub - c r c _ c v- : rv

L i t t l e ~ o’ ‘1- 0, 55 ca. been coca-I s-c . It se - s -co o ’s tr ’.at n ’ s - c s - c :ca ° r c - o n ‘ i - ca is-

; c - ” . - oi -s- .l ; t o n- - e v i s - S , ’ . r r - o - s-f S -a-c - -c, S l i t , ~ nd Sol s - - ’-aj [11’ 5] , cn 1 s - , - ’-,’ : t s- I nat

‘ S I ” ., - s , o o ’ . a l i ss-a ° - c , . the sc t ’-,,o , , I ar-J tech~n I - ,1u- -- S_ -f ~ r s- vI ng o r - L I e - c - , o c - -.r t . c - a:’.

n t - be on - -r Iul ‘. o , c o-si’s S :h a s- cs x . it is eves-n c— s -c - ce’ s-- cord e

‘ I S i s - = ‘I ~~ c i ’ _c~~~s-ton ‘ - e - o c , • be- solved w ith in the fr- s- c . on o r ’ :, :‘ f’ . - r :o,a.

s - c - - S thno’ , cr ,’ ( i i a o ’ t c s - a r ’ .i s ar 1 5 -o n r c S’t, [ 17 ’ 1 ) .

~s-i so - : , c - - o c - ro t ’ l y , w’ s ’ . on cs -n w a l to , sot nothing ab o ut the r I s - L i v e

0 . . 1- ” . - ‘ rccr, I , : l s ct i c and n . -a - 5 - ’ t , e r c c o n S s t i c ’ oci ’o g ’n i t b c ’ o s- , as-s- cl s-s-n ‘.1’. . the

r-c s-ti o rn so hi j ’ : -‘ ‘ w- ,’~.c ’ t i cs-” and s~ acc as measure s c o ol ’ cot s-ic I t - - c c ‘ t . y .

r- ’ ,c s - , b_ c is-i hi s area wo u l d be i tcrj ” rt an’t . Recent l y, H ‘ - cr :’ , . l aos-I . an si

[i ‘“ .1 ] w- -r c’ able t - .c Ls - o c w that any cc s-m 1-utaLan “t- go ’ rlng 5 (n)

i c c - ’ s-r i a c ’ s-~’t t ita j- e Tur i tog machine can be carrIed cr u l - i n S ( n ) 11ag t (n ’S

- ci - ‘ . Thus, at- co ast 0 ’ -r :oou i t i ta ~ -s- Tu ning cn accl’ ; ’l c o - c ’, c~ o s - - i l : - a , - , ‘so - :

‘. ‘ c , , ia t I c - r ec s-n l rc e thai’s- S t o , : . This is the -o ’II.y such m s’s-II ’S ci’s ‘-s--nYc.

147

_ _ _ $

-~~~. -- • ‘ —

~~ —

~~~~~~ — .- - —i --

~~~~~~~~

‘- ‘ — - ‘ - - r ’ ’ -—— - - —“ — “‘ —‘—- cc~~~~~~ c -’ ‘‘ ~~- , • ‘

sr i - : ’ -s- c , 1 n’ a - i c to ‘s - h - - çc = 7gs-s- ? question is t consi o ’i-s- r c t - i ’ a articul ar

~~- n’s- — c- .c-s- 1s-Sc - j r 0 lI ’s-c’s- os- c ’ s - c ,_c fOCi ‘ .0 s-s-las: of alg r .I ‘ItIs-r~: as-s- s-i. to .sl r- s -w that

-cc’,’-: g, s-lg.’n - l tSs - ’: Ia s - - i s i s - ,IItc :, lt ’s-s-l cIt -O s re,’s-u ,Lr e c - c ,, c - re than i’olt n n , - c ’ . , n~~l t ics-c.

Resujt,s ~‘ this ki on d has-- - - been Octal s-r od for the ‘.catisfiab’iIity r 0:, err

( - a , 11 [ 1 ‘~‘5 ] 5 , Icc- c ‘na - s ’ s - Icc _ c o o s t - _ c s - c I t e sos -c t u r s - b l e o c c (dOri s-tat [ i - c - - ] ) , and the

‘.c,rc ’t: S s c o t s - ’ ‘. rig c-’ , : - t s - c ’ o s - (‘- t , c c ’in . c, i,t [l’ s- - ‘ S .

-~~~ - t : o r ‘st _ c - Is i s -’ t o - ‘os-s-s- i d- -c ‘n . ,-del .c s-i’ cor,cc u t a t c i s-i s - i c - - s o

O h ’s-ti’s- i’s-s- ring “c a_ Sn c c - , . is-ic cc oss- s ibi l i ty is to s tuc ’ s~’ t O ’ . ’ : s - I , -: , - O ’ B oo_ can

s-r ’~~’o i ts . c r - - .‘ ccc -,0 ng 5c-o ‘I t s - o ur 0 00’ s- - ’ - c c . . For re- s-s-Its in t o t , ’ ‘cs-so - - a ,

c - s- s - .Ia’s-’ag’ [1- c-: - S . A r ’- ’, ’~’. - - 1 c nn ‘.1:1 i s - o h - - ‘, , ‘.-‘ . c~~e d -ac :s- s - n s- ’, . i s-c c S c o : c r ’ S t . ,

I.au,. ac-S :0 1 ’ s - co t S - o b t a I n ° I c e l s o - . ‘‘ — c s- ace i rad’.c- .’.I f r ’ c 4 u _ t . A so t s - c .Sc,’

5~o - ’_ -.cas ’_ - “ i n ’ s- - os-s-id: ‘ s - c c t i n - ’ ~ - -i s - r ’ l e gc .n_’’ c- s - I~~ a s- 1s- n’a o i s - i l ’ ,’ 5~~

, c c c ; o S o c a t r I ‘ci, 1 c ;- -s i ni . (vs - s I - nc , S [1 c-~51c , is-° ’ I ) .

Ave-rag ’- .‘ u - s - c a y _si .

Alt- h a’ :, st t ’ ’ 5 - ci’ ,c ’ rc cc - ‘c n o l n’s ’ct’.- s-nlaIt a s-I s - r l t ir’ o ,oc ut s I ’ s ” . t he

‘ - s o - -i ’: . 1 s, “ c - i c , t c-ti c - S .-‘ ‘t s- ” ‘ c o n g is-a : J - . ’. ’ - o i ’~’ ’” cI- — - .os-i , - -.- ici c la ‘.‘ . iS . d1v~’r s-ge— s- s -- ’ s - s -

a r o a _ c y c i I. j “ s- - s - c t l al, v , c s -~ - s ot a n t ‘ s- s - io’S , s - : - 0 ’ s - n . . The n-,:s:,c .I. S. so Di ’ c rc i °~s.’- a 1c , n

:, . . c ’s s-- [1’’ - iJ s i - - - ‘c - ’: s-ri “‘ ,,s - c , i cnnci gc’ 0t 5 o ’ . s ‘ - s - c s - co s-i .‘ s - ‘too t h s-i’ i lace i t - s o

- c ’c- ’ r - ~~- - — o . ’ c s - : ’ ’ a”i ’ c , y ’:~~s- -I “ s o ’ s - o S ; al~~ s o c - Sr - c: , . Sy i r s- [ l ’. r~~] has ce-v s - - c d

an 0( ns- (1, ‘c- s- 10 ” acJ- ’r - c~~ - t i nts - - a_ g o n ’. i ilcts ‘ - -n the alIl— ia ,rc sh 0- O s - - s t

s - d c , 1r ’ - h t - - , wO c - c’s B s - ‘ c d ’ s - c ’ s ’ , gi .c o : c - - r ’ , s - I s - -I -b y-c r [l s-~i’ ] m o d l t ’I ’ ..-i I s-,)

• ‘, ‘ : i s - i s i t, Sv e s- i s - s - s - ros s in o ) ( i ’ s I, cg n ) av er -i~lI’ : t .Lor ’ .c-. ichc c rr [l - . s- ( 1

s - s - c t. - i - -cr : ’ - ; q . O ( n log fl - on ’s ac - er - -s - c- ’ . t i cs-ac S r’ an sc t I c ’ s- ci s - s -’us-’c ,’ a ig - r . S c , ”,.

‘{a . c [ I . I ’ ’ ) , Doyl’s- and dl v- o , :t [1 ‘ - - ] , os - cr I - t ,- i ’ ;th and ~c , s ’. c I r S , 4 s - , , [ ‘ - ] I cc_ ce - -

a c - i - ‘ . r c - I Lb - b -cBs -v i c r s-I ’ c’ s-c t, un ‘ o n ; ‘s - ’g - r i t . hj ccsc 0 ’ -so - - ‘s--e r os - I i r 1-an - l i t~,’

d ’ : t r i c o -,r ’ - s -in S . -tuc lc cs-c rc ’ work irs- t5r 0, c ’s-”-sa i.’ c c - - c - : .

14 ”

-- —~~~~~~~~~~~~~~~~~~~~~~~ = . ~~~~~~~~~~~~~ ‘~~~

— “

Gill [ s c ’ ) J os - s - cd Rabin [ i s - -’ - ] have ro coco s-ed dti s-ot~r -: ’s’ k i ln- i II.’ ‘ ‘s- c - so ’ s-ge--

cs- s--c r:o- - -iei s - ct ’ s- - co o i s - o x i ty, in whoi ch the ~~~gor it-ho - . c : , c I’ss - - : us-c us - ’ rn~ cd . oc c‘~.

cis-oicess- . i- -is - ’ suc in an s-Ig:’s-’i ti _ c -cr, cc c - nc s - ’ coca ’ ,- be s-isle I c s-nv LSncil the -J:g -s - - i tb:.

cur’s: ‘s ’ s-c - s t ‘s - i Sil o . ‘ c ’ft’ I’ -’Ics-’S c ’ I n c ’ S _ c o s- n d,’s - cct ‘c- f the I s - s - c uS, d.ictribut l . - c c , 1 c - c o s - s -° ,ns-:

the os -vs - crags - c is tcc l ’c - .:cc s -r u ’S. over the i r i s s-nt hut over Use o .c _ cc’ IL - .l, e cc - cc c i us-. - .0 , 1 - n-s c

the algc’.rit h.’oc s- n i a S I i s - ’ e - s - o c c c l s-cs-, . Ai g, cs- ’itbrns cc-c l s-t c on - t e c ’Un ng r I o -c s-li t y-

( f l t r c s - c c ’- ’.n s- ca n s - i siolos-v uCj [1- ’ ’, ” ’ 1, n e - s ; - l r , l’ .~:- - } , c l c s - - i l s-in s-I - . -‘ u-st j cints-

(Rabin [l - ’.i’ J ” , and, hcs - :Ssl rr g ( C a r s - o r c e - s - s -i i’;e~~c- ’oan [ II. c- ”~~~ ) wb,cich cs - so ’, good

ion thi n’ c o o s - c c _ c .

C, c r rc t s - s - s - nt I c r s-c ’ s - c ’ s _ c s-s- r i co A15 r’.I t l noo , lj ’ o ’c c i s- c — ‘I fs

The ch, o , ’s- - : cs-f -cs-c :-,,ig n’i t hccc 10 ’ so - cs-n ’ Ice s-- c,- , 5 c - s - ‘ s - cc ’ s us- s- s - o s - 0 , so - s - ho- _ cc

acy s--~ t ,’n’c,, i c ro,n ,”s rn .i ng t o t s - ce . A sit’s- l~ ‘ s-I c s-’ I i . c : oos a ,y be h e l t e r - c ’ . l o st ers-’o’.:s-i..I c s - s-c e - -—

ccci os -cs - i r oLl ccc: , - S o ‘ s-n ’s a c ’ , ‘.“ ,‘ - ‘c: : 1, 1 - ’’ ’.’ ‘. , s - ’s - , L o ” rd - O nc - . ‘,-.‘s-t th a t ’ s-s ’s, os -F - : yc c-s- t , :-tis-”

runs-s-i m g t I c - - ic-ci t a iarg’ : so e- . r i : t cs - cs l I cc cs- s- t r . If tw- aig a: : Is- a s- - c the

‘ncytso’o - - r’Li c f l ,o ccc ” t ics - c e - -. Sc Or es -’, the s - - c-ct ant- r ao s - t s - os - ’c c: ,as- g s--es--n’s the

cIalc-: b u c ,v-s - - ’ c c ‘ I . - - . ‘‘as-”. - cs-’.r- - ”c ’u.l ‘socaIic,’c l c 0 ’ s- 0_ c- I - rc’ , c l c - c ’ - c c c c c s - s t s - u r l’ ’s-’c ’. ’.’ s- c - ,

and S m ,:- -c s- f l ’: U-H on ’s - - u - c , cs-l~~o -n’ l th~n_c W,”s-_cl ,oj be a ‘, rss - o O ’cII c-ont r i S ’-u t i - - n S

t rs-i~s-t.i c ’s - i - - - ‘roncc ut ,I mng . lois-cL cs-,s - s - c i l , v s ’ t s’ f cc- so s-n cs-u s-c l - - s - i ’ of o r I -I ts -c - cc ’ s - c s - c ’ s- n’ ,

in s-Iccr oct , i’s- ’ so ic - s - c ob _ c ( i~~uth [ 1 ’ . - ’ - , 1’” ’ ’ . . l ’c - c’ t l ) . A recent _co:-c as-:rl le - -I 11.1:

so-s - -se arch is- W’ i’ c’s i cy cr- a’s: [117’:], ‘w4, , ch - c - ,1 ccJ a, c . l ccs ’ s ie -c- : ’ : n ; i ‘ n O c ’ s - l o s - o f ’

c r . l - r l t y c- c s - - c-:: :‘_ c c d , ‘ji ’ , ’’ s’ts t b ’ s - S ~0 c - i I c , ’ o o ’ clocj, S ’ ’ . - - r , : r , , : ’ . : c S tn t .i cc , i _ ’

S ‘c - s t in no , . I , -o - i r , c s - oc ”.ss-t,an c - - c s - .

L w- ‘i r i s - so ow€cr I’ . - cs-a ds-.

Oci s’ ’ . c-i cc isi S 5 . 1 ’ ’ . i s- Irs , on’s ‘ o n - - c - i t t b ’.. I ’ . ‘,irn - , i c ’.rv S - - on -c c- s - c , s - n ’ . - ‘i. ato ’. ’ - ’ s-n’ , s -c i n c ’ s so- c ‘ S O i l . I

r - , u s-t i ’s- is kit o~s-’o, -~n - ‘ c i on’c , - ‘ . 0 _c r’ -o ’ : , S S O , exiH Soi ~s- go- ‘ o;

‘II,’- - r l t , lr- c ’ ’s - r ’ ’ iffi’j ‘ ‘ - , ‘ ± : - . Fsi c- ’ , r ’ ’ 5.1 . u g~ c s -s’t s ’ c o ’ s - - so ’ s-i S ‘ s-n ;S o s- . 1. c ,i r s-g - 0 , . n - o s .

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J I :-:r’-s- s- -,‘ i’ s-ui ’ I em tri’ns-I’, is -co c - ’ . cccrn l ut cd it . less’ i- s i ’ ~ o O (rs- It s-g O s - ) U s - ’’?

Can c’ ,s-L’c 1 : c , cscc , o r e - l w ‘t’i-s id-on : be c ’ ,-a c - s --. i on ‘i — s - s n ’ s - i s - - u s - ‘- ( a s-) 5, 1 - : , - ’ 1 o c s - — i i c - e - c s - o ’

I on- c’ s-s -ur n-is -.c i d c’t s- f’,cr ccl ”.’ ss- ‘ ‘ cw o cs-’us - I ’, - , ‘s - n ’ .’? : as 5 . - s ’ s - i n s -c (iIs s- ’.s t ? s - [ 1 - 3 ]

l i o n - i l . :c ’.s- 5 0 , - c ’s-” :s- ‘s aths ( c o - n , A s- i s . acn o h ive s- I [laI d J ) , d ’ o , I I s ’ .’ 5~~t J’~ i - 00

I t s - r i --.r c [ i ’ ’ ’~~] 1 , c-s - s - n -s-i ev - s-It s-cas -,i s-n _c ” sov’conss - ’s- t - r I c ~u,o c : s - I c - i s - ( , .It s ’ o ” .::ec-s- I u-r 1’ i ]

1 :-n n’s- , . 0 ros-~ - ‘rtic-s-s- -s-I d o s s- s-a c~tructuc’s--c- c arid Basic S-Ic -cs - Is - c -d c ’.’.

I ’S ’ . s- so- c-I s-is - ’ ‘i ’ t -s -’c }ir ; is-s-s-_ s and s-15 -c r - c l l:b.’ccs - - ‘s c l, l c - c ’, s .in i 01-s-c’s - i ‘ c-nr c, - ~. afl, .i 5

‘ t o ’ s - s i _ c’ , ‘s. ;-s- s- t i , c ,, t’ - s - s - i :’ c ’ ’ - : , t - ’ d w I th a j r - c- - ”- . Sc s- - ic ’s_ cs ‘ins - s - c

‘ .~s s . ’s -. r s - n ” s - a s- ’. I s-icc s - - i t t _ c - - c f - i’ “ c ’ -’ Ic’ s-d s , ’s -os - a ‘‘ - c a l c u l u s : c i ’ t o i t ’ t c r n’ s - c ’ - - : ’’

bc s - S I ’ S : -n c- - ‘ s-, s - i r s - c c - s e s-s- c - ’ - cc: : s - r I - n t , -, i nc-os - c - u s- so’. :cni t at i s -’n c “-s - c - S tc c’S ’ .oo ’ c s - n : .-

‘ n’ s-c gi’.’’.s-oc o r i - -s-os,? I’,’b,s-st cs-,cs -s - ,-c s - .’s _ cs - s - :s - iac -.a ss - s - s-”s-n ’ . ’ ’ ’ ,nn ’ o’.’. I s-- s- I - -so S o - s-c : s-to ’s tIer

O r a c-’ s - n - ’ ’ s - , o c on: ; I oca ti.ooi ’? Tb ,’. cs-n win n- -’ :.’u,’.t: s-cs- ’:; ‘s- it Y s - _ c s - s-On , s-s- s- ’ , ’,’ . 0 ~~

S I r s- so1,’ I c “ s o b u s s - s- - c c . ‘Th i s -is - s - S o - c o . ’ :s- ’.s- , ,i ’ ., c , ,s- s - ro s s - ch cn ,l ’t s - c s -n g’ l n g

o - lu -s-s c ”s-s -o’ ’lc-g c-”s - s -~ s-a’ - ’? ’ se s - ’c in s-.1 g -n’ l ths-’lc s- O , 0t o ’. ’s lec-s ,i

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- s - os - i l o s - s - c s -n: ] l ’ . c ’ . - l s - ’ . I ) .

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‘ s - s - os -i ci ce - I i l -u ’ - o , ” :. ‘s- i t - i r s - c r the ~s--iges s-s-cu o s - ’ : - n” .: 5 a lr . ’ ’, . on ’

s-I ’ d i i d i c ’s c s -S ‘s- - ec- ’t i cs - -e s - (the 011- - cs- In i s n I n - - - , - S . - ’- i l . i’ s- is ’ .,: e ’ ’s ac : ’ ,s - ’ - - ‘s-on n - - n - s - s o - c

c’ s s -n’ s - S I : t i c ’ , s- ’t v- c - l i ce , . ca se- so-- . r ’ - - c - ,- m ; t - ,coi - s-c ( v. ’s- , \ ( ‘ s i n e gi”a ci :1:

s-unii n’ , c c t o - ,c, ’ . If ( - , - , w S is an s - s - n c ’- ’ , ‘,‘ c’._c’, ’.cc w :c,so-.s- cit : e n , d : , l c s S : o-_c’s-,c

cs-s-”. ‘ s-’.t, ’ s-r c-. .c 0 0 0 . lb s- - ‘coge (a ’ . w~ j ’ . - c s - s - s . f r ’ a’ t - ’. ~ . ( s - f ~torJis - ’uct ’.

4 h e ”-o ’mg ’ ’ coa: s- ’ l ’ ’.’s-i .’ ‘ S ’ s ’ s - : on ‘. s . , A r; s - ’ o s - : is- i ’~~~~( l ’ , : ~~) : a c c n i gn’ - s- ’ .

of IT if ~s- ”~~~~V and ?c. ’ C c E . I ’ ,i : cs s- s - r rn l o”.g if \“ = V . A gra ~h

c - IT ’ = (‘ - , ‘ ‘ .ii ’ ) i_ c ocr c. c. c ’ - - c c s- c - ,’.,1’ ‘ 0 ’ -1 i i ’ thor ’s- Is or coo ’s- s c i r c g fr -un :. ‘,‘

cc ’. V :co’ S’s f ? o c , I ’ ( : - : . c - ’ 1 m E ’ I: ’ on_ cn O ”,cl ’,’ I c ’ c-c = f(a- ) and 1’ =

s’s-s-- sos-cs-c (s-’ , ’ n ’ ) ’ . o E . -‘I s-c . n ;-s -l ’l ’ - s - s o - c ’. :~ ’ _ c - s - a ? ’ c l _ c i f ’ the ’- o - c : I o ’ . g i:

- c c c ’ -10- 0 ‘ . A gc ’ c - ; -h I’ = (-t ’ .. ii - is or gus - .- s-c - I H ed cs - nc s-Is-”~j~’s s -f 0 I c ’

iso a sun s- 3m -c ’s - i c of a has-s- - c’s - - r s i c ’i c i ’ ’ -~” ’ s-i’ nIT

A . s - c’ s - c ’ s - s- 1 ‘ cc a’ in C I_ c~ a c ’. ” c - s - c ’ . ’ o c c - .H’ cs - . ig-s- .c (‘,‘ .a- 2 (s-’ . . s - ’ 1 .

Thi s- ~cc t-h ,.s- c’ s-i_c t . ,‘- , n , ’ os- , l ; . ‘ - s e ’s - (s-’ . . v 2 ) ~“c — i ’ ” ’

r ’s~ ar, ,i -, - c - - c j s-e~-

V s- V • ‘c s - s-c O . ‘ c- .v c c l d c’s-li. oc- tI’n eso - ‘ -ig ’: 5 s - cs - i - S ‘se-i’s - I - s - e s - . ‘i’}s- ’: ‘ ‘.0 ? , is . ioc s - c i.e

j O V~ ‘a”. -ici s t- ’° r’.-’I - c c - c s- -c- i t s c c i l ’ ’Ly v1 ‘mn.I v , c i , - ‘ ° I s a

co’ s- i_ cs I : ’ v i = “or • Ti n ’s- c- 1” o .c ’ . : s - ’,’,: ‘.s - ’ I s - , ,’ c ,ss-” c 0 ’ (0 - ‘ - .id t i s - . s- ‘.“i” c’, i O

0 0 ‘(0 = (V. 1- , sos - c m 00 ,_ c c-. o ’, , W ) ’ .’~~~~ ‘ S l o e - n c - o s - h-: i f “.-~~~~ -~ c ’ . cc , i ’t , O , e re - ‘u s - a

-: s -c - t b :‘s-’ ’s- V - on i n ‘I

,l’.o , c _c ” s ’ . ll s-’ e - ’ . c ’ ’s -j g s o s - i ; i c ‘ no ‘. ‘ -n r n ’ - e t - - ’ s - ’ i i f ’ I - O U- on 5 - - ’ ? . ‘ n ’ - os-ny ’s- -i’s- u- c’: t

‘cc’ : : s - i s - - c r ‘ : n ” , - , . A d i r - a c t s - c l ,s-- ,u ) ’, S c s’ S rs - ’so;~ l~- s-’ c s - c s - , n I f S ? c c s - ’ c -’ i s c_c

O cs-H n t o ’ 00’ ‘Ui~ ’ v e-rt ’ ”- o c - s - c : - S s - c - -i ’ ‘,‘ - ‘ n ’ t - - s - . ~~s-e cc n- c- s - cc ’ ‘. ‘oonios - c - - l c - - i ( ccl ’s’. ngly

‘ c os o ’. ’ - c . ’t - - S . oo c - ”.g r ’n : ic : -s -I c g r c-0 h ~~e c s - o i l - _ c o 0 0 : s - o c o u ’t,ed s-

(sO c 0 , 1 01 ,, c - lu c - - , o m ccc i - n n c s - i t . , ) . A ~r a ;h i _ c s - , ” ., ‘ c ’ 1 0 ’ i t u - O i , 00 I-c o0 ” .~s - c In

h j - L - _ci ’ . (w i S I r ‘s- s- t i c- . -: cc- s - ‘ j o - iSO - ‘ s - n - S c cdc -~’:s ’ :s-~’ .in~ h e cu~~ec) s lb _ c t no

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~ ~~so ’_ c , ’ T I s - a co~~~ect e-i , u s - s - i . r ’ . ’ s - e - , I ; ‘s ’s-, I- Hg s-h cs- s- , 1 o s - I n s r-

IOc l T C l s - c _ c . In a ar c s- ,- t ue - re is a un iqu e :i ncri_:l e a d . b _ c t ’ s s - c s -s - i c - s-nay lai r a’s’

, L l st ’i u u - t a- - s - r I I c - s - s . A root ed, u n d i r ec t e d ‘s- r e-c (T . r ) i s a ‘s.n ~s - os - w it h a

c - i s - t i c ’ s i s - s i s l r -ud ‘s -s - rI s-s- s’s r called the s- . - ‘ . A r-sou-t’.cd, olir-uctes- tm_cur It: a

s-U s - - s - s - c t ’ s - s - i u - r c c - ; cc T with a uni que ‘s-- - r s - - s - x r cuci c s-bat

(i ) c -b . c -r e i s c- s - s-s-tb f ’ r- co r rs - on’ t - .’. ~~iy s-s- t b -_ cr one - so s-

(i t~ each v- s- r d ’s--s - s - s o - c - - s t so ?n ~s- ’c exactly ~ ne edge leading to it :

(iii) r h a _ c a - ’ iges leading t~.- i t .

ton1 ’ ro . ’it - s - . i . u s-n ’, s - i c-” - -: d - c , o t n ’ - s - -,s- ( T , r ) cs - s - cc be c us -ivcs -i~t s - co? i t s - I a n ’ s - s-.s- . ’s,

s - I so -s - c t -_ c-i I s - - -s- ~~ c- di’s” ’ ‘ o t r i g c_ c c - s - i c cig’.s- (v, w~ s-u tht s-t v is o” cs-t a ’S s-s-ed I n c

‘ t I c s - ‘ n ’ ’ r to a

‘ ‘n a c - ’ ’ , - s - . s - i s o - s - - a - c - i ‘s c- ’ - . a v - c ’-, - : , w I s a -de cs-cer i lia ,nct- of a s-’er t -_ c c-s

(on i s ~~ c- a c - c c - s - c t so O l ’ n’ ) ii ’ thc . i’ - ’ is a ~~at ic Os-s-s-cc, v I c ‘~‘ . A v- s -sot ’s-s-s

a’ i _ c a chi ld s-f v (‘: U Os -lone - os -cr’ s-n t ci’ a ’ ) if (v. ’ni’) is an es-ige in tI s-s- ’ .

Tb ’s- c-, ‘ - i s - .sI t ’Ini t i ’cc n s- s ‘ux’t’.s-crs -d t ’s- c ’ 0 ’ - . . 1, cr ’ s d,I roo t s- - t n I s - u - c-co ‘00,1’ ciirec ”.cIscg tie-c :

-,u-igs-’c _ c s-f s-sb- ” SO’.” ’- os-c rs-l , o ’: . 1111’ (0 sic a~~ r’s-Us-, a , c ascrc s - in’i:’ Is-’s--c s-f ’ IT i s - a

r o o t ’:d t r’. ’.: c’ñs -ci ch l c c _ c .’ ; a n-cs - s-Ics g co ,o ’s g ’s - o s - ob of ( 0 .

A i ’ s - n’ s - i t i’ ‘ O s -

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a - -H S i s- a cc c-llectio n -o f cubs -u - I-s - s- p ‘

. 1 ’ S cu e - ir s-oh - i l. U C ,, = o ; and fl I” , = c?s- j f ~~~~~~~ If ~‘ , 1 j’J

o cr ’ s -c - t i i n n s ’ s- f C , j I s - c a r e - o I c s -’. s - o : ,’s- o ’ . c of ~~~ ‘ (anI s-f ‘Is- a

c-s - ’s-s- ’: - c n i r c - g of ~ ) i f . f - s -n - all S~ c ~i , there is s- alone ,/ ‘, ~ ~c ’ s-u s - i s- I

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‘4. AcSic ’.’ . n s - c n a i c O ’ . i_s - ’ .’s -° j . “ , ‘nn. :o H i h b s - s - r s - sc:h cs - i s - c i t d o a u - n ’ . i’ re’.s-l-:oi ‘,OcIilo c c , ”

tI cs - S, °,. s - b c s - c . ‘ ‘ . 11 ~~ s-~~ s-:

A. V . — - . . ‘. o f ’ ~~- c s - ’ s t . and 1. 5 - . ST1 I,o o na r n [l , r7’ S; 1 . The De s- igro cro c i Analy s i s

s- u ’ C io c ’ .s - i , O U es - ’ AI s-s- n I - i n s - o n : , Addi - ‘ co —I- ,’ - . : s-c s- ’ , i~s-’.auI c og , 0 - Sacs .

A. - . di ’.- - ‘_c rs - I T. 0. 1 0 ’ s - . a [1 s- 0 ’ I ’ J . “ A c’cini oc sus-c: .i.cl .’t cs-n o ,’ ,s- s-n’ i~ ‘ i’— s-c ’s- s- ’r : .s- s- ‘ tO n ’

~ :-a ’:’:I’ s ’ -l’ u- - n o t ’ : . - , ’ — s - n a ’ s - c - ‘ s- o s-a r s- ” . . “ ‘.V-l 2 . t ’ o s-onj ut . 1 , 3 1)5 -312 .

V . s- s -c s- - an ’s- i C. S. U1],nan [1 1)1. “ lb s - n - 111 , - i l o g : I ’or n- o s - n - .;s- ’ I : ’ ie ‘ s I t s - a

gr c -cr~’h :. - ‘ c s - ’ . - 0 . C”v- ’-nt i ’ . ’ .-0s-rs- r,u’ s- ’. A ’ S “ :s- ’ , ’: . s - n c Theory c ’ -‘ -c :s c uti s-r g, 177-i’5.

F’ . s - c . d1’s o ’’n 1 -- - J 0 ~~ - , - ‘ . s - o t i ’ ‘1 ! . o ”c- ’w’ Os - s - s - c ’ . ~:~

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Ic ’ . c ’ . ,-‘s- l , IL ’ c , s-cc s- -d ’T. C’ - ‘ Is-s-c- . “ A 1’. s-c’ c’ ’ ’.s’ , cj s-s-_ctc s-’s- fl,sion ’ -_s,a,. , I s : r ‘ ‘ c c ” .

Cl . 2 ’ S 1. ,- . 113- ‘ -i~~’..

s- I. A: ‘cr1 rs- s- s--,i , , . i j s - s - ks -oo : c [1 7~’ . ‘ ‘ H ’s- c ’: 1_ cs - s- s -cs - ’ c’ s- ’: i s - n s-u’ s-n I c c ’ ’ - ’

I T s - b l o c H . I - l a t i n . , to s - s o s - c s - s o .

Ac -s- :bc un- i ’s --so - c _ _ c s - i . 1. V. I s-ri’ t - i ’ [i - - a l . ‘c -I On ios-l ’.c ? i i c c g , g r s - s - c ? r s i t ’ s I i ’ s -’ ~ i -ac , ”

‘0 Is- c’ :. 0 ._ c l - i - S - oh. 1: , 51 1 5_ I l C I .

1’.. c-’ . os -a- is- i ’s us’c cc s- , c I S . s-’

. ‘ “ u - s - I [1 c : 5 j ~ “ hegul ,cr s-- -‘cia’ s-s- I c r - c- a~~’1ied t cs-

j at l c - s -’i n s - c h ’ s n s - so i t - c , , ” 2. Tc ’s.. S-lath s-. Aj.-~-iUs . 15, lo It -I ’ - .

IT . c s - h ’s- ’ . - O i l , an _ c , , . O_ cl’ c-vc iy [1’ c 0 ’5 1. “Relati’s-c i . : c s-I , i ’,’c -i s - c c- i t ’ the I = --

- 1ue st - i c , , ’ 0l I1A~’1 - IIl ’-non~ c- s c . I , ,

- . ‘ . iIOcI . .s- :oo - uo [11 i5 ’T ] . i~ ’ t c ” .s-” , i -, 1 r oc- gsoa ’s: ’ss-s-i rig, Princet - - - o n T i c - s I s-’ s- ’s- ’:i l y IT n’ - : :

F r l c , c- - ’ , - ’. : ‘ . . S c .

1- - - n : - - r [ ,L°~5 - ’ . 1 . “ - ‘sr i t b ’s- I , , i ‘oU- -gy s - f ’ the gem- t ic f’ I nc’. : l c - ’ : c - - ’ s - s c - ” . - , ”

I r e-c. I, o~t-. Ac-ad. 51ci. (0. 5 .A . l s - 5 , 1’ 0 -i-

1 . 5’ - ci ’ s- ”’ j i ‘ s- I J . “(0~ io - S - l i - c -r ’ . ‘s-s in [lr’c.i In theos-”, . - , 5 i -c - • hat • Ic -: ‘ c o o . nod .

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t n ’ s -e i sa . - . - ‘?s-s - c 1 - 1 1 - - ‘ , 1-hcw 7 - o s - k .

• ~? 1s-’srs-I ‘ s -c :’, , 55 . J. ~- ‘j, “ s- s- s- es-’ . and A. 5 - ,. Meyer [l’s-’.’( ’ . ] . ‘ A nota on the

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l ’s- o I L s - c oon: . ” - c c : ’ . i- lu - -u - c . li - lI -S E Ic s - - I c s-’_c ss - r ’s- .noj , ‘ n s - :: . s-cc -n ‘s- Us - s- s o ’

s- I ’ s- - c - o s - c l - - u - f l u - l u - s - s ‘ ‘ . 57— ’ It .

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E. :1 ’ s - u - r o o t and . ‘. s - . AUs-oouc [ 1C C ] . “ ‘ u- c- n’s log n Ug -nith c :. s - o r

J u - t e c’ ct ,:i s- c c - ; ~ ‘.,cI . i s - c ’ b blc_ ’ C o o ’ s - I c - c , ” I roc. - t i n i’_cmual Pr is -n cs - cI onn Cs - - s - cs -’. s-n ‘ l o s - f

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c , s - t , w c ’ -s - ’i s - c , ” s - . c-Its-l b- S . 5 . , 1—13 .

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l’s- . I-I. Ear s- [ IT , ‘s- [ ’A ] . “Iclc - ’o-ducb.iciiity s-s-’ss-s-s-ng cornnb ol nn cs - i. -r . l al s r -Ui u - c - , :, ” s- s-s-c-cs-i 1’s-c-s .1 on:

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11cc-s-us-cc I n ’’: : , flew ‘ ri ’s-, o-5— l -It .

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Conf. R ecoi r d , Ad -I 15’s-s-ni . o s- is - I r b o c u - i . pies of Pr s - -g rc.cn o ”.m,i n : s - s - i : ’ . 2 , s - l s -n ’ o~s-1s-’ . • -

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K. I’ - . 1 c c - n ’ s- i s - s - n , [ 1 ° - c . “ - I - - c s - n ’ o c - L J c u -’d s - 1u - .’ ’ ’ . ’,s- ,rS:s-”:s- ’I I ‘ . “ s - U : : s - iI ’ - l o j , ’ I i - - - .o LO t - ’: .

II. s-- . Tar ,~ ou-s- ( 1 ’ . 1c-’ .. , ’ ’ O ” :O O c . c-,c ‘ , ‘ c u - I n i n c - -s- s-’c n ’ ‘ - i C s-_ c - - — s - s - s - t i s - ,s-s- c-

c - - s - I s - s b ’ s- P s- h i s s- c - s - I n ’ : ‘ ‘ s - _ , ’’ 5 s - .~ ‘ . . ‘ ‘oi .S n ‘.‘s- s- s -o n ’ s - ’ s-,I As-s- S b s - s - c - s - - c. ‘ s - i ’ s - Is-s- c r y . I’

- o ., ’. c c ‘u- 0 - ‘ c i ’ .

‘7- ‘c- ’ . s_ c ’ s - : [1’ - ‘ ‘0 ] - ‘ o s-u’. ‘ ‘ A s -n s- s-’ .’

c e s -’ , [s - u - s - I o - ’ o c ,1

s- I u- - - :, ‘ ‘ ‘ - n ’ s - , .

‘Ic-s - n’s - Os - c -’ [1 ” .’ — “ i . “ “ cc - - ‘ r’o’c s - s - I ”c-L , c- ’ - o , c c j s s - n ’ c - n ’ , , w ,1 s - i , o s - s “ . ; n l h ’ o c - t . i , s - , L os - s- i n ’ ’

11, 0 : - i s - - 0 5 _ c s - s - I ,’ n ’ i i ’ - ” C’ n s - s- -~ 1 :-s - c - s - i - c s - b b s - c - lI :. , , o ci , c ,~,, s- - u - Ic - sc “ .

5; c- - u - s - s - - ct 1 ‘ o - , J s - c - S s . i s- s- ( 1 - 0 1) ,

c-IL. I . ° ‘lS 1~’:o’ s-o . [ 1’s-~~ - c - s - . “ A s ’ c c - .:t “ ..L c - ’ - r ’ I l i , c’o s-Cr ‘ i c - el,t ’ - o . i s - c - s- t I n , t c ‘or. s - .

s- n _ c - s - x s - c - ’- . , , ‘ c , . ,“ ACI ’c- l o s - S ’ s -n ’s - s - -c - i i c _ c A , 1, - i — , I. ’ .

‘5 . J s -~ ! fl ILn ’cs-c-s-’c ‘ I -’ A l - i s - ] . s - - i - c s - co . ’s-_ i e c - ’c-o’css - is- - ’. . - . - ,~u - l c ’ s - t s - i i s - , c -~ s s - ’ I . - I , ’ - c’ :’ , ’’ r ’ ’ c .

s- s-UI’ ‘. o . ’,,’s-’c-j . - c s - 0 ‘~~~ c -s - f o o t , i o c-~ Cy: I ‘ - “ I n’oi c c - s - o s - s ‘I- - , , - — - I

1.. 7. V s - i l l - -s-c -, : [1’.” ‘c c s - . J . ‘‘ :;,‘r n - - roni - ‘ - c . ’ - ’ :-: ’ — s - ’ o ” ’ -, ’ s- - c - , - s - s o c ‘ -s - i i n s- ‘ , , ,‘

- ‘s-i s - l u - ‘ I c - ’ . ’ C . C s-ns - c -~~ut c - ’r -in n , ‘ I’.’ , 5 c - .’ .m , . ” I c - - n i c c - - : i I s - , “

‘: 3. - s -s - c- [ ‘ I c- c - S I ] . ‘ s - c ‘U s - s - c - c, — S I c o - _ c r - s - s - s -- c - - c - i c- u -u_ ms- n , I ’ . - ‘ cs-s-o~ s- s- S ‘ c- c I ‘ ‘ c-_ i

c- s-c ~n os - 1 - x I ‘ ~, . “ 1 rca’ • c - I s - v c - ’ c . I l is -c-’ - 0 ‘ s - . I ‘c- ‘‘ ‘ “s-s - O s - . i s - , Ti n- : - cs-,- os - I ’ c ’ s- c- s-’s-I n s - t j__o s- i .

s- . . 0 5 • “.c- _cJ I ‘_ c , l [ 3 c- ’ . . “ ‘or” - ‘o s - n j s-’c °, c - n,rr .s- s- ’ - S ot t 0 c C 5 - ‘ .‘s- ns - - ‘r I o n -

o -mu i c - - s - oil s-, ’ I c - ’ ‘ s - . i n i - I s - , ” us - c f cs - I 1 .5 s : l s - c - ’s - I s- c s - o s - c s - i , - i ’ S C , - ‘ - O , s- n’ s - - c - u - C c- s - c s- c~t ’ ’ s-’

l l t u o S u -s - , I Is- s i s- c - c r , , t c-_ c c - s - i ’ I _ c - - s -c - s - n , ’ .

~~~~~~~~~~~~~~~~ ~~~~~~~~ .~~~~~~~ .~~~~~~ -

- - ‘ ~~~~ ‘ ‘~~~~~~~~~

. 0 ’ s- s - l ’ s - s - c - 0 5 , I 5 . ‘ ‘ s - s - c c - ,’ . cc - i o n s - , . . ‘,oi , ’l , ’ b 5 r o s - Hc, .’s~~k l . ‘ ‘c-

c - cs - ’ c _ c , - I s - I ’ i s - s - s - i - I s - ’ ~~ c - c - i t , 5 ,s- ‘ c - s- c- ‘-Ic c - I lls - s - s - c.- ’ I c c - s i : nlbu -’ s - .t s-” :. [u - . -

c- n O ‘s- c - i l - ‘ o . S o ’ s- [ I’., . ‘‘ A s-s-A ss- d O ’ s - ’, - ’ : cs-i’ . s-’ n ’ c sc-n C _c_i _c, c-_ , .ts- ” s- s-

- u s - s c - cc ,’ ’ , . ’ C . ’ o c - o -, . . , ’ ’ ’ , t s -.5) ‘ s - I c-s--o I l ’ .

c- c-, ‘ c s - c ’ 0 ’ ‘ ‘- 0 5 flc - c-’ s - s - ’.’ s - is - s -i c-s- _c - ’ n s ’ c . ’ c - i c o o . s - O c -c- _’ is-l u- s - l o b s - c s - s .’’ c - ’ . c Is - c - Ls- ’cs-

s- ’ ’ : . n - ~‘ c--n’ ’, s - - i s - I n c - s - I s _ c s - s - n s - Is - i s- s - O s - n c ’ s - os-n 0 0 - - ‘ O s - , — I_ i .

• l-1,l, c - c - s i ’ s - - c I ILs-s-I’t , . “‘ c - S n - c - s - s - - s - s t - : 5 5 . c-i s - c l ’s-lt d -a s - ’ c . c n : ’. ‘ 0 - s’ - r. the o , c .n” - . o ‘s- s--

on ’ co o n . t I C I c - - n O i s-s-cs- :.” 1 ” - ‘ . .A s-c ’ s - c -.’oc. i . Anc-r:u:c-.i Cyc-ccs- . s-c - - ‘ow ss - fo c - t I n , c ‘ ‘A

s - s - f - s -i’ C. ’ - ‘ 0 . - s - c - c , ~~~~

[I c- ° ~~~.

-- s - s s- - c 0 s- s -t .lon ’ s-~” ” .i c c - c - .tI ,’ , . ’ s- ” s - t c - ’ s - - ,-u r .1 _ cu - i n ’ - s ’s. ’ - ‘ .

“ ‘ s o ’s- s-. .

I1 ss-t . s - c - u - ’, i . s- . . , . . ~~~, 1,YJ -I , )Os-

s - - i c’ s - S c - [i s- l i . ‘ c - I S o s- s - ’ - ’ - c , ’ s- ’~~ c s - s - s- .s-os -i s-t1Us-,s-d’.o I c - _ c . :: . . ’ Au - ’,. s-cc o f l iA , :. ” ”t ,, Csn - ‘

A. ‘ ‘. ‘s ’ s-s- s -c - [i - c -I _ c c- . “ c- ’ s - . .. ’ . ( ‘~~ i s - C .t C s- c-

~ s-s-id~ ’ s -’ i s - s - _ c l i o ,r ‘ i s - - u - s- s - c -’ cco oio n .I :55 ,_cc’s

c c - ‘u - _ c 5 . I s - lc- ’ L u - ’ ’,:. .“ I s - c A~~. Sr -s- s . 7, - t I - c -or : ~~~. 21—23 .

C’ . ‘ ‘s-s- , [ S l o t ’ . ‘‘ ‘n o c-s - i s - ‘ s-’:c- s- - s - cc- c - s - L c - - ln us - ’ .l 1 - - I ’ s - c t c ’ ’.r C ioc C _ c j ” ’ ’ l ~~~t O , , ,

i r . .’. cs-o . l olcl g l i t ’S c’s-s- u - c - s d, c - s - A l l 1’ D’ics-l . s - - s - n c - 5 c - 3 n ’ s- s- i ”y oct Cs-c -c s -o ut s- n d , ~~ ‘ _ i . ‘S.

-‘c - . C . ‘n ’s - c - , Ill . “. A v i s - . ‘ c - s - c s - I i . L . l i i v ’s - , t [l s- ’ .t ] . s - c ,

I(s-s-~ , ,

‘. s- s- s-s~

I - c,’ ’ i . s - c - u s - n cs - ’ ., ‘ 1° - nb ’ . ’, s - - il l s - c n c _ c : . ’ . . ’ I c_ cu - . 7I s- s th i4J’_csj ’ _cL [C’’

i ’ s- - s-~ , ‘ ‘ c . ’ . , oc , ,; ‘ 0 5 I n s - c - ; , to c s - ’~ s- - ‘ s- I’ .

° . ‘{cO) c- c - s - . , s- ’ ’ O ’ (.L c- ‘c-’ l . -‘ - ‘ ‘ -‘,n - . 5 I . i - -rn ‘ c - s - s - _ I ‘ ‘..r : in g s- ’, 1 ’ s- ,o o t , ’ . ’S ’ o — ‘ l’s- c-s- ‘l, c i s - s - L’1”. , 0 0 ’ . s- c- s - , ’

s- A ’ . I , i s- ~_iu-c-

_ ,d,_,s-,..s-s-,..s-’s- .,~~~~~~~~s-~~~t ”_cO1 ~,s-,s-.,,,,,, ,s-1 s - , ~~~~~~~ ,,_s- . . - ~~~~~~~

“J

.‘ i ,,,_ , - - -.‘,,_ ,,, ‘

_ _ _ _— ~~~~~~ - .~~~~~~~~~~~~~~~~~~ - - -~~~~~~~~~~~~~~~~~~~~~ - -- s- - -~~~~~~~~~~

-- -~~~~~~~~~~~~~~~~

~~~~~~~~E1~~~~~~~~

tY

.~~~~~~~~

.:°°

500

sec sec sec s-cc see- sec

.0 1 .3 C” 1. ’ - 4 . 5 10i s -C u- f l s - og n i ‘

sec sec s - u - s - sec-,, s-ec s- c-s-Us-

iou n . 1-11 .25 .1 Is- 25 2,c- s - s - s sec sec sec r ec cs-os - fl

s, s-c-7 . 02 1 1,7 1 21 11 .7’s-u-c’ s-cc s-c - cc mm mm hr

00 . 5 1. 1 220 125 5 . 1 1’ses- hr days- cent cent

. 3 - 1 - 1 .1 2. 7 3 . 1 . ’oss - s - c . c-s-s- hr cent

51 75sec yr cent

~ fl 58 2.1-3 ”ms-in cent

Table 3.1. Scinn ing Tics-s-c i ’ I :t imatc -s-s .

(One s - I s - i ’ s = one rn1croc :econs-~ logarithoc.: are base s-a ’

L. ~~~~~~~~~~~~~~~~~~~ ~~~~~~~

•

.‘.~~~~~.

c- c- D

1.1” ,:c-: u- 1 s- c o ’ s-c- i i s c - c 1’.c- ccc’ l ’ s- s- es--

cs- “ oo ’:I ’s -x c’ct y ~ s- u-u- ’ ( S . ‘ s - c o i n t (7 . ’; i s - s - ’ 1 (17 s - I ’ s- c- ’: 1 (c I,’ c-: ’ c - O ’ s -~ (i cc - c s - n t )

1300 n o 1’ 1 c- ’ 1 i s - - - i o~~~

‘2 3 c-c ‘7 s- ‘s - I .s - c 1os - -~~ s - , ] . ) - s - ç i C ’ [. ‘s - I l 0’ s- c - d c -

c- ~~ s- “, .~~~\ 1’ s- ‘° . i , x l , s - ’ 515 .’ s - i ’

io2 i c c -~ ~ ,,

- ° ‘ i_c is-.- ’

I . s - c 5- C . 1~~~ 1 i-j ~ 1 .~ s- 1(s- ’ c- 2.i~~], ’ ” 1c~

.7- -~~~ s-n 22 3- 51’. 7 : 11,7 15-

c c - c-~~

‘[ 0 ) c - c - li’s- ]v , c c -

i s - I’ 2(T 3c -5 -;° 1

12 1-~ 20 25 ‘7’s- c -

Table 3. . b- ” s-s-’: .’i ’ s- o_c o. h i s s - c - c - of a s-° I - c -ILvah i ’,’ c rc - s - s -h i-c ’ ,.

(A factor of l~s-n I s - n o n’ s- - a s - c in ‘o , n cc i l , i n ’ :o ‘-s - u - n j C sr i ’ s - - c s - s - _ Os - c -is- t

s- ’ s - c - - : t , -r of I cL I s - s - c -’ i ’ s- s- u- in t i r i ’ . : s -.

70

L ~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

c. s - c- s - f - s - :0 0 , _ n ‘ n _ c - - s ( i - s - s - i t t s- ’r :’no ’. ar i a ,-: - s- J ’ . ..n 1 I r s - S_ cd s - - u - , , - s . - ’L i , I s - .

I J r i s - c - r d s - r c - : s - s - s-ou - ’t s-

- . r c-i -c- r u -A s - c ’s : .

s - i . -I1ra~’h:.

u - . Tr s- c-~ s .

2. s - i c- s-con ’: I s - c , .

a P i ‘-~~ is-- ‘ sn ’ s -

a. Cc - s -c u — ‘ ‘ . 1

s-s . s- 0, : n s - n b - i , — ‘ In - ct .

I . hiI ns - ’c - ’s - ‘s- - C — ‘ 1 r ,os-

ii . L u-al c-sc- 0) ’.s-31’Os- 0 u-n -

I,~ l’ s- t . i . o cc i ooa ’ t i. o s- - s-’s cs-c c- .~t5s - ‘ - ‘i:

c c . c s - n ’ s- -A .

C . As-ny ’ :ceu s-t :c -t ’ cs-n .

5. I’s-s -c - b ’s- c s - i n c - b u s - c -” - n is-n o l ’ s in ’ - .

a. s - s - s - t I c - C :s- . s- r c - : s, c ’~~

s- c . I s-s- r t s - t i _ n I’ s-C s - c s - ecs - o -.s - o s - t .

0s - . I s - b’s- - ca r s -c s- ’r as - od c-’oc , u - n nb

I

I ~c-~po~ ’s~ on ( ~ i

~c . c- ’i~r l nkcI c’sp ( u-- s-’,c- C.~~s-s-7 I s - i s --sac ’ c o i n s - s - c - : ) .

Tab le .1. T- s - c ’hm s-’ ques- s- r s-I~c-.s-d As-h g - n t hoc.:.

s-( 1

~~~~~~~~~~~~~~~~~~ ~~~~~~~ ,. s-.

~~~~~~~~~~ , ,.s- .. _ , - ‘ ‘

value link 1 link 2

(a ’S

V s - J U c - link 1 link 2i7T1i~i ~‘

-

_ _ _ _ _ _

‘

~~~~

L _ _

-

~~~~~~~[~1~L~11 -17i~I 3 I

(b) (c~

rs-igw,” .- ,.1. A l,, s - c - } ,c - _ ct st ~~~c- cI u r ’ - an ’ I i t : rej r ’ c : e n s - _ c ’s - t i - c c n by arrays.

( a) l’I c - c oo ’ u - - i 0 ’ u-c’ - s - I

(b) Linked st r i s -c ’l ou-r c-

( c ) Repre r entat ci - s - n by three arrays.

_ _ _ _ — -

F!,-, ~~~~~ -‘ _

~~-

~~~~~~~~~s-’

~,s- --

~~~~s-- -- —.—-.“ - ‘ ..— . -,--

~~~~-s-

~~~~’ s- --,, -s---—-—-,’ “-‘- - -•-•

s -s - I s- is -’

i I Sc - s - s - s - s - I 7

7 - + ‘s - c s - , l l = P

11 s - s -~

5

c-_c

t :c - 151

(1

s - s c- i. , ~~~~~~~~~ i - ’ s - r . s- cC, ’ c c ’ ’ . ’s - ’s- s- ’ ”j , i c - ” s - s - s - c - : ’s- , ‘ a u , :i is- ’sc ’, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ 1,

( s -c- S ,‘~‘n’a5,’ n’ ‘ c u -u s - c -c s - i t s - c -’.. s n .

( i s - _ c . s- s - n c -- , . c - ,~ s- ’ ‘ s - u- ‘ , c- s - n c ‘ c- f s

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ s - .~~~~~

“s-’. - ~‘ .“ ‘~~~~~‘“ ~~~~~ -‘—- ~~~~~~ ‘ --

~~~~~~~~~~~~~~ ~~~~~~

head 3

2

1

5

ta,il 5 0

l” ,I giir-s- 5.3. Re~ r u s s - - s - n t o s - t i n’s of list 3 , 2 , 1, 5 , c- . , 11 icy s - I - sc - s-ib15,s-

linked s-sctructure .

714

- -~~~~~~~ ‘ - -‘ ,: ,-

~~~~,•

~~~~~~~~~~~~~ -

- - ‘

_

[~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ]

(a) (ss- )

head

i ._ _4”iI . J ~~~~~ 3 J O J

2 ‘— -{~I ’ J 4 1 1. 1 1_~P1

~~ —~Fi~1 )( ~~~ i

• 1 ~~~ ~i ~i

11 ~~~ *21.1 ~ ‘ j 0 J

( c )

Figure lc- • 14 • R e p r - ’ :” c s - s - a t ci c”- s -’c- . c - 1 ’ a gra ~ch .

(a) h u - a s i s - .

(b) c-’c-dj ac ency cs - , - , t r i x .

(c ) Adj c~ 1’”. s-s- cc Ic- ’ 5’ t u-us-. ’ t ur s-s.

L~ .

.

.s-

’

..-. s-As-i ,, . ‘0,,. ,~~~~~’ .~~ ~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

~,,, ,,

.. ,, —

. ‘_ ~~~~~~~~~

parent

1 0

14~~~~~~~~~

T 9 1 s -

(

h1

(a) (b )

Figure 11 .5. Representat i on ‘ s - i ’ a tree.

(a) Tr ee.

(b) Pare nt ar ray for root 1.

76

~

~~~~~~~~~~~~~~~~~~~~~~~~ .~~~~~~ ,. ‘—s-

c- ‘ ,, .. _ ____

.-,-‘~~~~~~~~~~-‘ -~~~~~~~~~-“

“.‘.- -~~~~

A~~~, B ( c - )

3/ D

(

- A(5)

G ~~~~~

~~~ - C ( S ) J )

~~F / ‘ -

H I

\

\ I /

/

F (l

(a) (b )

Figure Ic- j - . Depth— fir s-’c - s - e s - s -jr c s - s _c - b ’ s-os -s s-n ’i si, u - - ” .’S c - ’ .c c- ’ s - ’ 5 s -~ s~.

(a) Graph.

(b ) spanning t r c - c c - s - g c - s - n i c - s - ri t ’ s - I s - ,~ 3~ - ‘ ‘

Vertices nocnbered a s - s- u -s- ; S

77

~~~ s- s-,.I , ‘~~~~~~-‘~~-c---. ~~~~~~~.

~~~~~~~~~~~~ _ _ _ _ _ _ _ _ _ - ‘

—~~ “‘ ‘ ‘ ‘ - “r

s-i C

(a)

I i ( ’ )

B~5~ ~7- s - i

I \

E ( I )

D( 1)

(b)

i” s - R c -.ns-” 11.7. 5 c - ’: i ls - — t ’ is - ’:i . s- ec s-s -’ ’ ’ i i of’ a s- .S c - ’ . r c - ’ c t e d s-s- ra I l .n .

(a) - b r _ c u I n .

( t o ) ‘i s-o s - s -o n ng t , r u - c --o Cs- - c oos - ra te s -I bj s - e o c - r s - i n .

1 s - s - e s f ln n’:,lcered as explored.

- .s-’ .. L—’,,s-. ,,,,. _~ —- ~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~ -—~~~~~ . .,.,.. , - 1,~ “ ~~~~~~~~~~~~~ ~~~~~~ __ . ~~~~~~~~~~~~

‘~~~~~~~~~~~~~ ‘ s - ” J ‘ ~~~~~~~~~~~~~~~~~~~~~~~~ ‘ ‘ -

~~~

A (2) B (2) H(2 ) T (2)

/\\ /G ( l ) D ( l ) / \~~~~~

/

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1~~~~~~~~~~~I( l ) I ( 1)

F ( O )

(a)

H ( 3 )

__-,~-— -*~

‘ I If E ( 2 )

G ( 2 ) ‘ E — — — ~~~~~~~~~

— --- —— — ~~s-

__~~~~~~~~~ F( 1S )

C ( 1)

//

D ( O ) —

(b)

Figure 11. . Brea dth- fi r’ .: ’t search. Level i nd i ca t ed in j arenthe r es.

(a) Search on ’ gro ss- in s - n Figure . . o

( b ) Search of grap h in Figure 1k ;;.

79

-~~~~~~~~ -~—~~~~ ‘ , s - s-,,~~~.. , r c -~~ . i— . -_- ~~‘

—— -‘ -,.—-. — ‘~“ ‘ “ ‘ ‘ ‘‘“ . “—

~~ — — - ‘ “ “ “ .“ “ ““'“'

~~~~~~“

~~“ —

1. D isc re te Fourier t ranc 1’ - c - i~~ (DFT):

n’ s-s-cur: .s- s-cs-i S

2. Matrix multi ili ca t ion (MM) :

u-es-cus-’sion.‘1 . Linear equations cfl a i’clan ar graph (LEG):

recur:’.I ccrn , decomposition by -c .’ .o s - s - nectivity , breadth- thu-at :e.-sr ’.s - : . .

S . c- ’s-lob al. flow s-ns-cals-j . 1 . : ( GFA) :

s-le coccsl -as-itt, o:c-n by s- ’ - ’ r rnec t7 ”c-dt-y , lat h cos-s-cr r e:s- :7-s-n , A s-s-i t h - s-,’lr :t c s -s - s c - s - -c - s - i c - .

5. t at ’.bc c-crn mat ching on :t r is-’ng:o ( i l - b ) :

data structur es.

St rong c onoot s-anent s ( S C ’ S :

‘lu -i - t i n - fi r s t cuss - s ’s - I s - .

i ianarity test ing (P T ) :

c-ju-c c . - t l i — s -.’s t rc ’t cs - s - cc -s-u-c-s-i n .

cs - . b-is-ed,~~~n - network 5 . 1’s- s-c (5- ’7~~~) :

auguentat i on , c r - s- a s-I t S’s-first s- ‘s- c_crcts-.

‘ c- . Gra s- I, “itching (CS-b

au~~~entation , breadth-first c’s-a rch , cycle shrink Ing .

10. Set union (SU) :

i ath cc cnuip r ’oc s- ..I ‘u - .

~L 1e 5 . 1 . Ten Tr ac ’taLl&c- ir ’:bl’.sscc’.: and 5-bcs -’thc’Js’ ’, ’ cr s- S olv i ng ‘CI ’ sc - s- nr c -.

— ~~~~~~~~~~~~~~~~~~ _s - . - .,,__~~~ ’—.c . . ‘ .s-,,. i__ s-

~~~~~~~~~~~~~~~~~~~~~~~~ ‘~~~-

— ‘ ‘ , “‘ “~“ I ‘

c - u - s - - s - s -ou r s -t i al

‘h n5

C-ru-on

0~n ”’-

-Ic-

1/~~ c 1 g n ±

1 1 , 1

1

n

is-~ °~ LEG GFA 11- ’ DC I T l-15t7 s- ic - ”

“ i gss-r ’c 5. 1. Recent complexfty improvements.

n = s .i sos - - (number of vert ices in graph lcr cc -b i o c- n s - n o c ) .

m = u - c - c c - s - - c - u s - c l par ameter (number of edges in gr c t 5cc in r . c- bic -s--s- n is - ) .

—~~~~ s- ’~~ .,,s- ~~~~~~~~~~~~~~~~ ~~., ~~~~~~~~~~~~ ~~~~~~~~~~~~ s-s-~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~

- _s- - s--,-~~~

— - ~~~~~~~~~~

- TT~__i~~~~ ~~~~~~~~~~~~~~~~ ~~1~~~~~~__

N -

I. ~~~~~~~~~~ ~~~~~~ — — —. -

. — —

L ~~~~~~~~~ ~~~_. — — - - .

~~—

/- -- —

_ _ _ _ ~~~ ‘ ‘ Jr

~‘1I s - c -’ ’ 5 . . , s - c -.~ c - c s - s o , o ’ . : 1 ” ’

‘1 ‘ ‘1 ’ sc - ‘ n ” c- f e C u - c - I ~~~~~~ 0’ -s - il’ , ‘i s - i , .

0. -c-

L ~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~ . ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ •

. ~~ s- . . s-~~~~~~~~~~~~~~~s- . ,~~~~~~~~~ , , ~~~~~~~~~~~~~~~~~~

-- - - ~~~~~~~~ “ ~~~~~ ‘- ‘ -~~~ - “ —~~~~~~~~

(a) (b)

Figure 5 . 3 . Ku r a ton ws-Gci :.i Ls-5r a, sc’. ’ .

(a) K5.

(b ) K~~~ .

s- s-7

-

~

--

~

s-- - —- ~~~~~~~~~~~~~~~ - - . ,- ---.- _ -s-~~~~~~~~~~- --- ~~~~~~~~~~~~~~~~~~~~~~ L. , , ~~~~~~~~ — .

s-t

o - A( t - ’)

Figure 5.1, . i- s - c - s - i r- . sets-’ s-i s- I c - c , c- ’ Set A f - i , :c , C . 0 , C. 5 .

i s - i ‘1 5 n ’ - - - . s-Si:: -, ’ of ’ ‘c c-s- c,, S . c-i s-is-’s-s - n I tS iu ’o- c: .

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— .—.± . ,,

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uni an ( A , B ) :

~~~~~~~~~~~~~~

A (x±y(a)

if x < y

A ( x+y)

( b )

Fi gure 5.5. Irs -c -n i, c-c ’s ’ sc - o”n b s - , a t j s -r n - s - f ’ acG c - c s .

(a ) Ba: i,’s ‘s - , ’ t

(b) Weight ing heuristic .

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C

7

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