I Dinh ly 5.4.2 Khi P^ NP, moi thuat todn xap xi thdi gian da thiic giai bai todn M A X C L Q U E deu co hi^u suat tuyet doi bang 0 0
5.5 Phan tich xac suat cac thuat toan
Phan tieh thuat toan dg thay dUdc do phiie tap cua thuat toan la mot doi hoi chinh dang khi xay dung thuat toan va la mot phan quan trong cua ly thuygt do phiic tap t i n h toan noi chung. Trong
phan tich trvldng hdp xdu nhdt (worst-case analysis) doi v6i
thuat toan, mot hinh thiic phan tich thong thudng, t a khao sat mOt each t i m i qua t r i n h t i n h toan cua thuat toan trgn m5i du: kien bat ky thuoc tap tat ea cac dfl- kien bai toan vdi cting mot "kich cd" rigng
biet va xac dinh do philc tap cua thuat toan, cu thg la dp phUc
tap theo tnCdng hdp xdu nhdt {worst-case complexity), bang
each dua ra nhuing danh gia mang t i n h ly thuyet doi vdi licang toi
da cac tai nguyen (nhu thdi gian va bo nhd) mk thuat toan can
sijf dung trong moi qua t r i n h tinh toan trgn tdng du: kien bai toan. Nhu vay, do phiic tap cua thuat toan giai bai toan cho trudc thg
hien dp phiic tap cua thuat toan tTnng tnidng hdp xdu nhdt
5.5 Phan tich xac sudt cac thuat toan 339
( m worst-case) va cung la dO phirc tap cua thuat toan tren m6i dH
Men hat ky cua bai toan, tiic do phiic tap (thdi gian va khong gian)
cua thuat toan trong moi tnidng hdp {in any case).
Tuy nhien, theo hinh thiic phan tich nay, dO phiic tap cua thuat toan chi bieu lO tuong doi chinh xac luong thdi gian (khong gian) ma
thuat toan c^n sii dung trong nhvcng trudng hap xdu nhdt, nhung
rat C O the Idn hon nhieu so vdi luang thdi gian (khong gian) ma
thuat toan thiic su can den trong da phan cac trudng hap khdc
con lai, va do do doi khi khong the hien duoc day du gia t r i sii
dung cua thuat toan. Khong i t thuat toan giai bai toan nao do dil r^ng c6 do phiic tap thdi gian ham mu, nhung lai to ra kha hieu qua khi dUdc ap dung dg giai cac bai toan thiic te, tiic thuat
toan ty chi can den thdi gian da thiic. Bdi vi, cac bai toan thiic te thudng chi la cac bai toan con va c6 thg khong chiia nhiing dH
kien xdu nhdt cua bai toan tdng quat da chọ T h i du, thuat toan
don hinh cua Dantzig {Dantzig's simplex algorithm) giai bai toan
quy hoach tuyen t i n h cho dii c6 do phiic tap thdi gian ham mu, nhung van duoc xem la thuat toan hieu qua nhat doi vdi cac bai toan thiic te, hieu qua hon nhieu so vdi cac thuat toan thdi gian da thiic da dUdc biet dgn, chang han nhu thuat toan ellipsoid cua
Khachiyan {Khachiyan's ellipsoidal algorithm) va thuat toan xa anh cua Karmarkar {Karmarkar's projective algorithm).
Dg bigu 15 ro hon gia t r i thiic sii ciaa thuat toan, ciing vdi viec nghign ciiu do phiic tap theo trudng hop xau nhat, mOt hinh thiic
phan tich khac duoc sii dung, cu thg la phan tich xac sudt
{probabilistic analysis), nhSm xac dinh dp phiic tap trung blnh
{average-case complexity) cua thuat toan. Do la do phiic tap duoc
tinh t r u n g binh cho moi dfl kien bai toan vdi cung mot "kich cd". Do phiic tap t r u n g binh c6 thg duoc tinh theo xac suat, tiic la tinh trung binh theo mot phan bo xac suat nao do tren tap tat ca cac d i i kien vdi cung "kich cd", va thong thudng nhat la t i n h t r u n g binh
340 Cac giai phap
n^o do CO do phurc tap ham mu (trong trudng hop xau nhat) va lai CO do phlic tap trung binh da thiic, thi rat c6 thg thuat toan
ky CO do phlic tap da thiic tren phin Idn cac dii kien ciia bai toan hay tren "hau het" cac dii kien. Thuat toan don hmh cho bai toan quy hoach tuyen tinh duoc chiing minh la c6 do phiic tap th5i gian trung binh da thiic, vi the no to ra kha hieu qua doi vdi cac bai toan thuc te nhu tren da neụ
Nhu vay, nghien ciiu do phiic tap th5i gian trung binh cua thuat toan la viec lam thiet thuc. Noi rieng, thay vi khong hy vong c6 duoc thuat toan thdi gian da thiic doi vdi bat ky bki toan NP-day du, viec tim kiem thuat toan vdi do phiic tap thdi gian trung binh da thiic cho bai toan NP-day du nao do la mot giai phap diing dan
va CO y nghia nhat dinh v6 ling dung thuc tign.
Tudng tu, doi vdi c^c bai toan toi Uu NP-kho, hieu suat tuyet doi ciia thuat toan xap xi, vira duoc dinh nghIa trong Phan 5.4,
chu yeu the hien hieu suat cua thuat toan trong trUdng hdp xau nhat. Bdi vay, dg thay dudc hieu suat thuc su cua thuat toan xap
xi, ta CO thg diing khai niem "hieu suat hau chac chan", dUdc de
xuat trong [18] nham bigu lo hieu suat ciia thuat toan tren "hau het m6i du: kien bai toan". Viec danh gia hieu suat hau chac chan ciia thuat toan xap xi dUdc thuc hien bang phudng phap phan tich xac suat. Kgt qua nghign ciiu thu duoc chiing to r^ng, d6i vdi nhieu bai toan toi uu NP-kho ma moi thuat toan xap xi deu c6 hieu suat tuyet doi bang oo, nhung trong do c6 nhiing thuat toan kha don gian (nhu kigu thuat toan tham lam) lai c6 hieu suat hau chac chan rat nho, tham chi bang 1. Dieu nay c6 nghIa la, trgn hau het m6i dii kien bai toan, nhiing thuat toan ay cho ta nghiem rat gan vdi nghiem toi uu (xem [18, 19, 20]).
Bay gid ta chinh x^c hoa cac khai niem ngu trgn, chu yeu la gidi thieu each xac dinh "do phiic tap thdi gian trung binh", "do phlic tap thdi gian hau khSp noi" va "hieu suat hau chac chan" cua thuat toan.
5.5 Phan tich xac suat cac thuQ,t toan 341
Gia sijf n la mot bai toan vdi tap dii kien Dn (IT c6 t h i Ih mot
bai toan NP-day du va cung cd thg la mot bai todn toi uu NP-kho).
Ta ky hieu Dfj"^ la tap tat ca cac du: kien bai todn vdi "kích cd" n
(do dai bigu dign hoac tham so co ban cua dii kien, chang han nhu
so dinh cua do thi, neu IT la bai toan trgn do thi). 6 day ta gia
thiet Ang, doi vdi moi n > 1, tap Dfj"^ la hiiu han va \D^^^\o
khi n oọ Gia sii (9„ la bien ngtu nhien lay gia t r i trgn D{J"\i
do trgn tap D^^ cd mot phan bo xac suat duoc xac dinh bdi cac xac suat Pv[d^ = d], trong do Pr[(9„ = d] > 0 doi vdi moi d e DJJ"^
va E^g^(n)Pr[5„ = d] = 1. Phan bo ay la phan bS xac sudt deu,
khi Pr[5„ = d] = 1/1 DJ^^ I d6i vdi moi d e D^^l
Gia siJt la thuat toan giai bai toan IT. Ta nhic lai mot vhi ky
hieu lien quan: tA^id) \a thdi gian tinh toan ciia An trgn dii kien d;
khi n la bai toan toi uu va An la thuat toan xap xi, ta ky higu
OTPn{d) la gia t r i cua nghiem toi uu cua IT doi vdi d vk An{d) la
gia t r i cua nghiem ma Au tim duoc khi thi hanh trgn d.
Bay gid ta goi X„ = X(5„) la mot bien so khong am tren D^^^
lay gia t r i X{d), trong do X{d) la mot trong cdc gia t r i tAuid),
OTPn{d)vkAn{d), khi = d vdi xac suat la Pr[a„ = d]. Dai ludng
nglu nhign Xn cd ky vong (expectation) va phuang sai (variance):
E(X„) = J2 P'l^n = d] X{d), Var(X„) = E{Xl) - (E(X„))'.
Trong trudng hop khi X„-lay gia tri tAn{d), dai lUdng E{Xn) cho ta do phacc tap thdi gian trung binh (average time complexity) cua thuat toan An va duoc ky hieu la t^{n), tiic
tZ{n)= PAdn = d]tAn{d).
Neu cd ham /(n) sao cho t^{n) < /('")> thi ta noi r^ng thuat toan
342 Cac giai phap
thiic, ta noi r^ng thuat toan An c6 do phttc tap thdi gian trung
blnh da thiic {average polynomial time complexity).
SiJf dung cac dai luong E(X„) va Var(X„), khi X „ lay gia t r i
tAM, OTPu{d) va An{d), ta c6 thg danh gia tdng dai luong tuong
ling tAxiid), OTPu{d) va Au{d) tren hau hit mdi dvl kien bai
toan {almost every problem instance) dua vho mot vai bat dang
thiic quen thuoc cua ly thuyet xac suat, nhu bat dang thiJc Markov Pr[£ < E(0 sl> 1 - - , doi vdi moi s > 1;
'- s vh bat dang thiic Chebyshev
P r [ K - E ( 0 | > t J -< , d6i t^ v6i moi t > 0,
trong do ^ la dai luong ngtu nhien bat kỵ
Ve khai niem "hau het" noi chung, doi vdi mot tinh chat Q cho
triidc, ta noi r^ng "hau hit mdi dH kien cua bai toan U c6 tinh chat Q", hay "tinh chat Q dmc thoa man hdu khdp ndi" {almost
everywhere), n l u :
Pv[d e DJi"^ : d CO tinh chat Q] —> I, khi n —> oo;
trong triidng hop thong thirdng, khi tren D^^ c6 phan bo xdc suat
deu, t h i doi hoi
{d\de va d CO tinh chat Q]
D (n) 1, khi n —> oọ
Sau day, nhu nhiing t h i du minh hoa, ta ap dung phuang phap neu tren dg lam sang to hieu qua thuc su cua cac thuat toan giai mot so bai toan lien quan den bai toan CLIQUE quen thuoc. Cu thg,
danh gia hieu suat cua thuat toan tham lam tren h&u hit mdi du:
kien cua bai toan M A X - C L I Q U E va chilng to do phUc tap thdi gian
trung binh da thiic cua mot Idp cac bai toan con NP-day du cua bai
toan C L I Q U E . De don gian, ta gia thiet rang trgn tap cac du: kien
"kich ca" n cua bai toan c6 phan bo xdc suat deụ
5.5 Phan tich xdc suat cac thuat toan 343