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1、A Simple Min-Cut Algorithm MECHTHILD STOER Televerkets Forskningsinstitutt, Kjeller, Norway AND FRANK WAGNER Freie Universita t Berlin, Berlin-Dahlem, Germany Abstract. We present an algorithm for finding the minimum cut of an undirected edge-weighted graph. It is simple in every respect. It has a s
2、hort 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 speaki
3、ng, the algorithm consists of about uVu nearly identical phases each of which is a maximum adjacency search. Categories and Subject Descriptors: G.L.2 Discrete Mathematics: Graph Theorygraph algorithms General Terms: Algorithms Additional Key Words and Phrases: Min-Cut 1. Introduction Graph connecti
4、vity is one of the classical subjects in graph theory, and has many practical applications, for example, in chip and circuit design, reliability of communication networks, transportation planning, and cluster analysis. Finding the minimum cut of an undirected edge-weighted graph is a fundamental alg
5、orithmical problem. Precisely, it consists in finding a nontrivial partition of the graphs vertex set V into two parts such that the cut weight, the sum of the weights of the edges connecting the two parts, is minimum. A preliminary version of this paper appeared in Proceedings of the 2nd Annual Eur
6、opean Symposium on Algorithms. Lecture Notes in Computer Science, vol. 855, 1994, pp. 141147. This work was supported by the ESPRIT BRA Project ALCOM II. Authors addresses: M. Stoer, Televerkets Forskningsinstitutt, Postboks 83, 2007 Kjeller, Norway; e-mail: mechthild.stoernta.no.; F. Wagner, Instit
7、ut fu r Informatik, Fachbereich Mathematik und Informatik, Freie Universita t Berlin, Takustrae 9, Berlin-Dahlem, Germany; e-mail: wagnerinf.fu- berlin.de. Permission to make digital/hard copy of part or all of this work for personal or classroom use is granted without fee provided that the copies a
8、re not made or distributed for profit or commercial advantage, the copyright notice, the title of the publication, and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery (ACM), Inc. To copy otherwise, to republish, to post on servers, or to
9、redistribute to lists, requires prior specific permission and/or a fee. q 1997 ACM 0004-5411/97/0700-0585 $03.50 Journal of the ACM, Vol. 44, No. 4, July 1997, pp. 585591. The usual approach to solve this problem is to use its close relationship to the maximum flow problem. The famous Max-Flow-Min-C
10、ut-Theorem by Ford and Fulkerson 1956 showed the duality of the maximum flow and the so-called minimum s-t-cut. There, s and t are two vertices that are the source and the sink in the flow problem and have to be separated by the cut, that is, they have to lie in different parts of the partition. Unt
11、il recently all cut algorithms were essentially flow algorithms using this duality. Finding a minimum cut without specified vertices to be separated can be done by finding minimum s-t-cuts for a fixed vertex s and all uVu 2 1 possible choices of t V s and then selecting the lightest one. Recently Ha
12、o and Orlin 1992 showed how to use the maximum flow algorithm by Goldberg and Tarjan 1988 in order to solve the minimum cut problem in time 2(uViEulog(uVu2/uEu), which is nearly as fast as the fastest maximum flow algorithms so far Alon 1990; Ahuja et al. 1989; Cheriyan et al. 1990. Nagamochi and Ib
13、araki 1992a published the first deterministic minimum cut algorithm that is not based on a flow algorithm, has the slightly better running time of 2(uViEu 1 uVu2loguVu), but is still rather complicated. In the unweighted case, they use a fast-search technique to decompose a graphs edge set E into su
14、bsets E1, . . . , Elsuch that the union of the first k Eis is a k-edge-connected spanning subgraph of the given graph and has at most kuVu edges. They simulate this approach in the weighted case. Their work is one of a small number of papers treating questions of graph connectivity by non-flow-based
15、 methods Nishizeki and Poljak 1989; Nagamochi and Ibaraki 1992a; Matula 1992. Karger and Stein 1993 suggest a randomized algorithm that with high probability finds a minimum cut in time 2(uVu2loguVu). In this context, we present in this paper a remarkably simple deterministic minimum cut algorithm w
16、ith the fastest running time so far, established in Nagamochi and Ibaraki 1992b. We reduce the complexity of the algorithm of Nagamochi and Ibaraki by avoiding the unnecessary simulated decomposition of the edge set. This enables us to give a comparably straightforward proof of correctness avoiding,
17、 for example, the distinction between the unweighted, integer-, rational-, and real-weighted case. This algorithm was found independently by Frank 1994. Queyranne 1995 generalizes our simple approach to the minimization of submodular functions. The algorithm described in this paper was implemented b
18、y Kurt Mehlhorn from the Max-Planck-Institut, Saarbru cken and is part of the algorithms library LEDA Mehlhorn and Na her 1995. 2. The Algorithm Throughout the paper, we deal with an ordinary undirected graph G with vertex set V and edge set E. Every edge e has nonnegative real weight w(e). The simp
19、le key observation is that, if we know how to find two vertices s and t, and the weight of a minimum s-t-cut, we are nearly done: THEOREM2.1.Let s and t be two vertices of a graph G. Let G/s, t be the graph obtained by merging s and t. Then a minimum cut of G can be obtained by taking the smaller of
20、 a minimum s-t-cut of G and a minimum cut of G/s, t. 586M. STOER AND F. WAGNER The theorem holds since either there is a minimum cut of G that separates s and t, then a minimum s-t-cut of G is a minimum cut of G; or there is none, then a minimum cut of G/s, t does the job. So a procedure finding an
21、arbitrary minimum s-t-cut can be used to construct a recursive algorithm to find a minimum cut of a graph. The following algorithm, known in the literature as maximum adjacency search or maximum cardinality search, yields the desired s-t-cut. MINIMUMCUTPHASE(G, w, a) A 4 a while A ? V add to A the m
22、ost tightly connected vertex store the cut-of-the-phase and shrink G by merging the two vertices added last A subset A of the graphs vertices grows starting with an arbitrary single vertex until A is equal to V. In each step, the vertex outside of A most tightly connected with A is added. Formally,
23、we add a vertex z y A such that wA, z!5 max$wA, y!uy y A%, where 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, that is, the two vertices are replaced by a new vertex, and any edges from the two vertices to a
24、 remaining vertex are replaced by an edge weighted by the sum of the weights of the previous 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 is called the cut-of-the-phase. The lightest of these cuts-of-the-phase is
25、the result of the algorithm, the desired minimum cut: MINIMUMCUT(G, w, a) while uVu . 1 MINIMUMCUTPHASE(G, w, a) if the cut-of-the-phase is lighter than the current minimum cut then store the cut-of-the-phase as the current minimum cut Notice that the starting vertex a stays the same throughout the
26、whole algorithm. It can be selected arbitrarily in each phase instead. 3. Correctness In order to proof the correctness of our algorithms, we need to show the following somewhat surprising lemma. LEMMA3.1.Each cut-of-the-phase is a minimum s-t-cut in the current graph, where s and t are the two vert
27、ices added last in the phase. PROOF.The run of a MINIMUMCUTPHASEorders the vertices of the current graph linearly, starting with a and ending with s and t, according to their order of addition to A. Now we look at an arbitrary s-t-cut C of the current graph and show, that it is at least as heavy as
28、the cut-of-the-phase. 587A Simple Min-Cut Algorithm We call a vertex v ? a active (with respect to C) when v and the vertex added just before v are in the two different parts of C. Let w(C) be the weight of C, Av the set of all vertices added before v (excluding v), Cvthe cut of Av v induced by C, a
29、nd w(Cv) the weight of the induced cut. We show that for every active vertex v wAv, v!# wCv! by induction on the set of active vertices: For the first active vertex, the inequality is satisfied with equality. Let the inequality be true for all active vertices added up to the active vertex v, and let
30、 u be the next active vertex that is added. Then we have wAu, u!5 wAv, u!1 wAu Av, u!5:a Now, w(Av, u) # w(Av, v) as v was chosen as the vertex most tightly connected with Av. By induction w(Av, v) # w(Cv). All edges between Au Avand u connect the different parts of C. Thus they contribute to w(Cu)
31、but not to w(Cv). So a# wCv!1 wAu Av, u!# wCu! 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 cut-of-the-phase is at most as heavy as C. 4. Running Time As the running time of the algorithm MINIMUMCUTis essentially equal to the add
32、ed running time of the uVu 2 1 runs of MINIMUMCUTPHASE, which is called on graphs with decreasing number of vertices and edges, it suffices to show that a single MINIMUMCUTPHASEneeds at most 2(uEu 1 uVu log uVu) time yielding an overall running time of 2(uViEu 1 uVu2log uVu). The key to implementing
33、 a phase efficiently is to make it easy to select the next vertex to be added to the set A, the most tightly connected vertex. During execution of a phase, all vertices that are not in A reside in a priority queue based on a key field. The key of a vertex v is the sum of the weights of the edges con
34、necting it to the current A, that is, w(A, v). Whenever a vertex v is added to A we have to perform an update of the queue. v has to be deleted from the queue, and the key of every vertex w not in A, connected to v has to be increased by the weight of the edge vw, if it exists. As this is done exact
35、ly once for every edge, overall we have to perform uVu EXTRACTMAXand uEu INCREASEKEY operations. Using Fibonacci heaps Fredman and Tarjun 1987, we can perform an EXTRACTMAXoperation in 2(loguVu) amortized time and an INCREASEKEY operation in 2(1) amortized time. Thus, the time we need for this key s
36、tep that dominates the rest of the phase, is 2(uEu 1 uVuloguVu). 588M. STOER AND F. WAGNER 5. An Example FIG. 1.A graph G 5 (V, E) with edge-weights. FIG. 2.The graph after the first MINIMUMCUTPHASE(G, w, a), a 5 2, and the induced ordering a, b, c, d, e, f, s, t of the vertices. The first cut-of-th
37、e-phase corresponds to the partition 1, 2, 3, 4, 5, 6, 7, 8 of V with weight w 5 5. FIG. 3.The graph after the second MINIMUMCUTPHASE(G, w, a), and the induced ordering a, b, c, d, e, s, t of the vertices. The second cut-of-the-phase corresponds to the partition 8, 1, 2, 3, 4, 5, 6, 7 of V with weig
38、ht w 5 5. FIG. 4.After the third MINIMUMCUTPHASE(G, w, a). The third cut-of-the-phase corresponds to the partition 7, 8, 1, 2, 3, 4, 5, 6 of V with weight w 5 7. 589A Simple Min-Cut Algorithm ACKNOWLEDGMENT.The authors thank Dorothea Wagner for her helpful re- marks. REFERENCES AHUJA, R. K., ORLIN,
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43、 25th ACM Symposium on the Theory of Computing (San Diego, Calif., May 1618). ACM, New York, pp. 757765. FIG. 5.After the fourth and fifth MINIMUMCUTPHASE(G, w, a), respectively. The fourth cut-of-the- phase corresponds to the partition 4, 7, 8, 1, 2, 3, 5, 6. The fifth cut-of-the-phase corresponds
44、to the partition 3, 4, 7, 8, 1, 2, 5, 6 with weight w 5 4. FIG. 6.After the sixth and seventh MINIMUMCUTPHASE(G, w, a), respectively. The sixth cut-of-the- phase corresponds to the partition 1, 5, 2, 3, 4, 6, 7, 8 with weight w 5 7. The last cut-of-the-phase corresponds to the partition 2, V 2; its
45、weight is w 5 9. The minimum cut of the graph G is the fifth cut-of-the-phase and the weight is w 5 4. 590M. STOER AND F. WAGNER MATULA, D. W.1993.A linear time 2 1eapproximation algorithm for edge connectivity. In Proceedings of the 4th ACMSIAM Symposium on Discrete Mathematics ACM, New York, pp. 5
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47、RAKI, T.1992b.Computing edge-connectivity in multigraphs and capaci- tated graphs. SIAM J. Disc. Math. 5, 5466. NISHIZEKI, T.,ANDPOLJAK, S.1989.Highly connected factors with a small number of edges. Preprint. QUEYRANNE, M.1995.A combinatorial algorithm for minimizing symmetric submodular functions. In Proceedings of the 6th ACMSIAM Symposium on Discrete Mathematics ACM, New York, pp. 98101. RECEIVED APRIL1995;REVISED FEBRUARY1997;ACCEPTED JUNE1997 Journal of the ACM, Vol. 44, No. 4, July 1997. 591A Simple Min-Cut Algorithm