1 Minimum Cut Problem
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1 CS 6 Lctur 6 Min Cut and argr s Algorithm Scribs: Png Hui How (05), Virginia Dat: May 4, 06 Minimum Cut Problm Today, w introduc th minimum cut problm. This problm has many motivations, on of which coms from imag sgmntation. Imagin that w hav an imag mad up of pixls w want to sgrgat th imag into two dissimilar portions. If w think of th pixls as nods in th graph and add in dgs btwn similar pixls, th min cut will corrspond to a partition of th pixls whr th two parts ar most dissimilar. Lt us start with th dfinition of a cut. A cut S of a graph G = (V, E) is a propr subst of V (S V, S, S V ). Th siz of a cut w.r.t. S is th numbr of dgs btwn S and th rst of th graph (V \ S). In th xampl blow, whr S is th st of black nods and V \ S is th st of whit nods, th siz of th cut is. Figur : Cut siz = (Imag Sourc: Wikipdia) Th minimum cut problm (abbrviatd as min cut ), is dfind as followd: Input: Undirctd graph G = (V, E) Output: A minimum cut S that is a partition of th nods in G into S and V \ S that minimizs th numbr of dgs running across th partition. Intuitivly, w want to dstroy th smallst numbr of dgs possibl. To find th min-cut, a trivial solution is to numrat ovr all O( n ) substs. Howvr, this is way too tim consuming. Thus w prsnt a mor fficint algorithm, as follows. argr s Algorithm A not on randomizd algorithms: argr s Algorithm is a randomizd algorithm. It is diffrnt from th randomizd algorithms that w hav sn bfor. Th randomizd algorithms w v sn so far (such as Quicksort and th HW qustion to find colinar points) hav good runtims in xpctation, but may occasionally run significantly longr than this (and may not vn b guarantd to trminat!). Nvrthlss, ths algorithms provid a guarant that th solution producd upon trmination is always corrct. Algorithms with th proprtis abov ar known as Las Vgas algorithms. This is not th cas for argr s Algorithm. argr s Algorithm is a randomizd algorithm whos runtim is dtrministic; that is, on vry run, th tim to xcut will b boundd by a fixd tim function in th siz of th input (i.. a worst-cas runtim bound), but th algorithm may rturn a wrong answr with a small probability. Such an algorithm is calld a Mont Carlo algorithm.
2 . Finding a Min-Cut Lik som of th othr graph algorithms w v sn bfor, argr s Algorithm will us th notions of suprnods and suprdgs. A suprnod is a group of nods. A suprdg conncting two suprnods X and Y consists of all dgs btwn a pair of nods, on from X and on from Y. Initially, all nods will start as thir own suprnod and vry suprdg just contains a singl dg. Th intuition bhind argr s Algorithm is to pick any dg at random (among all dgs containd in suprdgs), mrg its ndpoints, and rpat th procss until thr ar only two suprnods lft. Ths suprnods dfin th cut. Algorithm : Intuitivargr(G) whil thr ar mor than suprnods: do Pick an dg (u, v) E(G) uniformly at random; Mrg u and v; Output dgs btwn rmaining two suprnods Th goal of this lctur will b to show that this simpl algorithm can b mad to work with good probability. In what follows, w will us th following notation: W will rfr to th nods within a suprnod u as V (u), and th st of dgs running btwn two suprnods u, v as E uv (this is th suprdg btwn u and v). Hr is how w initializ th algorithm: Algorithm : Initializ(G) Γ ; // th st of suprnods F ; // th st of sts of dgs forach v V do v nw suprnod; V ( v) {v}; Γ Γ { v}; forach (u, v) E do E uv {(u, v)}; F F {(u, v)}; and hr is how w mrg two suprnods: Algorithm 3: Mrg(a, b, Γ) //Γ is th st of suprnods with a, b P x nw suprnod ; V (x) V (a) V (b); //mrg th vrtics of a and b forach d Γ\{a, b} // O(n) itrations do E xd E ad E bd ; //O() opration using linkd lists Γ (Γ\{a, b}) {x}; Now, w can prsnt th argr s algorithm in full dtail. Algorithm 4: argr(g) Initializ(G); //Γ is th st of suprnods, F is th st of suprdgs whil Γ > do (u, v) uniform random dg from F ; Mrg(ū, v, Γ); //u ū, v v F F \Eū v; Rturn on of th suprnods in Γ and E xy ; //Γ = {x, y}
3 Th following xampl illustrats on possibl xcution of argr s Algorithm. a b d c a b d c a b c d a b c d Th xampl abov happns to giv th corrct minimum cut, but only bcaus w carfully pickd th dgs to contract. Thr ar many othr choics of dgs to contract, so it s possibl w could hav ndd with a cut with > dgs. Th runtim of th algorithm is O(n ) sinc ach mrg opration taks O(n) tim (going through at most O(n) dgs and vrtics), and thr ar n mrgs until thr ar suprnods lft. 3 Analysis W prov that w obtain th corrct answr with sufficintly high probability undr uniformly random slction of dgs. Claim. Th probability that argr s algorithm rturns a min-cut is at last ( n ). Proof. Fix a particular min-cut S. If argr s algorithm picks any dg across this cut to do a mrg on, thn S will not b output. Howvr, if all dgs that th algorithm slcts ar not across th cut, thn S will b output. P (argr outputs S ) =P ( st dg not across S ) P ( nd dg not across S st dg not across S ) P ((n ) th dg not across S st, nd,..., (n 3) th dgs all not across S ) W say that an dg is good (w.r.t. S ) if it is not across th cut S (V \S ). Not that P ( st dg is good) = m n n n whr is th numbr of dgs across th min-cut S, and m is th total numbr of dgs in th graph. Not that if th min-cut siz is, thn m n. This is bcaus any nod v in th graph is a cut of siz dg(v), and so for all v, dg(v). Thus, m = v dg(v)/ n/. W now show that in th multigraph, aftr th first j dgs ar mrgd, if non of ths dgs ar across S, thn th min-cut siz is still : Call th multigraph aftr th first j dgs ar mrgd, G j. Evry cut in G j is a valid cut in G, so th min cut of G j has valu at last that of S. If th first j dgs ar not across S, thn for any suprnod x in G j, th dgs in V (x) must b on th sam sid of th min cut S (ithr all in S or all in V \S ). Bcaus of this, S is a valid cut in G j as wll, and th siz of th min cut of G j is th sam as th min cut siz of G. Considr j, thn dfin P j as th probability that th jth dg is good, givn that th first j 3
4 dgs ar also good: P j = P (j th dg good st, nd,..., (j ) th dgs all good) = numbr of dgs in multigraph numbr of suprnods Th first quality is a dfinition. Th scond quality coms from th fact that th first j dgs ar good, so that th min cut is still a valid cut in G j, so that th probability that th jth dg is good is th probability that non of th min cut dgs ar pickd. Th third quality holds bcaus (just as in our arlir argumnt) vry suprnod rprsnts a cut, so that vry suprnod must hav dgr, and hnc th numbr of dgs in th multigraph is at last th numbr of suprnods /. Evry call to th Mrg procdur dcrass th numbr of suprnods by, thus th numbr of supr nods aftr j mrgs is n j +. Thn, for all j, P j (n j + ) = n j + n j + = n j n j + Thus, w v provd that P j n j n j+. But what w rally car about is P P P n. Using th xprssion of P j calculatd, w hav P (argr outputs S ) = P P P n n n n 3 n n 4 n = n(n ) = ( n ). W obtaind th quantity abov sinc all numrators xcpt for cancl and all dnominators xcpt for n(n ) cancl. A succss probability of /Θ(n ) might sm vry small. Howvr, w ll s that w can boost this probability to an arbitrarily larg probability, by prforming rpatd indpndnt trials of argr s algorithm. Lt C > 0 b an arbitrarily larg constant. Algorithm 5: Amplifidargr(G) //C is a constant ( ) n Run C ln n indpndnt argr procdurs, kp track of min cut so far, output th bst on. Th runtim of Amplifidargr is clarly O(n 4 log n) sinc w run O(n log n) trials of an O(n ) tim algorithm. Claim. P (Amplifidargr is corrct) n C whr C is th constant usd in th amplification algorithm. Rmark (Usful fact). For x > 0, ( x) x. 4
5 Proof. P (Amplifidargr is incorrct) = P (argr is incorrct for all th C = (P (argr is incorrct)) ( n ) C n n = ( n ) ln n C C ln n n ln n ( ) n ln n indpndnt runs) ( ) C ln n = n C. Rmark (Gnral Way Of Boosting Th Succss Rat of Mont Carlo Algorithms). If an algorithm is corrct with probability (w.p.) P, w can run it c (/P ) ln n tims and output th bst rsult found, so that th amplifid algorithm is corrct w.p. n c. 4 argr-stin Algorithm Whil w may b happy with our polynomial tim algorithm for finding th min cut as compard to th trivial xponntial algorithm, Ω(n 4 ) is slowr than w d lik. In particular, this algortihm would not b practical on larg ntworks ncountrd in many of today s applications. Can w do bttr? In fact, thr is a bttr way to run argr s Algorithm than running n indpndnt trials. This algorithm is known as th argr-stin algorithm, and dtaild in this papr. Hr is th psudocod of th argr-stin Algorithm: Algorithm 6: argrstin(g) n G ; if n < thn argr(g) ; ls Run argr s procdur twic, ach tim until (S, ) argrstin(g ) ; (S, ) argrstin(g ) ; Output min{(s, ), (S, )} n suprnods rmain; gt two multigraphs G, G ; Whil w don t includ th fairly tchnical proof, th following bound can b shown for th probability 5
6 of succss of argr-stin. P (argr-stin is corrct) Θ(log n). So, by our rmark abov, if w run O(log n) trials, th nw amplifid algorithm will b corrct w.p. poly(n) Now w ar xprts in divid-and-conqur. W thus prform th runtim analysis: ( ) n T (n) = O(n ) + T, by th Mastr s Thorm, T (n) = Θ(n log n). Thus th final runtim of th AmplifidargrStin algorithm is Θ(n log n) O(log n) = O(n log 3 n). 6
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