We present an algorithm for finding the minimum cut of an edge-weighted graph.
It is simple in every respect. It has a short and compact description, is easy
to implement and has a surprisingly simple proof of correctness. Its runtime
matches that of the fastest algorithm known. The runtime analysis is
straightforward. In contrast to nearly all approaches so far, the algorithm
uses no flow techniques. Roughly spoken the algorithm consists of about |V|
nearly identical phases each of which is formally similar to Prim's minimum
spanning tree algorithm

Stoer, Mechtild

Wagner, Frank

We present an algorithm for nding the minimum cut of an edge-weighted graph. It is simple in every respect  It has a short and compact description  is easy to implement and has a surprisingly simple proof of correctness  Its runtime matches that of the fastest algorithm known  The runtime analysis is straightforward  In contrast to nearly all approaches so far  the algorithm uses no ow techniques  Roughly spoken the algorithm consists of about jV j nearly identical phases each of which is formally similar to Prim&apos;s minimum spanning tree algorithm

Mechtild Stoer

Frank Wagner

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A simple min cut algorithm

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SERIE B  INFORMATIKA Simple Min Cut AlgorithmMechtild Stoer Frank WagnerB May AbstractWe present an algorithm for nding the minimum cut of an edgeweighted graph It issimple in every respect It has a short and compact description is easy to implementand has a surprisingly simple proof of correctness Its runtime matches that of thefastest algorithm known The runtime analysis is straightforward In contrast tonearly all approaches so far the algorithm uses no ow techniques Roughly spokenthe algorithm consists of about jV j nearly identical phases each of which is formallysimilar to Prims minimum spanning tree algorithm KonradZuseZentrum fur Informationstechnik Berlin Heilbronner Str   Berlin Germany email	 stoerzibberlinde  Institut fur Informatik Fachbereich Mathematik und Informatik Freie Universitat Berlin Takustrae  BerlinDahlem Germany email	 wagnermathfuberlinde  OverviewGraph connectivity is one of the classical subjects in graph theory and has manypractical applications e	 g	 in chip and circuit design reliability of communicationnetworks transportation planning and cluster analysis	 Finding the minimum cutof an edgeweighted graph is a fundamental algorithmical problem	 Precisely itconsists in nding a nontrivial partition of the graphs vertex set V into two partssuch that the sum of the weights of the edges connecting the two parts is minimum	The usual approach to solve this problem is to use its close relationship to the maximum ow problem	 The famous MaxFlowMinCutTheorem by Ford and Fulkerson FF showed the duality of the maximum ow and the socalled minimum stcut	 There s and t are two vertices which are the source and the sink in the owproblem and have to be separated by the cut i	e	 they have to lie in dierent partsof the partition	 Until recently all cut algorithms were essentially ow algorithmsusing this duality	 Finding a minimumcut without specied vertices to be separatedcan be done by nding minimum st cuts for all OjV j pairs of vertices and thenselecting the lightest one	 Gomory and Hu GH reduced this to OjV j pairs ofvertices and thus OjV j maximum ow computations	Recently Hao and Orlin HO showed how to use the maximum ow algorithm byGoldberg and Tarjan GT the fastest so far in order to solve the minimum cutproblem as fast as the maximum ow problem i	e	 in time OjV jjEj logjV jjEj	In the same year Nagamochi and Ibaraki NIa published the rst minimum cutalgorithm that is not based on a ow algorithm has the slightly better running timeof OjV jjEj  jV jlog jV j but is still rather complicated	 In the unweighted casethey use a fast search technique to decompose a graphs edge set E into subsetsE     E such that the union of the rst k Eis is a kedgeconnected spanningsubgraph of the given graph and has size at most kjV j	 They simulate this approachin the weighted case	 Their work is one of a number of papers treating questions ofgraph connectivity by nonowbased methods NP NIa M	In this context we present in this paper a remarkably simple minimumcut algorithmwith the optimal running time established in NIb	 We reduce the complexityof the algorithm of Nagamochi and Ibaraki by avoiding the unnecessary simulateddecomposition of the edge set	 This enables us to give a comparably straightforwardproof of correctness avoiding e	 g	 the distinction between the unweighted integerrational and realweighted case	 The AlgorithmThroughout the paper we deal with an ordinary undirected graph G with vertex setV and edge set E	 Every edge e has positive real weight we 	In order to describe the idea of the algorithm we start by reminding the reader ofPrims minimum spanning tree algorithmMinimumSpanningTreeGw aA Simple Min Cut Algorithm A fagT  while A  Vadd to A the most loosely connected vertexadd to T the connecting edgeA subset A of the graphs vertices grows starting with an arbitrary single vertexuntil A is equal to V 	 In each step the vertex outside of A most loosely connectedwith A is added	 Formally we add a vertexz  A such that wa z  minfwb y j b  A y  A by  Egwhere wb y is the weight of edge by and az is the connecting edge	At the end the set of all the connecting edges then forms a minimum spanning tree	The simple minimum cut algorithm we describe here consists of jV j   phases eachof which is very similar to Prims algorithmMinimumCutPhaseGw aA fagwhile A  Vadd to A the most tightly connected vertexstore the cutofthephase and shrink G by merging the two vertices added lastA subset A of the graphs vertices grows starting with an arbitrary single vertexuntil A is equal to V 	 In each step the vertex outside of A most tightly connectedwith A is added	 Formally we add a vertexz  A such that wA z  maxfwA y j y  Agwhere wA y is the sum of the weights of all the edges between A and y	At the end of each such phase the two vertices added last are merged i	e	 the twovertices are replaced by a new vertex and any edges from the two vertices to aremaining vertex are replaced by an edge weighted by the sum of the weights of theprevious two edges	 Edges joining the merged nodes are removed	The cut of V that separates the vertex added last from the rest of the graph iscalled the cutofthephase	 The lightest of these cutsofthephase is the result ofthe algorithm the desired minimum cut	MinimumCutGw awhile jV j  MinimumCutPhaseGw aif the cutofthephase is lighter than the current minimum cutthen store the cutofthephase as the current minimum cutNotice that the starting vertex a stays the same throughout the whole algorithm	 An Example w   Figure  A graph G  VE with edgeweights	 ab cdefst  Figure  The graph after the rst MinimumCutPhaseGw a a   and theinduced ordering a b c d e f s t of the vertices	 The rst cutofthephase corresponds to the partition fg f      g of V with weight w  	  ab cdes tFigure  The graph after the second MinimumCutPhaseGw a and the induced ordering a b c d e s t of the vertices	 The second cutofthephase corresponds to the partition fg f      g of V with weight w  	A Simple Min Cut Algorithm   ba dstcFigure  After the thirdMinimumCutPhaseGw a	 The third cutofthephasecorresponds to the partition f g f     g of V with weight w  	 bs tca   tsbaFigure  After the fourth and fth MinimumCutPhaseGw a respectively	The fourth cutofthephase corresponds to the partition f  g f    g	 Thefth cutofthephase corresponds to the partition f   g f   g with weightw  	aV n    st tsFigure  After the sixth and seventh MinimumCutPhaseGw a respectively	The sixth cutofthephase corresponds to the partition f g f     g withweight w  	 The last cutofthephase corresponds to the partition fg V n fgits weight is w  	 The minimum cut of the graph G is the fth cutofthephaseand the weight is w  	 CorrectnessThe core of the proof of correctness is the following somewhat surprising lemma	Lemma Each cutofthephase is a minimum st cut in the current graph where sand t are the two vertices added last in the phaseAssuming that the lemma holds we can show by a simple case distinction that thesmallest of these cutsofthephase is indeed the minimum cut we are looking for	This is done by induction on jV j	 The case jV j   is trivial	 If jV j   look at therst phase If G has a minimum cut that is at the same time a minimum st cutthen according to the lemma the cutofthe phase is already a minimum cut	 If notthen G has a minimum cut with s and t on the same side	 Therefore a minimumcut of G  the input graph of phase  that diers from G by the merging of s andt is a minimum cut of G	 Now by induction the lightest of the cutsofthephases to jV j   is such a minimum cut of G 	 Notice that the application of phases to jV j   to G is the same as the application of the complete algorithm to G 	Finally we show the claimed property of the cutofthephase	 The run of aMinimumCutPhase orders the vertices of the current graph linearly starting witha and ending with s and t according to their order of addition to A	 Now we lookat an arbitrary st cut C of the current graph and show that it is at least as heavyas the cutofthephase	A Simple Min Cut Algorithm We call a vertex v  a active with respect to C when v and the vertex added justbefore v are in dierent parts of C	 Let wC be the weight of C Avthe set of allvertices added before v excluding v Cvthe partition of Av fvg induced by Cand wCv the weight of the induced cut i	e	 the sum of the weights of the edgesgoing from one part of the induced partition to the other	We show that for every active vertex vwAv v 	 wCvby induction on the set of active verticesFor the rst active vertex the inequality is satised with equality	 Let the inequalitybe true for all active vertices added up to the active vertex v and let w be the nextactive vertex that is added	 Then we havewAw w  wAv w  wAwnAv w  Now wAv w 	 wAv v as v was chosen as the vertex most tightly connectedwith Av	 By induction wAv v 	 wCv	 All edges between AwnAvand w connectthe dierent parts of C	 Thus they contribute to wCwbut not to wCv	 So 	 wCv  wAwnAv w 	 wCwAs t is always an active vertex with respect to C we can conclude that wAt t 	wCt which says exactly that the cutofthephase is at most as heavy as C	 Running TimeAs the running time of the algorithm MinimumCut is essentially equal to theadded running time of the jV j   runs of MinimumCutPhase which is calledon graphs with decreasing number of vertices and edges it suces to show thata single MinimumCutPhase needs at most OjEj  jV j log jV j time yielding anoverall running time of OjV jjEj jV jlog jV j	The key to implementing a phase eciently is to make it easy to select the nextvertex to be added to the set A the most tightly connected vertex	 During executionof a phase all vertices that are not in A reside in a priority queue based on a keyeld	 The key of a vertex v is the sum of the weights of the edges connecting it tothe current A i	e	 wA v	 Whenever a vertex v is added to A we have to performan update of the queue	 v has to be deleted from the queue and the key of everyvertex w not in A connected to v has to be increased by the weight of the edge vw ifit exists	 As this is done exactly once for every edge overall we have to perform jV jExtractMax and jEj IncreaseKey operations	 Using Fibonacci heaps FTwe can perform an ExtractMax operation in Olog jV j amortized time and aIncreaseKey operation in O amortized time	Thus the time we need for this key step that dominates the rest of the phase isOjEj jV j log jV j	Notice that this runtime analysis is very similar to the analysis of Prims minimumspanning tree algorithm	AcknowledgementThe authors thank Dorothea Wagner for her helpful remarks	ReferencesFT M L Fredman and R E Tarjan Fibonacci heaps and their uses inimproved network optimization algorithms Journal of the ACM   FF L R Ford D R Fulkerson Maximal ow through a network Canadian Journal on Mathematics   GT A V Goldberg and R E Tarjan A new approach to the maximumow problem Journal of the ACM   GH R E Gomory Multiterminal network ows Journal of the SIAM  HO X Hao and J B Orlin A faster algorithm for nding the minimum cutin a graph rd ACMSIAM Symposium on Discrete Algorithms  M D W Matula A linear time  approximation algorithm for edge connectivity  Proceedings of the th ACMSIAM Symposium on Discrete Mathematics NIa H Nagamochi and T Ibaraki Linear time algorithms for nding asparse kconnected spanning subgraph of a kconnected graph Algorithmica  NIb H Nagamochi and T Ibaraki Computing edgeconnectivity in multigraphs and capacitated graphs SIAM Journal on Discrete Mathematics  NP T Nishizeki and S Poljak Highly connected factors with a small number of edges Preprint P R C Prim Shortest connection networks and some generalizations BellSystem Technical Journal   

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