randomizing quick sort

7/24/2019 Randomizing Quick Sort

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Analysis of AlgorithmsCS 477/677

Randomizing Quicksort

Instructor: Gorg !"is

(Appendix C.2 , Appendix C.3)

(Chapter 5, Chapter 7)


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#

Randomizing Quicksort

$ Randomly %rmut th lmnts of th in%utarray "for sorting

$ &R ''' modify th (AR)I)I&* %rocdur

+At ach st% of th algorithm , -chang lmnt

A[p],ith an lmnt chosn at random from A[pr]

+ )h %i.ot lmnt x = A[p]is ually likly to " anyon of th r p + 1lmnts of th su"array


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0

Randomizd Algorithms

$ *o in%ut can licit ,orst cas "ha.ior+ 1orst cas occurs only if , gt 2unlucky3 num"rs

from th random num"r gnrator

$ 1orst cas "coms lss likly+ Randomization can *&) liminat th ,orstcas "ut

it can mak it lss likly5


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4

Randomizd (AR)I)I&*

Alg.:RA*&I89(AR)I)I&*(A, p, r)

i RA*&(p, r)

-chang A[p] A[i]

return (AR)I)I&*(A, p, r)


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Randomizd Quicksort

Alg. :RA*&I89Q;IC


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6

=ormal 1orstCasAnalysis of Quicksort

$)>n? @ ,orstcas running tim)>n? @ ma- >)>? )>n?? >n?

B C / C n4B

$ ;s su"stitution mthod to sho, that th runningtim of Quicksort is &>n#?

$ Guss )>n? @ &>n#?

+ Induction goal: )>n? cn#

+ Induction hy%othsis: )>k? ck#for any kD n


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1orstCas Analysis of Quicksort

$(roof of induction goal:)>n? ma- >c# c>n?#? >n? @

B nB

@ c ma- ># >n?#? >n?

B nB$ )h -%rssion # >n?#achi.s a ma-imum o.r th

rang B nB at on of th nd%oints

ma- ># >n ?#? @ B# >n B?#@ n#+ #>n + B?B nB

)>n? cn#+ #c>n + B? >n?

cn#


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R.isit (artitioning

$ Foars %artition+ Slct a %i.ot lmnt xaround ,hich to %artition

+ Gro,s t,o rgions

A[pi] xx A[jr]

A[pi] x x A[jr]

i H


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Anothr 1ay to (AR)I)I&*>Jomutos %artition + %ag B46?

$ Gi.n an array AK %artition th

array into th follo,ing su"arrays:

+ A %i.ot lmnt x = A[q]

+Su"array A[p..q-1] such that ach lmnt of A[p..q-1]is smallrthan or ual to x>th %i.ot?

+ Su"array A[q+1..r]K such that ach lmnt of A[p..q+1]is strictly

gratr than x>th %i.ot?

$ )h %i.ot lmnt is not includd in any of th t,o

su"arrays

A[pi] x A[i+1j-1] > x

% i iB rHB

unkno,n

%i.ot

H


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BL

9-am%l

at th ndK

s,a% %i.ot


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BB

Anothr 1ay to (AR)I)I&* >contd?

Alg': (AR)I)I&*(A, p, r)

x A[r]

i p - 1

forj p to r - 1

do if A[ j ] x then i i + 1

-chang A[i]M A[j]

-chang A[i + 1]M A[r]

return i + 1

Chooss th last lmnt of th array as a %i.otGro,s a su"array N%''iO of lmnts -Gro,s a su"array NiB''HBO of lmnts P-Running )im: >n?K ,hr n@r%B

A[pi] x A[i+1j-1] > x

% i iB rHB

unkno,n

%i.ot

H


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B#

Randomizd Quicksort>using Jomutos %artition?

Alg. :RA*&I89Q;IC


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B0

Analysis of Randomizd Quicksort

Alg. :RA*&I89Q;IC


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B4

(AR)I)I&*

Alg': (AR)I)I&*(A, p, r)

x A[r]

i p - 1

forj p to r - 1

do if A[ j ] x then i i + 1

-chang A[i]M A[j]

-chang A[i + 1]M A[r]

return i + 1

&>B? constant

&>B? constant

of com%arisons: k"t,n th %i.ot andth othr lmnts

Amount of ,ork at call k: c k


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B

A.ragCas Analysis of Quicksort

$ Jt @ total number of comparisonsperformedin all calls to PAR!!"#$

$ )h total ,ork don o.r th entire-cution ofQuicksort is

&>nc?@&>n?

$ *d to stimat 9>?

k

k

X X=


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B6

R.i, of (ro"a"ilitis


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B7

R.i, of (ro"a"ilitis

>discrt cas?


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BE

Random aria"ls

Def.:(Discrete) random variable X: afunction from a sample space S to the real

numbers.

+ It associats a ral num"r ,ith ach %ossi"l outcomof an -%rimnt'

>H?


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B

Random aria"ls

9'g': )oss a coin thr tims

dfin @ 2num"rs of hads3

C ti ( " "iliti


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#L

Com%uting (ro"a"ilitis;sing Random aria"ls


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#B

9-%ctation

$ 9-%ctd .alu >-%ctationK man? of a discrtrandom .aria"l is:

2[3] = 4xx r53 = x6

+ 2A.rag3 o.r all %ossi"l .alus of random .aria"l


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##

9-am%ls

9-am%l: @ fac of on fair dic

9NO @ BB/6 #B/6 0B/6 4B/6 B/6 6B/6 @ 0'

9-am%l:


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Indicator Random aria"ls

$ Gi.n a sam%l s%ac S and an .ntAK , dfin th indicator

random variableITAU associatd ,ith A:

+ ITAU @ B if A occurs

L if A dos not occur

$ )h -%ctd .alu of an indicator random .aria"l 3A=/5A6 is:

E[XA] = Pr {A}

$ (roof:

9NAO @ 9NITAUO @ B (rTAU L (rTVU @ (rTAU


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#4

A.ragCas Analysis of Quicksort

$ Jt @ total number of comparisonsperformedin all calls to PAR!!"#$

$ )h total ,ork don o.r th entire-cution of

Quicksort is

&>n?

$ *d to stimat 9>?

k

k

X X=


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*otation

$ Rnam th lmnts of A as zBK z#K ' ' ' K znK ,ith

zi"ing th ith smallst lmnt

$ fin th st 8iH@ TziK ziABK ' ' ' K zHU th st of

lmnts "t,n ziand zHK inclusi.

BL6B40E 7#

zBz# z zE zz0 z4 z6 zBL z7

) t l * " f C i


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)otal *um"r of Com%arisonsin (AR)I)I&*

iB n

=Xi nB

=

1

1

n

i

+=

n

ij 1 ijX

$ )otal num"r of com%arisons %rformd "yth algorithm:

$ fin iH@ I Tziis com%ard to zH U

9 t d * " f ) t l


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#7

9-%ctd *um"r of )otalCom%arisons in (AR)I)I&*

$ Com%ut th expected %alue of &$

=][XE

"y linarityof -%ctation

th -%ctation of iHis ual

to th %ro"a"ility of th .nt2ziis com%ard to zH3

=

= +=

1

1 1

n

i

n

ij

ijXE [ ]=

= +=

1

1 1

n

i

n

ij

ijXE

= +=

=1

1 1

}Pr{n

i

n

ij

ji ztocomparedisz

indicatorrandom .aria"l

C i i (AR)I)I&*


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Com%arisons in (AR)I)I&* :&"sr.ation B

$ 9ach %air of lmnts is com%ard at most onceduring th ntir -cution of th algorithm

+ 9lmnts ar com%ard only to th %i.ot %oint5

+ (i.ot %oint is -cludd from futur calls to (AR)I)I&*

Com%arisons in (AR)I)I&*:


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Com%arisons in (AR)I)I&*:&"sr.ation #

$ &nly th %i.ot is com%ard ,ith lmnts in "oth%artitions5

zB

z#

z

zE

z

z0

z4

z6

zBL

z7

BL6B40E 7#

8BK6@ TBK #K 0K 4K K 6U 8EK@ TEK K BLUT7U%i.ot

9lmnts "t,n diffrnt %artitionsar n.r com%ard5


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Com%arisons in (AR)I)I&*

$ Cas B: %i.ot chosn such as: 8i < x < 8j

8iand 8j,ill n.r " com%ard

$ Cas #9 8i or8jis th %i.ot 8i and8j,ill " com%ard

+ only if on of thm is chosn as %i.ot "for any othrlmnt in rang 8

ito8

j

'6B40E 7#

zBz# z zE zz0 z4 z6 zBL z7

8BK6@ TBK #K 0K 4K K 6U 8EK@ TEK K BLUT7U

Pr{ }?i jz is compared to z


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0B

S ,hy

z#z4zB z0zz7zz6

z# ,ill n.r "com%ard ,ith z6sinc z >,hich"longs to Nz#K z6O?

,as chosn as a

%i.ot first 5


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0#

(ro"a"ility of com%aring zi,ith zH

@ B/> H i B? B/> H i B? @ #/> H i B?

ziis com%ard to zH

ziis th first %i.ot chosn from 8iH

@(rT U

(rT U

zH is th first %i.ot chosn from 8 iH(rT U

&R

$)hr arj i + 1lmnts "t,n 8i

and 8j

+(i.ot is chosn randomly and ind%ndntly

+)h %ro"a"ility that any %articular lmnt is th firston chosn is 1:( j - i + 1)


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*um"r of Com%arisons in (AR)I)I&*

1 1 1 1

1 1 1 1 1 1 1

2 2 2[ ] (lg )1 1

n n n n i n n n

i j i i k i k i

E X O nj i k k

= = + = = = = =

= = < = + +

)lg( nnO=9-%ctd running tim of Quicksort usingRA*&I89(AR)I)I&* is 0(n!n)

= +=

=1

1 1

}Pr{][n

i

n

ij

ji ztocomparediszXE

9-%ctd num"r of com%arisons in (AR)I)I&*:

>st k@Hi? >harmonic sris?

Altrnati. A.rag Cas


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Altrnati. A.ragCasAnalysis of Quicksort

$ S (ro"lm 7#K %ag B6

$ =ocus on th -%ctd running tim ofach indi.idual rcursi. call to QuicksortK

rathr than on th num"r of com%arisons%rformd'

$;s Foar %artition in our analysis'

Altrnati. A.rag Cas


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>i''K any lmnt has th sam %ro"a"ility to " chosn as %i.ot?

Altrnati. A.rag Cas


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06


Altrnati. A.rag Cas


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07


Altrnati. A.rag Cas


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Altrnati. A.rag Cas


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B:nB s%lits ha. #/n %ro"a"ility all othr s%lits ha. B/n %ro"a"ility

Altrnati. A.rag Cas


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4L


Altrnati. A.ragCas


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4B


rcurrnc for a.rag cas: Q>n? @


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4#

(ro"lm

$Considr th %ro"lm of dtrmining ,hthr anar"itrary sunc T-BK -#K '''K -nU of nnum"rscontains r%atd occurrncs of som num"r'Sho, that this can " don in W>nlgn? tim'

+ Sort th num"rs$ W>nlgn?

+ Scan th sortd sunc from lft to rightK chcking,hthr t,o succssi. lmnts ar th sam

$ W>n?+ )otal

$ W>nlgn?W>n?@W>nlgn?


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9- #'06 >%ag 07?

$ Can , us !inary Sarch to im%ro. InsrtionSort >i''Kfind th corrct location to insrt ANHOX?

Alg.:I*S9R)I&*S&R)(A)

forj ; to ndo e&Y A[ j ]

Insrt A[ j ]into th sortd sunc A[1 . . j -1]

i j - 1

hile i > and A[i] > e&do A[i + 1] A[i] i i 1

A[i + 1] e&


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9- #'06 >%ag 07?

$ Can , us "inary sarch to im%ro. InsrtionSort >i''Kfind th corrct location to insrt ANHOX?

+ )his ida can rduc th num"r of com%arisonsfrom &>n? to &>lgn?

+ *um"r of shifts stays th samK i''K &>n?

+ &.rallK tim stays th sam '''

+ 1orth,hil ida ,hn com%arisons ar -%nsi.

>'g'K com%ar strings?


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4

(ro"lm

$Analyz th com%l-ity of th follo,ing function:

*(i)

if i+

then return '

return (2*(i-'))

Rcurrnc: )>n?@)>nB?c

;s itration to sol. it '''' )>n?@W>n?


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(ro"lm

$ 1hat is th running tim of Quicksort ,hn allth lmnts ar th samX

+ ;sing Foar %artition"st cas

$ S%lit in half .ry tim$ )>n?@#)>n/#?n)>n?@W>nlgn?

+ ;sing Jomutos %artition ,orst cas

$ B:nB s%lits .ry tim$ )>n?@W>n#?

randomizing quick sort

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