jewrÐa upologismoÔ - intelligence: lab...

ΠΛΗ 401 – Θεωρια Υπολογισμου – 2006 – 15η Διαλεξη

JewrÐa UpologismoÔUpologistik Poluplokìthta

M. G. Lagoud�khc Τμημα ΗΜΜΥ, Πολυτεχνειο Κρητης SelÐda 1 apì 37


Epan�lhyh

• Prìblhma plakìstrwshc

� upologismìc me plakÐdia sto epÐpedo (tiling computation)

• Idiìthtec anadromik¸n glwss¸n

� Turing-aparijm simec gl¸ssec� lexikografik� Turing-aparijm simec gl¸ssec� to je¸rhma tou Rice



AnakefalaÐwsh• Kanonikèc gl¸ssec

� paragwg : kanonikèc ekfr�seic� anagn¸rish: peperasmèna autìmata

• Gl¸ssec qwrÐc sumfrazìmena� paragwg : grammatikèc qwrÐc sumfrazìmena� anagn¸rish: autìmata stoÐbac

• Anadromikèc gl¸ssec� anagn¸rish: mhqanèc apìfashc Turing

• Anadromikèc sunart seic� upologismìc: mhqanèc Turing grammatikèc

• Anadromik� aparijm simec gl¸ssec� paragwg : grammatikèc qwrÐc periorismoÔc� anagn¸rish: mhqanèc hmiapìfashc Turing



Probl mata

• KathgorÐec problhm�twn

� epilÔsima probl mata� mh epilÔsima probl mata

• EpilÔsima probl mata

� praktik� efiktoÐ algìrijmoi� mh praktik� efiktoÐ algìrijmoi

• Apìdosh algorÐjmwn

� upologistik poluplokìthta� posotikopoÐhsh thc jèshc twn Church kai Turing



Par�deigma: Mhqanèc Antigraf c



S mera• Upologistik poluplokìthta

� qronik poluplokìthta� rujmìc aÔxhshc� kl�seic isodunamÐac O

• H kl�sh P� poluwnumik poluplokìthta� orismìc thc kl�shc P� idiìthtec thc kl�shc P

• Probl mata� apìfashc� beltistopoÐhshc



Algìrijmoi kai Poluplokìthta• Algìrijmoc

� leptomer c kai saf c akoloujÐa stoiqeiwd¸n bhm�twn� dèqetai k�poia eÐsodo pou kwdikopoieÐ èna prìblhma� par�gei k�poio apotèlesma pou dÐnei lÔsh sto prìblhma

• Upologistik poluplokìthta� qronik poluplokìthta (time complexity) � taqÔthta� qwrik poluplokìthta (space complexity) � mn mh

• EktÐmhsh apìdoshc� peiramatik an�lush me upojetik� pragmatik� dedomèna� majhmatik an�lush gia ektÐmhsh qronik c poluplokìthtac� majhmatik an�lush gia ektÐmhsh qwrik c poluplokìthtac



Qronik Poluplokìthta

• Diaisjhtik�

� o arijmìc twn bhm�twn (basik¸n entol¸n) tou algorÐjmou� ekfr�zetai wc sun�rthsh tou megèjouc thc eisìdou� k�je basik entol ekteleÐtai se stajerì qrìno� basikèc entolèc: anajèseic, sugkrÐseic, pr�xeic, ...

• Majhmatik�

� sun�rthsh T : N 7→ N

� n, to mègejoc thc anapar�stashc thc eisìdou� T (n), o arijmìc twn bhm�twn tou algorÐjmou� sun jwc exet�zoume th qeirìterh perÐptwsh (worst-case)



Mègejoc thc Eisìdou

• Taxinìmhsh

� to m koc thc lÐstac twn stoiqeÐwn proc taxinìmhsh

• Pollaplasiasmìc duadik¸n arijm¸n

� to m koc thc duadik c touc anapar�stashc

• EÔresh monopatioÔ se gr�fo

� o arijmìc twn kìmbwn kai twn akm¸n tou gr�fou

• Suntaktik an�lush

� m koc sumboloseir�c eisìdou� mègejoc anapar�stashc grammatik c qwrÐc sumfrazìmena



Par�deigma• Algìrijmoc

� function Summation(sequence) returns integer

local sequence: array of integers, sum: integer

sum ← 0

for i ← 1 to Length(sequence) do

sum ← sum + sequence[i]

end

return sum

• Mègejoc eisìdou� n, to m koc thc akoloujÐac

• Qronik poluplokìthta� T (n) = 4n + 2




� function Find 13(sequence) returns integerlocal sequence: array of integers, pos: integerpos ← 1while sequence[pos] 6= 13 and pos ≤ Length(sequence) do

pos ← pos + 1end

if pos ≤ Length(sequence) then return pos else return 0


• Qronik poluplokìthta� T (n) = 6n + 4



Asumptwtik Sumperifor�• Sunart seic

� f(n) = 1.000.000 · n� g(n) = 10 · n3

� h(n) = 2n

• Poia sun�rthsh eÐnai megalÔterh?� mikr� n: h(n) < g(n) < f(n)

� meg�la n: f(n) < g(n) < h(n)

• Asumptwtik sumperifor� (n →∞)� kaj¸c n →∞, k�poioc ìroc wc proc n kuriarqeÐ� sthn T (n) = 3n2 + 5n + 2, o kurÐarqoc ìroc eÐnai o n2

� sthn T (n) = 3n + 1000n5 + 105, o kurÐarqoc ìroc eÐnai o 3n



O Sumbolismìc O(·) (Big-O Notation)

• T�xh (order) sun�rthshc� èstw sun�rthsh f : N 7→ N, tìte h t�xh O(f) thc f eÐnai:

O(f) ={

g : N 7→ N : ∃c > 0, d > 0,∀n ∈ N, g(n) ≤ cf(n) + d}

• Sqèsh isodunamÐac ³� f ³ g ⇐⇒ f ∈ O(g) kai g ∈ O(f)

� h ³ eÐnai sqèsh isodunamÐac� diamèrish twn sunart sewn f : N 7→ N se kl�seic isodunamÐac

• Rujmìc aÔxhshc (rate of growth) sun�rthshc� h kl�sh isodunamÐac miac sun�rthshc f wc proc th sqèsh ³� dhl¸netai wc O(·) me thn aploÔsterh sun�rthsh thc kl�shc



Poluwnumikìc Rujmìc AÔxhshc• f(n) = 31n2 + 17n + 3

� f ∈ O(n2), epeid f(n) ≤ 48n2 + 3

� n2 ∈ O(f(n)), epeid n2 ≤ f(n) + 0

� 31n2 + 17n + 3 ³ n2 =⇒ 31n2 + 17n + 3 ∈ O(n2)

• Polu¸numa� f(n) = adn

d + ad−1nd−1 + · · ·+ a1n + a0, me ai ≥ 0, ad > 0

� f ³ nd =⇒ f ∈ O(nd)

• Prìtash� Ta polu¸numa Ðdiou bajmoÔ èqoun ton Ðdio rujmì aÔxhshc.� An to polu¸numo g èqei megalÔtero bajmì apì to f , tìte o

rujmìc aÔxhshc tou g eÐnai megalÔteroc apì autìn tou f .



Ekjetikìc Rujmìc AÔxhshc

• f(n) = 3 · 2n + 31n2 + 3

� f 6∈ O(n2)

� f ∈ O(2n) kai 2n ∈ O(f(n))

� 3 · 2n + 31n2 + 3 ³ 2n =⇒ 3 · 2n + 31n2 + 3 ∈ O(2n)

• Prìtash

� Oi ekjetikèc sunart seic èqoun megalÔtero rujmì aÔxhshcapì k�je poluwnumik sun�rthsh.

• Parat rhsh

� oi nn, n!, 2n2, 22n èqoun akìma megalÔtero rujmì aÔxhshc




� function Summation(sequence) returns integer

local sequence: array of integers, sum: integer

sum ← 0

for i ← 1 to Length(sequence) do

sum ← sum + sequence[i]

end

return sum


• Qronik poluplokìthta� T (n) = 4n + 2 =⇒ T (n) ∈ O(n) � grammik




� function Find 13(sequence) returns integerlocal sequence: array of integers, pos: integerpos ← 1while sequence[pos] 6= 13 and pos ≤ Length(sequence) do

pos ← pos + 1end

if pos ≤ Length(sequence) then return pos else return 0


• Qronik poluplokìthta� T (n) = 6n + 4 =⇒ T (n) ∈ O(n) � grammik



H Kl�sh P



To Prìblhma tou Planìdiou Pwlht Travelling Salesman Problem – TSP

• Dedomèna� ènac pwlht c sthn pìlh c1

� èna sÔnolo {c1, . . . , cn} apì pìleic� ènac n× n pÐnakac d me fusikoÔc arijmoÔc

dij h apìstash metaxÔ pìlewn ci kai cj

• ZhtoÔmeno� to suntomìtero dromolìgio pou pern�ei ap' ìlec tic pìleic� amfimonos manth antistoiqÐa π : {1, 2, . . . , n} 7→ {1, 2, . . . , n}� πi eÐnai h i-ost pìlh tou dromologÐou� minπ c(π) = dπ1π2 + dπ2π3 + . . . + dπn−1πn + dπnπ1



An�lush TSP

• Algìrijmoc

� exètash k�je pijanoÔ dromologÐou kai eÔresh bèltistou

• An�lush

� pl joc dromologÐwn (n− 1) · (n− 2) . . . 3 · 2 · 1 = (n− 1)!

� gia k�je dromolìgio n pr�xeic (prosjèseic apost�sewn)� poluplokìthta O(n!)

• Sthn pr�xh

� n = 10, n! = 3628800, 1 deuterìlepto� n = 20, n! = 2432902008176640000, perÐpou 20000 qrìnia� n = 40, n! ≈ 2141, merik� disekatommÔria èth



PraktikoÐ UpologismoÐ

• PraktikoÐ algìrijmoi

� logarijmikoÐ, O(log n)

� grammikoÐ, O(n)

� poluwnumikoÐ, O(nd)

• Mh praktikoÐ algìrijmoi

� ekjetikoÐ, O(cn)

� dipl� ekjetikoÐ, O(dcn)

� paragontikoÐ, O(n!)

� �lloi, O(nn)



PraktikoÐ UpologismoÐ

• PraktikoÐ algìrijmoi

� logarijmikoÐ, O(log n)

� grammikoÐ, O(n)

� poluwnumikoÐ, O(nd) � prìtuph mhqan Turing

• Mh praktikoÐ algìrijmoi

� ekjetikoÐ, O(cn)

� dipl� ekjetikoÐ, O(dcn)

� paragontikoÐ, O(n!)

� �lloi, O(nn)



H Kl�sh P• Poluwnumik� fragmènh mhqan

� mia mhqan Turing M = (K, Σ, δ, s,H) eÐnai poluwnumik�

fragmènh, an up�rqei polu¸numo p(n) ¸ste na isqÔei:

gia k�je eÐsodo x, den up�rqei sunolik kat�stash C ¸ste

(s, .tx) `p(|x|)+1M C

� h mhqan termatÐzei p�nta met� apì p(|x|) to polÔ b mata

• Poluwnumik� apofasÐsimh gl¸ssa� apofasÐsimh apì mia poluwnumik� fragmènh mhqan Turing

• H kl�sh P� ìlec oi poluwnumik� apofasÐsimec gl¸ssec



Idiìthtec Kl�shc P• Posotik eklèptunsh jèshc Church-Turing

� praktikoÐ algìrijmoi � poluwnumik� fragmènec mhqanèc Turing

� realistik� epilÔsima probl mata � kl�sh P• Kl�sh P

� to posotikì an�logo twn anadromik¸n glwss¸n

• Je¸rhma

� H kl�sh P eÐnai kleist wc proc th sumpl rwsh.

• Apìdeixh

� antistrof y kai n thc M , to poluwnumikì fr�gma paramènei



Idiìthtec Kl�shc P• Je¸rhma

� Oi kanonikèc gl¸ssec an koun sthn kl�sh P .

• Apìdeixh� kanonik gl¸ssa L =⇒ peperasmèno autìmato M

� to M anagnwrÐzei mia sumboloseir� w se qrìno O(|w|)• Je¸rhma

� Oi gl¸ssec qwrÐc sumfrazìmena an koun sthn kl�sh P .

• Apìdeixh� gl¸ssa qwrÐc sumfr. L =⇒ grammatik qwrÐc sumfr. G

� o alg. dun. progr. anagnwrÐzei thn w se qrìno O(|w|3|G|)



Idiìthtec Kl�shc P• Je¸rhma

� H parak�tw anadromik gl¸ssa den an kei sthn kl�sh P :

E = {“M”“w” : h M dèqetai thn w met� apì 2|w| to polÔ b mata}

� to posotikì antÐstoiqo thc H (prìblhma termatismoÔ)

H = {“M”“w” : h mhqan Turing M termatÐzei me eÐsodo w}• Apìdeixh

� parìmoia me thn apìdeixh gia to prìblhma termatismoÔ� teqnik thc diagwniopoÐhshc



Kritik Kl�shc P• Erwt mata

� swst èkfrash thc ènnoiac tou praktikoÔ algorÐjmou?? eÐnai h kalÔterh apìpeira mèqri stigm c

� praktikìc ènac algìrijmoc me poluplokìthta n100 10100n2?? akraÐa fainìmena pou de sunant¸ntai sthn pr�xh

� mh praktikìc ènac algìrijmoc me poluplokìthta nlog log n?? akraÐa fainìmena pou de sunant¸ntai sthn pr�xh

� giatÐ ìqi poluplokìthta mèshc perÐptwshc?? èqei megalÔtera probl mata amfisb thshc



Probl mataApìfashc kai BeltistopoÐhshc



Probl mata Apìfashc (Decision Problems)

• Decision Problem for L

� 'Estw alf�bhto Σ, gl¸ssa L ⊆ Σ∗, sumboloseir� x ∈ Σ∗.An kei h x sthn L?

• Halting Problem

� 'Estw mhqan Turing M kai sumboloseir� eisìdou w.Apodèqetai h M thn w?

• Reachability

� 'Estw ènac kateujunìmenoc gr�foc G ⊆ V × V , ìpou V =

{v1, . . . , vn} eÐnai èna peperasmèno sÔnolo kìmbwn, kai vi, vj ∈V . Up�rqei monop�ti an�mesa stouc kìmbouc vi kai vj?



Anapar�stash Problhm�twn Apìfashc• Anapar�stash

� qr sh kat�llhlhc gl¸ssac wc proc k�poio alf�bhto

• Halting Problem 6∈ P� H = {“M”“w” : h mhqan M termatÐzei me eÐsodo w}

? “M” kwdikopoÐhsh mhqan c Turing M

? “w” kwdikopoÐhsh sumboloseir�c w

• Reachability ∈ P� R = {κ(G)b(i)b(j) : up�rqei monop�ti apì ton vi ston vj}

? κ(G) anapar�stash tou gr�fou G (pÐnakac geitnÐashc)? b(i) h duadik anapar�stash tou akèraiou i

? b(j) h duadik anapar�stash tou akèraiou j



Gr�foi Euler

• Eulerian Cycle

� 'Estw gr�foc G. EÐnai o G Euler? Up�rqei ston G kleistìmonop�ti pou pern�ei apì k�je akm akrib¸c mia for�?

� o gr�foc (a) eÐnai Euler, o gr�foc (b) den eÐnai



Gr�foi Euler

• Je¸rhma

� 'Enac gr�foc G ⊆ V × V eÐnai gr�foc Euler ann:? gia k�je zeÔgoc koruf¸n u, v ∈ V , oi opoÐec den eÐnai

apomonwmènec, up�rqei monop�ti apì th u sth v

? se k�je koruf o arijmìc twn eiserqomènwn akm¸n isoÔ-tai me ton arijmì twn exerqomènwn akm¸n

• Eulerian Cycle ∈ P� oi parap�nw idiìthtec elègqontai se poluwnumikì qrìno



Gr�foi Hamilton

• Hamiltonian Cycle 6∈? P� 'Estw gr�foc G. Einai o G Hamilton? Up�rqei kÔkloc ston

G pou pern�ei ap' ìlec tic korufèc akrib¸c mÐa for�?

� oi gr�foi (a) kai (b) eÐnai Hamilton

� �gnwsto an Hamilton Cycle ∈ P ìqi



Probl mata BeltistopoÐhshc• BeltistopoÐhsh (optimization)

� elaqistopoÐhsh megistopoÐhsh upì k�poiouc periorismoÔc� p.q. to prìblhma tou planìdiou pwlht (TSP)

• Metatrop � prìblhma beltistopoÐhshc =⇒ prìblhma apìfashc� par�metroc B: �nw k�tw fr�gma sth sun�rthsh tim c� apìfash: up�rqei lÔsh me tim ≤ B (min) ≥ B (max)?

• Je¸rhma� An k�poio probl ma apìfashc den mporeÐ na lujeÐ se poluw-

numikì qrìno, tìte oÔte to antÐstoiqo prìblhma beltistopoÐ-hshc mporeÐ!



BeltistopoÐhsh wc Apìfash• TSP 6∈? P

� 'Estw akèraioc n ≥ 2, n × n pÐnakac apost�sewn dij, kaiakèraioc B ≥ 0. Up�rqei antimet�jesh π : {1, 2, . . . , n} 7→{1, 2, . . . , n} ¸ste c(π) = dπ1π2 + dπ2π3 + . . . + dπnπ1 ≤ B?

• Independent Set 6∈? P� 'Estw mh kateujunìmenoc gr�foc G ⊆ V × V kai akèraioc

K ≥ 2. Up�rqei C ⊆ V me |C | ≥ K, ¸ste gia ìlec tic koru-fèc vi, vj ∈ C, den up�rqei akm metaxÔ vi kai vj?

• Clique 6∈? P� 'Estw mh kateujunìmenoc gr�foc G ⊆ V × V kai akèraioc

K ≥ 2. Up�rqei C ⊆ V me |C | ≥ K, ¸ste gia ìlec tickorufèc vi, vj ∈ C, up�rqei akm metaxÔ vi kai vj?



BeltistopoÐhsh wc Apìfash• Vertex Cover 6∈? P

� 'Estw mh kateujunìmenoc gr�foc G ⊆ V × V kai akèraiocK ≥ 2. Up�rqei C ⊆ V me |C | ≤ K, ¸ste to C na kalÔpteiìlec tic akmèc tou G?



Melèth

• SÔggramma

� Harry R. Lewis kai QrÐstoc Q. PapadhmhtrÐou,StoiqeÐa JewrÐac UpologismoÔ, Ekdìseic Kritik , 2005.

� Enìthtec 1.6, 6.1�6.2


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