ORIE 633 Network Flows September 27, Lecture 8

Transcription

1 ORIE 633 Network Flows September 7, 007 Lecturer: David P. Williamso Lecture 8 Scribe: Gema Plaza-Martíez 1 Global mi-cuts i udirected graphs 1.1 Radom cotractio Recall from last time we itroduced the radom cotractio algorithm. Radom Cotractio (Karger 93) Util V Pick edge (i, j) at radom with probability proportioal to its capacity. Cotract (i, j). The ituitio for the algorithm is that the global mi-cut i a graph has capacity that is small relative to the capacity of the graph, ad so we are ulikely to choose a edge that is i a particular global mi-cut. Last time, we showed the followig. Theorem 1 Let S be a give global mi-cut. The Pr[Radom Cotractio cotracts o edge i S ] 1 (. We start off the lecture by observig that we ca derive from this a upper boud o the umber of global mi-cuts. Theorem There are at most ( global mi cuts. Proof: Notice that evets i which we do t cotract edges i two differet global mi cuts are disjoit. Let S1, S,..., S p be all global mi cuts. Let C i be the evet that we do t cotract ay edge i Si. We showed i last lecture that P r[c i] ( 1. Sice these p evets are all disjoit P r[c i ] 1, so we must have p (. i1 A example of a graph with ( global mi cuts is a ode cycle, where each edge has the same capacity, sice the ay pair of edges forms a global mi-cut. We ow tur to the questio of how to take the Radom Cotractio algorithm ad tur it ito a algorithm that returs a global mi-cut with high probability. Theorem 3 We ca implemet radom cotractio i O( ) time (O() per cotractio). Theorem 4 We ca fid a global mi cut with probability 1 1 i time O(4 log ). 8-1

2 Proof: Suppose we repeat Radom Cotractio ( log times. The usig the fact that 1 x e x, we have ( ) ( Pr[we do t fid mi cut] 1 ( 1 log ( e 1 ) ( ( log e log 1. So Pr[we do fid mi cut] Recursive radom cotractio The ruig time of this algorithm is ot very competitive, but Radom Cotractio (RC) has some features that we like, so we ll try to exploit those to fid a better algorithm. Oe of the drawbacks of RC is that the probability of cotractig a edge i a cut is low for large graphs ad higher as the graph gets smaller. Oe idea is to make the graph smaller by usig RC, ad the use some other algorithm. I this case, we ll use as our other algorithm two rus of RC. Our algorithm is thus called Recursive Radom Cotractio (RRC). We ll do two iteratios of cotractig the graph dow to roughly a factor of 1/ of its previous size, the call RRC o the remaiig graph. We retur the smaller of the two cuts foud i the two iteratios. Recursive Radom Cotractio (Karger, Stei 96) If V 6, fid a mi cut by exhaustive eumeratio Else For i 1 to Ru RC to get H i of 1 + V vertices C i RRC(H i ) Retur smaller of C 1, C Now we ca start to aalyze the algorithm. Lemma 5 RRC rus i O( log ) time. Proof: The ruig time o vertices is give by the recurrece T () T ( 1 + ) + O( ) Solvig this recurrece we get T () O( log ). ( ) Lemma 6 RRC fids a global mi cut with probability Ω 1 log. 8-

3 Corollary 7 Ruig RRC O(log ) times we fid a mi cut with probability 1 1. To prove Lemma 6, we ll eed the followig Lemma 8 For a give global mi-cut S, Pr[o edge i δ(s ) is cotracted whe graph is cotracted from to t vertices] ( t (. Proof: We recall from last time that the probability that o edge i δ(s ) is cotracted i the ith iteratio give that o edge was cotracted i previous iteratios is at least 1 /( i + 1). Thus the desired probability is at least t i1 ( 1 ) i + 1 t ( ) i 1 i + 1 i1 t+1 ( ) j Now we ca prove Lemma 6. Proof of Lemma 6: Give a graph of size t, the probability that o edge i δ(s ) is cotracted whe reduced to 1 + t vertices is at least 1+ t 1+ t 1 t(t 1) 1. Let P (t) be the probability that RRC does t cotract ay edge i δ(s ) i a graph of t vertices. The P (t) 1 P r[some edge i δ(s ) is cotracted i either iteratio 1 or ] ( ( 1 1 Pr[o edges are cotracted whe goig to t vertices]p 1 + t )) 1 (1 1 ( P 1 + t )). We kow that P (t) 1 for t 6. Skippig the algebra eeded to work out this recursio, we get ( ) 1 P (t) θ. log t So far we ve see the followig global mi0cut algorithms: MA orderigs: O((m + log )) RRC: O( log ) Hao-Orli: O(m log ) j ( t (. j 8-3

4 There is still aother global mi-cut algorithm that we have t covered: Karger: O(m log c ) Õ(m) Cotractios are ot easy to implemet, so though they look like the simplest algorithms, they are ot the clear wier. There is a comparative study of all these four algorithms by Chekuri, Goldberg, Karger, Levie ad Stei (1996). The two clear wiers i their study are MA orderigs ad Hao-Orli, ad they say that Hao-Orli is the most robust of their codes; this is probably i part because there are a lot of good implemetatios of push/relabel aroud. This shows that a better theoretical ruig time does t ecessarily imply that a algorithm is better i practice. Efficiet algorithms for max flow.1 Blockig flow We ll work ow toward oe more algorithm for computig a maximum flow. It is based o the cocept of a blockig flow. Defiitio 1 A flow f is blockig if i G, every s t path has some arc saturated. Every maximum flow is obviously also a blockig flow. Is every blockig flow a maximum flow? No. The followig example illustrates this: However, blockig flows are still useful i order to compute maximum flows. We give a algorithm below. Let d i distace from a ode i to the sik t. Defiitio A arc (i, j) is admissible w.r.t. flow f if (i, j) A f ad d i d j + 1. Note that a arc beig admissible meas it is o a shortest path to t. Next time we ll discuss the followig algorithm. 8-4

5 Diic s algorithm (1970) f 0 While s t path i G f Compute distace d i from i to t i G f Fid blockig flow f i graph G f with oly admissible arcs, capacity u f ij f f + f 8-5