Lecture 2: Karger s Min Cut Algorithm


 Derick Lester
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1 priceto uiv. F 3 cos 5: Advaced Algorithm Desig Lecture : Karger s Mi Cut Algorithm Lecturer: Sajeev Arora Scribe:Sajeev Today s topic is simple but gorgeous: Karger s mi cut algorithm ad its extesio. It is a simple radomized algorithm for fidig the miimum cut i a graph: a subset of vertices S i which the set of edges leavig S, deoted E(S, S) has miimum size amog all subsets. You may have see a algorithm for this problem i your udergrad class that uses maximum flow. Karger s algorithm is elemetary ad ad a great itroductio to radomized algorithms. The algorithm is this: Pick a radom edge, ad merge its edpoits ito a sigle superode. Repeat util the graph has oly two superodes, which is output as our guess for micut. (As you cotiue, the superodes may develop parallel edges; these are allowed. Selfloops are igored.) Note that if you pick a radom edge, it is more likely to come from parts of the graph that cotai more edges i the first place. Thus this algorithm looks like a great heuristic to try o all kids of reallife graphs, where oe wats to cluster the odes ito tightlykit portios. For example, social etworks may cluster ito commuities; graphs capturig similarity of pixels may cluster to give di eret portios of the image (sky, grass, road etc.). Thus istead of cotiuig Karger s algorithm util you have two superodes left, you could stop it whe there are k superodes ad try to uderstad whether these correspod to a reasoable clusterig. Today we will first see that the above versio of the algorithm yields the optimum mi cut with probability at least /. Thus we ca repeat it say 0 times, ad output the smallest cut see i ay iteratio. The probability that the optimum cut is ot see i ay repetitio is at most ( / ) 0 < 0.0. Ufortuately, this simple versio has ruig time about 4 which is ot great. So the we see a better versio with a simple tweak that brigs the ruig time dow to closer to. The idea is that roughly that repetitio esures fault tolerace. The reallife advice of makig two backups of your hard drive is related to this: the probability that both fail is much smaller tha oe does. I case of Karger s algorithm, the overall probability of success is too low. But if ru part of the way util the graph has / p superodes, the chace that the micut has t chaged is at least /. So you make two idepedet rus that go dow to / p superodes, ad recursively solve both of these. Thus the expected umber of istaces that will yield the correct micut is =. (Uwrappig the recursio, you see that each istace of size / p will geerate two istaces of size /, ad so o.) Simple iductio shows that this wise repetitio is eough to brig the probability of success above / log. As you might suspect, this is ot the ed of the story but improvemets beyod this get more hairy. If aybody is iterested I ca give more poiters. Also this algorithm forms the basis of other algorithms for other tasks. Agai, talk to me for poiters.
2 CSE 03: Probability ad statistics Witer 00 Topic 4 Radomized algorithms, II 4. Karger s miimum cut algorithm 4.. Clusterig via graph cuts Suppose a mail order compay has the resources to prepare two differet versios of its catalog, ad it wishes to target each versio towards a particular sector of its customer base. The data it has is a list of its regular customers, alog with their purchase histories. How should this set of customers be partitioed ito two coheret groups? Oe way to do this is to create a graph with a ode for each of the regular customers, ad a edge betwee ay two customers whose purchase patters are similar. The goal is the to divide the odes ito two pieces which have very few edges betwee them. More formally, the miimum cut of a udirected graph G =(V,E) isapartitiooftheodesitotwo groups V ad V (that is, V = V V ad, V V = ), so that the umber of edges betwee V ad V is miimized. I the graph below, for istace, the miimum cut has size two ad partitios the odes ito V = {a, b, e, f} ad V = {c, d, g, h}. a b c d e f g h 4.. Karger s algorithm Here s a radomized algorithm for fidig the miimum cut: Repeat util just two odes remai: Pick a edge of G at radom ad collapse its two edpoits ito a sigle ode For the two remaiig odes u ad u,setv = {odes that wet ito u } ad V = {odes i u } A example is show i Figure 4.. Notice how some odes ed up havig multiple edges betwee them Aalysis Karger s algorithm returs the miimum cut with a certai probability. To aalyze it, let s go through a successio of key facts. Fact. If degree(u) deotes the umber of edges touchig ode u, the degree(u) = E. u V 4
3 CSE 03 Topic 4 Radomized algorithms, II Witer 00 a b c d e f g h a c d e g h a c d e gh a c 4 edges to choose from Pick b f (probability /4) 3 edges to choose from Pick g h (probability /3) edges to choose from Pick d gh (probability /6) e dgh c 0 edges to choose from Pick a e (probability /0) ae dgh c 9edgestochoosefrom Pick ab ef (probability 4/9) abef dgh 5edgestochoosefrom Pick c dgh (probability 3/5) abef cdgh Doe: just two odes remai Figure 4.. Karger s algorithm at work. 4
4 CSE 03 Topic 4 Radomized algorithms, II Witer 00 To see this, imagie the followig experimet: for each ode, list all the edges touchig it. The umber of edges i this list is exactly the lefthad sum. But each edge appears exactly twice i it, oce for each edpoit. Fact. If there are odes, the the average degree of a ode is E /. This is a straightforward calculatio: whe you pick a ode X at radom, E[degree(X)] = Pr(X = u)degree(u) = degree(u) = E u V u where the last step uses the first Fact. Fact 3. The size of the miimum cut is at most E /. Cosider the partitio of V ito two pieces, oe cotaiig a sigle ode u, ad the other cotaiig the remaiig odes.thesizeofthiscutisdegree(u). Sice this is a valid cut, the miimum cut caot be bigger tha this. I other words, for all odes u, (size of miimum cut) degree(u). This meas that the size of the miimum cut is also the average degree, which we ve see is E /. Fact 4. If a edge is picked at radom, the probability that it lies across the miimum cut is at most /. This is because there are E edges to choose from, ad at most E / of them are i the miimum cut. Now we have all the iformatio we eed to aalyze Karger s algorithm. It returs the right aswer as log as it ever picks a edge across the miimum cut. Ifitalwayspicksaocutedge,thethisedgewill coect two odes o the same side of the cut, ad so it is okay to collapsethemtogether. Each time a edge is collapsed, the umber of odes decreases by. Therefore, Pr(fial cut is the miimum cut) = Pr(first selected edge is ot i micut) Pr(secod selected edge is ot i micut) ( )( )( ) ( )( ) 4 3 = = ( ) The last equatio comes from oticig that almost every umerator cacels with the deomiator two fractios dow the lie. Karger s algorithm succeeds with probabililty p /. Therefore, it should be ru Ω( )times,after which the smallest cut foud should be chose. Those who are familiar with miimum spaig tree algorithms might be curious to hear that aother way to implemet Karger s algorithm is the followig: Assig each edge a radom weight Ru Kruskal s algorithm to get the miimum spaig tree Break the largest edge i the tree to get the two clusters (Do you see why?) Over the decades, the ruig time of Kruskal s algorithm has bee thoroughly optimized via special data structures. Now this same techology ca be put to work for cuts! 43
5 = = ()( ) =. I order to boost the probability of success, we simply ru the algorithm times. The probability that at least oe ru succeeds is at least ( ) Settig = c l we have error probability / c. e. It s easy to implemet Karger s algorithm so that oe ru takes O( ) time. Therefore, we have a O( 4 log ) time radomized algorithm with error probability /poly(). A faster versio of this algorithm was devised by Karger ad Stei [4]. The key idea comes from lookig at the telescopig product. I the iitial cotractios it s very ulikely we cotracted a edge i the miimum cut. Towards the ed of the algorithm, our probability of cotractig a edge i the miimum cut grows. From the earlier aalysis we have the followig. For a fixed miimum cut (S), the probability that this cut survives dow to vertices is at least /. Thus, for = / we have probability / of succeedig. Hece, i expectatio two trails should su ce. Improved algorithm: From a multigraph G, if G has at least 6 vertices, repeat twice:. ru the origial algorithm dow to / + vertices.. recurse o the resultig graph. Retur the miimum of the cuts foud i the two recursive calls. The choice of 6 as opposed to some other costat will oly a ect the ruig time by a costat factor. We ca easily compute the ruig time via the followig recurrece (which is straightforward to solve, e.g., the stadard Master theorem applies): T () = + T (/ ) = O( log ). Sice we succeed dow to / with probability /, we have the followig recurrece for the probability of success, deote by P (): P () P (/ + ). This solves to P () = log. Hece, similar to the earlier argumet for the origial algorithm, with O(log ) rus of the algorithm, the probability of success is /poly(). Therefore, i O( log 3 ) total time, we ca fid the miimum cut with probability /poly(). Before fiishig, we observe a iterestig corollary of Karger s origial algorithm which we will use i the ext lecture to estimate the (u)reliability of a etwork. 3
6 Corollary 5 Ay graph has at most O( ) miimum cuts. This follows from Lemma 3 sice that holds for ay specified miimum cut. Note, we ca also eumerate all of these cuts by the above algorithm. Refereces [] A. V. Goldberg ad R. E. Tarja. A ew approach to the maximumflow problem. J. Assoc. Comput. Mach., 35(4):9 940, 988. [] J. Hao ad J. B. Orli. A faster algorithm for fidig the miimum cut i a graph. I Proceedigs of the Third Aual ACMSIAM Symposium o Discrete Algorithms (Orlado, FL, 99), pages 65 74, New York, 99. ACM. [3] D. R. Karger. Global micuts i RNC, ad other ramificatios of a simple micut algorithm. I Proceedigs of the Fourth Aual ACMSIAM Symposium o Discrete Algorithms (Austi, TX, 993), pages 30, New York, 993. ACM. [4] D. R. Karger ad C. Stei. A ew approach to the miimum cut problem. J. ACM, 43(4):60 640,
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