S. Even and R. E. Tarjan. Network flow and testing graph connectivity. SIAM J. Comput., 4:507--518, 1975.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....A) and s, t V , let (D) denote the minimum length of an s t path in D. If no such path exists, set (D) If we choose the paths P 1 , P l in such a way that (D k 1 ) D k ) then the number of iterations clearly is not larger than (as (D k ) for each k) In fact, as Even and Tarjan [1975] noted, in that case there are the following better bounds on the total number N of iterations: Theorem 4.2. If (D k 1 ) D k ) for each k N , then N . If moreover D is simple, then N . Proof. Let k : # A #. Hence D k contains A A pairwise arc disjoint s t ....

....d(v i ) d(v i 1 ) 1; if a 1 i is an arc of D, then d(v i 1 ) d(v i ) 1, since a 1 belongs to one of the P j . Now at least one of the a i is not an arc of D (as P 1 , P l is blocking in D) Hence m d(v m ) d(t) a contradiction. This gives the following result of Even and Tarjan [1975]: t paths can be found in time O( A ) If D is simple, the paths can be found also in time O( V A ) Proof. Directly from Corollary 4.3a and Theorem 4.2. The vertex disjoint case. If we are interested in vertex disjoint paths, the results can be sharpened. Note that if D = V, A) is ....

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S. Even, R.E. Tarjan, Network flow and testing graph connectivity, SIAM Journal on Computing 4 (1975) 507--518.

....route only O( m) pairs in the residual graph. The key observation underlying our analysis of the greedy algorithm is that when there are no short paths, the capacity of the graph to route additional pairs is more strongly diminished than suggested by the above counting argument. Even and Tarjan [4] showed that in a simple unit capacity directed graph, for any two vertices s and t, the maximum flow from s to t is ) where is the shortest path distance from s to t. Our analysis of the greedy algorithm is based on the following question. Given a unit capacity graph G, let S be the ....

S. EVEN AND R. E. TARJAN. Network flow and testing graph connectivity. SIAM Journal on Computing, Vol. 4, 1975, pp. 507-518.

....jFj 4jT j. This involves the creation of a lookup table of jFj entries for incrementing elements (Conway s Zech logarithm , e.g. We can also use Dinitz algorithm [7] to find many paths in time O(jEj) For large h this also yields improved asymptotic time bounds for the flow computation part [9]. Function LIF(V , E, s, T ) h : min t2T min fjCj : C is s t cutg = min t2T jmax flow from s to tj insert a new source s into V help to establish the invariant insert h parallel edges fe1 ; ehg from s to s into E let f denote a set of h edge disjoint paths from s ....

S. Even and E. Tarjan. Network flow and testing graph connectivity. SIAM J. Comput., 4:507--518, 1975.

.... is constant, the expected cost for finding a linearly independent m e is O(1) Delta O(h Delta jT j) O(h Delta jT j) We can also use Dinitz algorithm [5] to find many paths in time O(jEj) For large h this also yields improved asymptotic time bounds for the flow computation part [6]. If the table is considered too large, one can resort to the polynomial representation of field elements. In this case, no table is needed at the cost of additional factors in running time that are polylogarithmic in jT j. 7 The most expensive part is to compute a t (c) for all t and all c ....

S. Even and E. Tarjan. Network flow and testing graph connectivity. SIAM J. Comput., 4:507--518, 1975.

....we would like to achieve f( O(1) witit linear space. 1.1 Previous Results on Path and Connectivity Queries In this section, we overview previous results on path and connectivity queries. First, we consider algorithms that do not exploit preprocessing. Using network flow techniques [17], a patit query can be answered in 0 (m ) time for arbitrary and in 0 ( m) time for any fixed . Regarding planar graphs, it has been recently shown that a path query can be performed in O (tz) time for any [49] Faster query time can be achieved if preprocessing is allowed. For : ....

S. Even and R. E. Tarjan. Network flow and testing graph connectivity. SIAM J. Cornput., 4:507 518, 1975.

....a 2 approximation of the minimum vertex cut. To our knowledge, there was no approximation algorithm for this problem known, the best algorithm to compute the exact solution takes time O(l3n 3 2 2n2) 10, 1, 17] This is an improvement of if x and of a3 n , otherwise. Even and Tarjan [6] give an algorithm to compute the minimum vertex cut r; a, b) between two given nodes a and b in time O(mx ) which we refer to as the pair algorithm. Galil s minimum vertex cut algorithm [10] makes 0( 2 n) calls to the pair algorithm. Our algorithm reduces the number of calls to n d using ....

S. Even and R. E. Tarjan, "Network flow and testing graph connectivity", SlAM Journal on Computing, 4, 1975, 507 518.

....vertex coloring of G, we assign color i to vertices u i and v i , 1 i k, and color arbitrarily each of the remaining vertices with one of the colors k 1; k 2; jV j Gamma 2k. Computing the maximum bipartite matching can be done in O(jEjjV j ) as follows from Even and Tarjan [12]. We summarize the above arguments in the following theorem: Theorem 3.18 The vertex sum coloring problem can be solved on cobipartite graphs in O(jEjjV j ) time. Note that Jansen [22] independently has proved this theorem using the same technique. Edge sum coloring It is quite natural to ....

S. Even and R.E. Tarjan "Network flow and testing graph connectivity" SIAM J. Comp., 4, 507-518 (1975).

....experiments regarding the question when perfect balance L max = N=D for integer N=D is achievable. It looks like for N D d2:3 log De perfect balance can be achieved in 90 of all cases. 4 Fast Scheduling For very large D, the worst case bounds for maximum flow computations( Omega 3=2 , [18]) might become too expensive, since eventually, the scheduling time exceeds the access time. Therefore, we will now explain, why slightly modified maximum flow algorithms can actually find a schedule with L max = dN=De 1 efficiently with high probability. In Section 4.4 even faster linear time ....

Even, S., and Tarjan, E. Network flow and testing graph connectivity. SIAM J. Comput. 4 (1975), 507--518.

....In directed graphs, the first result (about the average length) still holds but not the second. We show that we may not be able to avoid paths of length Omega Gamma n) There are various algorithms to decide the edge connectivity of graphs and to find edge disjoint paths between vertices. See [6] and [15] for sequential algorithms and [13] for a parallel parallel algorithm. All these algorithms ignore the lengths of the edge disjoint paths. Using the approach of minimum cost flow, we propose an algorithm that finds the edge disjoint paths claimed above. It runs in time O(m) In a edge ....

S.Even and R.E.Tarjan, "Network Flow and Testing Graph Connectivity", SIAM J. on Computing, 4:507-518, 1975.

....requires that no subset of the (a, b) vertex separator is an (a, b) vertex separator. In the rest of the paper we will say separator and minimum separator for an (a, b) vertex separator and a minimum (a, b) vertex separator, respectively. 2 We briefly review an algorithm by Even and Tarjan [17, 18] for finding minimum vertex separators. The algorithm is given two vertices, a, b, and an undirected graph, G. It transforms G into a directed graph, G, that has two vertices (corresponding to input and output) for each original vertex of G, directed edges connecting the corresponding input ....

....for lack of space. It is supplied in the full paper. 6 Theorem 3. 4 Procedure triangulate(G, #, k) finds a triangulation of G of clique number # 3k 1, if the treewidth of G is at most k 1, in time O(2 2k 2 k 3 2 V 5 2 ) using the min (a, b) vertex separator algorithm of [18]) or O(2 2k 2 k 7 2 V 2 ) using the min (a, b) vertex separator algorithm given by [22, 23] PROOF Lemmas 3.1 and 3.2 guarantee the correctness of the procedure. We are left to show the time bound. We will bound the number of times that procedure 1 2 vtx sep is called by O( V ....

Shimon Even and R. Endre Tarjan. Network flow and testing graph connectivity. SIAM Journal on Computing, 4(4):507--518, December 1975.

....finding a minimum weight bipartite matching and for finding a minimum cost flow in a network with zero one capacities, if the weights are polynomially bounded integers. 1 Introduction Bipartite matching and related problems have been studied extensively in the contexts of both sequential (e.g. [9, 8, 19, 36, 33]) and parallel (e.g. 1, 21, 24] computation. Though recent research produced RNC algorithms for these problems, i.e. randomized parallel algorithms that run in expected polylogarithmic time on a polynomial number of processors, no sublinear time deterministic parallel algorithms were known. ....

....n) time using BFS(n; m) processors. Observe that maximal sets of node disjoint paths correspond to blocking flows in matching networks (described in detail in the next section) Thus, by using the Maximal Paths procedure to find a blocking flow at each iteration of Dinitz s maximum flow algorithm [8, 9], we can compute a maximum bipartite matching in sublinear time. In the subsequent sections we will show more efficient algorithms for bipartite matching and related problems; these algorithms do not use the Maximal Paths algorithm. Depth First Search Another application of the Maximal Paths ....

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S. Even and R. E. Tarjan. Network Flow and Testing Graph Connectivity. SIAM J. Comput., 4:507--518, 1975.

....2 nm nF (n; m) time algorithm to increase the edge connectivity of an undirected graph by ffi, where n and m are the number of vertices and edges in G, respectively, and F (n; m) is the time to perform one maximum flow on G. The best known bound for F (n; m) is O(minfn 2 3 m;m 3 2 g) ET75] They first devised an O(nm) time algorithm to optimally increase the edge connectivity of an undirected graph by 1. By applying the basic algorithm ffi times, they showed that by carefully choosing the edges added in each iteration, they could optimally increase the edge connectivity of any ....

S. Even and R. E. Tarjan. Network flow and testing graph connectivity. SIAM J. Comput., 4:507--518, 1975.

....of a free form surface by another type necessary. ffl Automatic tools that create CAD models from input devices like computer tomographs or 3D scanners do not have any knowledge about the topology of the workpiece. Errors and an inaccurate precision of measurement may then lead to gaps. See [3] for a detailed discussion of these issues. Existing approaches. Some restrictions on the input format as well as much cleaner CAD models sometimes allow the use of a distance measure in combination with a threshold value. Sheng and Meier examined in [8] the case when the surface is the boundary ....

....[1] G. Barequet and M. Sharir. Filling gaps in the boundary of a polyhedron, in Computer Aided Geometric Design 12 (1995) 207 229. 2] J. H. Bhn and M. J. Wozny. A Topology Based Approach for Shell Closure, in Geometric Modeling for Product Realization, IFIP Transactions B 8 (1993) 297 319. [3] G. M. Fadel and C. Kirschman. Accuracy issues in CAD to RP translations, in Internet Conference on Rapid Product Development. MCB University Press, 1995. http: www.mcb.co.uk services conferen dec95 rapidpd fadel fadel.htm. 4] G. Krause. Interactive finite element preprocessing with ISAGEN, in ....

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S. Even and R. E. Tarjan, Network flow and testing graph connectivity, SIAM J. Comput. 4 (1975), pp. 507--518.

....our implementation. 1. Introduction Despite intensive research for more than three decades, problems related to flows in networks still motivate cutting edge algorithmic research. In a recent development, Goldberg and Rao [4] combined elements of the algorithms of Dinitz [2] and Even and Tarjan [3] with new arguments to obtain a simple algorithm, the binary blocking flow (BBF) algorithm, for computing maximum flows in networks with integer capacities. On networks with n vertices, m edges, and integer capacities bounded by U , the algorithm runs in O(m log(n 2 =m) log U) time, where, here ....

....g(e 0 ) r f (e 0 ) 2 Delta, and the anticycle property implies that jgj Delta. As observed by Goldberg and Rao, the fact that every edge in the residual graph joining two adjacent layers is of residual capacity less than 3 Delta means that one can use the arguments of Even and Tarjan [3] to show that after cde rounds, the residual value of f has dropped below fi b F . Here c is a positive constant that depends on ff and fi. In a very simple implementation of the BBF algorithm, one could therefore end a phase after cde rounds and choose Delta : bfi Deltac for the next round. ....

S. Even and R. E. Tarjan, Network flow and testing graph connectivity, SIAM J. Comput. 4 (1975), pp. 507--518.

....possible to add an edge while still maintaining a matching. We now survey the heuristics we test which are related to maximum matching. MBM. We consider first computing an actual maximum size matching. The fastest algorithm to compute a maximum size matching runs in O(n 1=2 m) time worst case [9, 10]. A drawback of maximum matching is that is can yield instability after admissible but non uniform traffic even on a 3 Theta 3 switch [22] Another problem with MBM is that the relevant algorithms are sequential in nature, requiring a centralized controller with knowledge of the state of each of ....

S. Even and R. E. Tarjan, "Network flow and testing graph connectivity," SIAM Journal on Computing, vol. 4, pp. 507--518, 1975.

....IS flows in acyclic networks with at most n nodes and m arcs each. In the cases of unit arc capacities and unit node capacities the number of blocking IS flows that we need to construct becomes O(m 1=2 ) and O(n 1=2 ) respectively, similarly to that for the corresponding usual networks [8, 17, 18]. This paper is organized as follows. Section 2 gives basic definitions. Sections 3 and 4 are devoted to theoretical results, considering combinatorial and linear programming aspects of ISflows, respectively. Section 5 describes the algorithm for the MSFP based on finding a good initial solution. ....

....the unit arc capacity and unit node capacity networks. To combine these into one case, for a node x 2 V , define the capacity u(x) of x to be the minimum of values P y: x;y)2E u(x; y) and P y: y;x)2E u(y; x) and define Delta(N ) X (u(x) x 2 V Gamma fs; s 0 g) It was shown in [8, 18] that the number q of big iterations of the blocking flow method does not exceed 2 p Delta(N ) In particular, if all arc capacities are ones, then q = O(m 1=2 ) while if all node capacities u(x) x 6= s; s 0 , are ones, then q = O(n 1=2 ) the latter generalizes the case of networks ....

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S. Even and R. E. Tarjan. Network Flow and Testing Graph Connectivity. SIAM J. Comput., 4:507-- 518, 1975.

....we would like to achieve f(n) O(1) with linear space. 1.1 Previous Results on Path and Connectivity Queries In this section, we overview previous results on k path and k connectivity queries. First, we consider algorithms that do not exploit preprocessing. Using network flow techniques [17], a k path query can be answered in O(m p n) time for arbitrary k, and in O(n m) time for any fixed k. Regarding planar graphs, it has been recently shown that a k path query can be performed in O(n) time for any k [49] Faster query time can be achieved if preprocessing is allowed. For k = ....

S. Even and R. E. Tarjan. Network flow and testing graph connectivity. SIAM J. Comput., 4:507--518, 1975.

....minimum degree of G is at least k, then there exist k edge disjoint s t paths, each of length O(n k) We view G as a directed graph by replacing each undirected edge by two oppositely oriented directed edges. Our proof of the first theorem is based on a maximum flow algorithm of Even and Tarjan [2]; for our purposes, we need only consider its global structure. The algorithm of [2] runs in phases numbered 1, 2, In phase d, a residual graph is maintained as a layered directed graph: the endpoints of each edge lie either in the same layer or in adjacent layers, and the distance from s ....

....of length O(n k) We view G as a directed graph by replacing each undirected edge by two oppositely oriented directed edges. Our proof of the first theorem is based on a maximum flow algorithm of Even and Tarjan [2] for our purposes, we need only consider its global structure. The algorithm of [2] runs in phases numbered 1, 2, In phase d, a residual graph is maintained as a layered directed graph: the endpoints of each edge lie either in the same layer or in adjacent layers, and the distance from s to t is equal to d. The algorithm finds augmenting s t paths of length d in this ....

S. Even, R. Tarjan, "Network flow and testing graph connectivity," SIAM J. Computing, 4(1975), pp. 507--518.

.... and equation (b) can be used to find an underapproximation for the F i values when i b (n 1) The number of minimum cardinality cutsets for undirected graphs can be calculated in polynomial time using an algorithm by Ball and Provan [4] This algorithm makes use of results of Even and Tarjan [17] and Bixby [5] The number of spanning trees can be efficiently counted using a result due to Kirchhoff [29] A more recent paper by Brooks, Smith, Stone and Tutte [7] also presents this theory in a computational form. Knowing these values gives the following bounds for all terminal reliability: ....

S. Even and R.E. Tarjan, "Network Flow and Testing Graph Connectivity", SIAM Journal on Computing, Vol. 4, pp. 507-518, 1975.

....this paper we consider the undirected maximum flow problem in a network with unit capacities and no parallel edges. Until recently, the fastest known way to solve this problem was using a reduction to the directed problem with unit capacities and no parallel arcs. Karzanov [8] and Even and Tarjan [2] have shown that Dinitz s blocking flow algorithm [1] applied to the directed problem, runs in O(min(m 1 2 , n 2 3 )m) time. Here n and m are the number of input vertices and edges, respectively. Recently, Karger [6] developed two randomized algorithms for the undirected problem, with ....

....H # can be found in O(m) time. Dinitz s algorithm [1] for finding maximum flows in undirected graphs repeatedly augments the current flow by a blocking flow in the graph induced by the residual arcs on shortest paths from s to t. Based on the following two lemmas, Karzanov [8] and Even and Tarjan [2] have shown that Dinitz s algorithm terminates in min(n 2 3 , m 1 2 ) iterations. Note that these lemmas hold for both directed and undirected flows. Lemma 2.1. Given a network G with no parallel edges and a flow f , the residual flow is at most (2n D f ) 2 [2, 8] Lemma 2.2. Given a ....

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<F4.319e+05> S. Even and R. E. Tarjan,<F3.298e+05> Network flow and testing graph connectivity,<F3.808e+05> SIAM J. Comput., 4 (1975), pp. 507--518.

....this paper we consider the undirected maximum flow problem in a network with unit capacities and no parallel edges. Until recently, the fastest known way to solve this problem was using a reduction to the directed problem with unit capacities and no parallel arcs. Karzanov [7] and Even and Tarjan [2] have shown that Dinitz s blocking flow algorithm [1] applied to the directed problem, runs in O(min(m 1=2 ; n 2=3 )m) time. Here n and m is the number of input vertices and edges, respectively. Recently, Karger [6] developed two randomized algorithms for the undirected problem, with ....

....0 can be found in O(m) time. Dinitz s algorithm [1] for finding maximum flows in undirected graphs repeatedly augments the current flow by a blocking flow in the graph induced by the residual arcs on shortest paths from s to t. Based on the following two lemmas, Karzanov [7] and Even and Tarjan [2] have shown that Dinitz s algorithm terminates in min(n 2=3 ; m 1=2 ) iterations. Lemma 2.1 [2, 7] Given a network G with no parallel edges and an undirected flow f , the residual flow is at most (2n=D f ) 2 . Lemma 2.2 [2, 7] Given a network G a flow f , the residual flow is at most m=D f . ....

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S. Even and R. E. Tarjan. Network Flow and Testing Graph Connectivity. SIAM J. Comput., 4:507--518, 1975.

....and Karp s algorithm time bound is O (m p n) This is the most efficient sequential algorithm known, provided G is not dense. When the graph is dense it is perhaps better to use an algorithm due to Alt, Blum, Mehlhorn and Paul [3] which has complexity O(n 3=2 p m= log n) Even and Tarjan [15] observed the similarity between Hopcroft and Karp s algorithm and Dinic s algorithm [10] 2.3 Kim and Chwa s Parallel Algorithm ffl Authors: Kim and Chwa [27] ffl Date: 1987. ffl Type: parallel. ffl Complexity: O (n log n log log n) ffl Processors: O Gamma n 3 = log n Delta . ....

S. Even and R. E. Tarjan. Network flow and testing graph connectivity. SIAM J. Comput., 4(4):507--518, Dec 1975.

....values. Traditionally, only a special case of this problem has been considered: unit capacity graphs with no parallel edges (called simple graphs) Until recently, the best known algorithm for this special case used the blocking flow method of Dinitz [2] which Karzanov [15] and Even and Tarjan [3] showed runs in O(mminfn 2=3 ; m 1=2 ; vg) time. Here n is the number of nodes, m is the number of edges, and v is the value of the maximum flow. Note that for graphs with no parallel edges m n 2 and for simple graphs v n, so the above bound is O(n 8=3 ) In an exciting new result, ....

....best for practical purposes. The first algorithm works for the capacitated case as well, running in O(m nv 3=2 ) time. 1 f (n) O(g(n) if 9c such that f (n) O(g(n)log c n) Source Year Time bound Capacities Directed Deterministic Ford Fulkerson [4] 1956 O(mv) p p p Even Tarjan [3] 1975 O(mminfn 2=3 ; m 1=2 g) p p Karger [13] 1997 O(m 2=3 n 1=3 v) Goldberg Rao [7] 1997 O(mminfn 2=3 ; m 1=2 glogv) p p p Goldberg Rao [8] 1997 O(n p nm) p Karger [14] 1998 O(v p nm) p this paper 1998 O(m nv 3=2 ) p this paper 1998 O(nm 2=3 v 1=6 ) p this ....

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S. Even and R. E. Tarjan. Network Flow and Testing Graph Connectivity. SIAM Journal on Computing, 4:507--518, 1975.

....or on the boundary of C. To analyze the complexity, we note that Phase 1 and Phase 2 are just maximum cardinality matching and maximal matching for bipartite graphs, which can be done in time O(n 2:5 ) either by the O(n 2:5 ) time algorithm of [11] or by the O( p nm) time algorithm of [7], where m = O(n 2 ) is the number of edges in the graphs) In Phase 3, we consider an arbitrary edge e 2 E 0 and the disks (C i and R i ) of radii r=2 and r centered at its endpoints, identify the corresponding good bad triangles, and perform local swappings, in O(n) time per swap. Finally, ....

S. Even and R. E. Tarjan. Network flow and testing graph connectivity. SIAM J. Comput., 4:507--518, 1975.

....in a general graph in his famous paper Paths, trees and flowers . A few years earlier, Ford and Fulkerson [19] in their research on network flow, raised the problem of matchings in bipartite graphs and presented an algorithm. The fastest algorithms today are attributed to Even and Tarjan [15] for maximum bipartite matching and to Micali and Vazirani [44] for general graphs. Both algorithms achieve O( q jV jjEj) time, but, while the Even and Tarjan algorithm is simple to implement, the Micali and Vazirani algorithm is complex and not considered practical. In this section we will ....

S. Even and R.E. Tarjan. Network flow and testing graph connectivity. SIAM J. Computing, 4, 507-518, 1975. REFERENCES 38

....some problem classes are faster by a moderate constant factor. 1. Introduction Maximum unit capacity flow and bipartite matching are two closely related combinatorial problems that have been extensively studied from the point of view of theoretical analysis of algorithms. See e.g. 19] 20, 21] [11], 3] 12] The best bound for the former problem is O(m 1:5 ) 11, 20] and for the latter O( p nm log(n 2 =m) log n) 12] We are aware of three previous computational studies ( 8] 5] and [25] of bipartite matching algorithms and no computational studies of unit capacity flow ....

....Maximum unit capacity flow and bipartite matching are two closely related combinatorial problems that have been extensively studied from the point of view of theoretical analysis of algorithms. See e.g. 19] 20, 21] 11] 3] 12] The best bound for the former problem is O(m 1:5 ) [11, 20] and for the latter O( p nm log(n 2 =m) log n) 12] We are aware of three previous computational studies ( 8] 5] and [25] of bipartite matching algorithms and no computational studies of unit capacity flow algorithms. In this paper we study computational efficiency of bipartite ....

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S. Even and R. E. Tarjan. Network Flow and Testing Graph Connectivity. SIAM J. Comput., 4:507--518, 1975.

....in hardware. 2.2 Maximum Size Matching A desirable algorithm would be one that finds the maximum number of matches between inputs and outputs. This would provide the highest possible instantaneous throughput in a given time slot. There are several algorithms for achieving a maximum size match [8, 11, 17, 28], but these are not suitable for this application for two reasons. First, a maximum size match can take a long time to converge and, second, it can lead to starvation of an input output flow under certain traffic patterns. Referring to Figure 2 we see that even a simple traffic pattern can lead to ....

Even, S., Tarjan, R.E. "Network flow and testing graph connectivity" SIAM J. Comput., 4 (1975), pp.507-518.

....be more specific, the number of distinct equivalence classes for a set of k elements is P k i=1 S k;i , where S k;i is a Stirling number of the second kind. If k = 2, 3, 4, 5, 6, 7, and 8, the number of distinct equivalence classes is 2, 5, 15, 52, 203, 877 and 4140, respectively. Even and Tarjan[8] describe an algorithm to determine graph connectivity. If there are n vertices and m edges, the algorithm takes O( p nm 2 ) time. For this application, however, it would be prudent if the two resulting subproblems were split as evenly as possible. An edge separator of a graph is a set of ....

S. Even and R. E. Tarjan, "Network flow and testing graph connectivity," SIAM J. Comput. 4 (1975) 507-518.

....if neither path contains an inner vertex of the other path. The problem is to find as many pairwise vertex disjoint (s, t) paths as possible. The general vertex disjoint Menger problem, directed or undirected, can be solved by reduction to a bipartite maximum flow problem with unit capacities [3, 5]. When using the famous labeling algorithm [1, 4] this approach yields an O(km) algorithm, where k is the maximum number of (s, t) paths and m is the number of edges. When the algorithm of Malhotra, Kumar, and Maheshwari is used instead, O(n 1 2 m)is achieved, where n is the number of ....

S. Even and R. Tarjan, Network flow and testing graph connectivity, SIAM J. Comput., 4 (1975), pp. 507--518.

....updates of node distances or prices has been critical to obtaining the best running times in practice. Several algorithms for the bipartite matching problem run in O( p nm) time. 1 Hopcroft and Karp [15] first proposed an algorithm that achieves this bound. Karzanov [16] and Even and Tarjan [5] proved that the blocking flow algorithm of Dinitz [4] runs in this time when applied to the bipartite matching problem. Two phase algorithms based on a combination of the push relabel method [13] and the augmenting path method [7] were proposed in [12, 19] Feder and Motwani [6] give a graph ....

....when Gamma max 2 [0; k] and Gamma max 2 [n; n k] The algorithm expends O(km) work during these periods. To study the behavior of the algorithm during the remainder of its execution, we introduce a combinatorial lemma that is a special case of a well known decomposition theorem [7] see also [5]) Lemma 4.2. Any integral pseudoflow f 0 in the residual graph of an integral preflow f in a matching network can be decomposed into cycles and simple paths that are pairwise nodedisjoint except at the endpoints of the paths. Each path takes one of the following forms: ffl from s to t; ffl ....

S. Even and R. E. Tarjan. Network Flow and Testing Graph Connectivity. SIAM J. Comput., 4:507--518, 1975.

....Problem Given: A graph G = V; E) jV j = n and vertices s and t. Problem: Find a maximum set of (resp. k) internally vertex disjoint paths connecting s and t. A maximum number of vertex disjoint paths connecting s and t in undirected graphs can be computed by solving a maximum unit flow problem [2, 3]. If the graph is planar, this yields an O(n 3 2 ) resp. O(kn) algorithm, where k is the number of paths. The approach presented in [32] leads to an algorithm of running time O(n log n) It is based on divide and conquer techniques similar to the methods given in [8] and [24] Recently, ....

S. Even and R. E. Tarjan (1975). Network flow and testing graph connectivity. SIAM J. Comput., 4, 507--518.

....a worst case bound of O(n 1:5 p m= log n) It is well known that bipartite matching is just a special case of the maximum flow problem. In fact, Hopcroft and Karp s algorithm can be seen as a version of Dinic s algorithm for maximum flow [Din70] specialized for unweighted bipartite graphs [ET75]. Experimental results for the maximum flow problem [DM89, AS92a] have shown that Goldberg s algorithm for maximum flow [Gol87, GT88b] is the fastest in practice for a large number of input classes, outperforming Dinic s algorithm (the second fastest) by a large margin. Given these results, it is ....

S. Even and R. E. Tarjan. Network flow and testing graph connectivity. SIAM J. Comput., 4(4):507--518, 1975.

....or on the boundary of C. To analyze the complexity, we note that Phase 1 and Phase 2 are just maximum cardinality matching and maximal matching for bipartite graphs, which can be done in time O(n 2:5 ) either by the O(n 2:5 ) time algorithm of [7] or by the O( p nm) time algorithm of [5], where m = O(n 2 ) is the number of edges in the graphs) In Phase 3, we construct C i and R i in O(1) time, identify good bad triangles and perform local swappings, in O(n) time. Finally, as analyzed before, radius r and circle C are computed in O(n 2 ) time. Theorem 18 Given a complete ....

S. Even and R. E. Tarjan. Network flow and testing graph connectivity. SIAM J. Comput., 4:507--518, 1975.

.... 2 ) min( p n; We give an insertionsonly algorithm for maintaining a (2 ffl) approximation of the minimum vertex cut with amortized insertion time O(n=ffl) 1 Introduction Computing the connectivity of a graph is a fundamental problem that has achieved a lot of attention (see for example [1, 2, 6, 8, 9, 10, 11, 15, 16, 18, 20]) In this paper we study the problem of maintaining the connectivity of the graph during modifications of the graph. Let G = V; E) be an undirected, unweighted multigraph. Two vertices x and y of G are k edge connected if there exist k pairwise edge disjoint paths connecting x and y. A graph G ....

....Algorithm 5.1 A Static Algorithm We give an algorithm that in time O(n 2 min( p n; approximates the vertex connectivity of a graph (not a multigraph) as follows: If bffi=2b, then the algorithm outputs , otherwise, it outputs bffi=2b. Our algorithm uses ideas from [18] Even and Tarjan [8] give an algorithm to compute the minimum vertex cut (a; b) between the nodes a and b in time c(min(m p n; m) for some constant c, which we call the pair algorithm (PA(a,b) The exact minimum vertex cut algorithm [11] makes O( 2 n) calls to PA. Let ffi 2 be bffi=2c. The basic approach of ....

S. Even and R. E. Tarjan, "Network flow and testing graph connectivity", SIAM Journal on Computing, 4, 1975, 507--518.

....path from the source to the sink in the residual graph. The value of a flow is the total flow into the sink. The residual flow value is the difference between the maximum and the current flow values. capacities directed undirected unit O(min(n 2=3 ; m 1=2 )m) Karzanov [35] Even Tarjan [16] integral O (nm) Dinitz [14] Gabow [19] real O (nm) Galil Naamad [20] Table 1: The best bounds known in 1980. Arrows show problem reductions: left arrow means the problem reduces to the problem on the left; double arrow means the problem is equivalent to that on the left. 3 History ....

....(nm) maximum flow algorithm. See also [19] Galil and Naamad [20] developed data structures to speed up flow computations and obtained an O (nm) algorithm. We use E instead of O when stating expected time bounds. For the unit capacity case, Karzanov [35] and independently Even and Tarjan [16] show that Dinitz blocking flow algorithm solves the maximum flow problem in O(m 3=2 ) time on multigraphs and in O(min(n 2=3 ; m 1=2 )m) time on simple graphs. Table 1 shows the state of the maximum flow time bounds in 1980. Note the gap between the unit capacity and the other cases. The ....

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S. Even and R. E. Tarjan. Network Flow and Testing Graph Connectivity. SIAM J. Comput., 4:507--518, 1975.

....we speed up the running times of the bipartite matching algorithm (Hopcroft Karp [HK] or Dinic [Din] by a factor of k. This is achieved by modifying these algorithms to take advantage of the new representation. Similar speed ups are attained for the vertex connectivity algorithms of Even Tarjan [ET] and Galil [Gal] the edge connectivity algorithm of Matula [Mat] and Mansour Schieber [MS] and the all pairs shortest paths algorithm. Also, there has been some recent work on vertex connectivity [CT, NI] which involves the computation of socalled sparse certificates for k connectivity. Our ....

....per unit of flow, an O(Km ) bound per phase, and an O(nm ) bound for the entire edge connectivity algorithm. 2 4.4 Vertex Connectivity Consider now the problem of computing the vertex connectivity of a graph. Let c G denote the connectivity of the graph G. The algorithm of Even Tarjan [ET] computes vertex connectivity in time O(c G n 1:5 m) We improve this result as follows. Note that there is a significant improvement in the running time only for very large value of connectivity, at least n 1 Gammao(1) Theorem 4.4 The modified version of the Even Tarjan algorithm can ....

S. Even and R. E. Tarjan, "Network flow and testing graph connectivity," SIAM Journal on Computing, vol. 4 (1975), pp. 507-518.

.... k. Moreover the algorithm of Section 3 is a realization of their algorithm. Hopcroft and Karp gave an implementation of their algorithm only for the case of bipartite graphs, in which case their algorithm is essentially equivalent to the maximum flow algorithm of Dinic [5] see Even and Tarjan [6], and Tarjan [16] Micali and Vazirani [11] gave an algorithm for the maximum cardinality matching problem for general graphs. Their algorithm can be summarized as follows: Given a matching M , find a maximal set of disjoint minimum length augmenting paths relative to M , say 5(M) which may ....

S. Even and R. E. Tarjan, Network flows and testing graphs for connectivity, SIAM J. Computing, 4 (1975), 100-112.

....1 Introduction Graph connectivity has become a widely studied component of graph theory. To date, most of the work in this area has dealt with determining if an undirected graph contains at least two, three, or more vertex or edge disjoint paths between every pair of vertices (see, e.g. [5, 7, 8, 11, 12, 14, 15, 17, 20, 21]) or if a directed graph possesses analogous properties (see, e.g. 5, 16] This paper investigates a different sort of connectivity problem: determining if a graph has at most some number of distinct paths between every pair of vertices. In particular, we say a directed graph is singly ....

S. Even and R. E. Tarjan. Network flow and testing graph connectivity. SIAM Journal on Computing, 4(4):507--18, 1975.

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S. Even and R. E. Tarjan. Network flow and testing graph connectivity. SIAM J. Comput., 4:507--518, 1975.

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S. Even and R.E. Tarjan. Network flow and testing graph connectivity. SIAM J. Comput., 4:507--518, 1975.

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Even, S., & Tarjan, R.E. (1975), `Network flow and testing graph connectivity,' SIAM J. Computing 4, 507--518.

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S. Even and R. E. Tarjan, Network Flow and Testing Graph Connectivity, Siam Journal on Computing, 4, 1975, 507-518.

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S. Even and R.E. Tarjan. "Network flow and testing graph connectivity". SIAM Journal on Computing, 4:507--518, 1975.

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S. Even and R. E. Tarjan, Network flow and testing graph connectivity, SIAM Journal on Computing, 4 (1975), pp. 507--518.

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S. Even and E. Tarjan. Network flow and testing graph connectivity. SIAM J. Comput., 4:507--518, 1975.

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S. Even and R. E. Tarjan. Network flow and testing graph connectivity. SIAM Journal on Computing, 4:507--518, 1975.

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S. Even and R. E. Tarjan. Network flow and testing graph connectivity. SIAM Journal on Computing, 4:507--518, 1975.

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Even, S.; Tarjan, R.E. "Network flow and testing graph connectivity", SIAM J. Comput., 4 (1975), pp.507-518.

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S. Even, and R. E. Tarjan, "Network flow and testing graph connectivity," SIAM J. Computing, vol. 4, 1975, pp. 507-518.

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Even, S.; Tarjan, R.E. "Network flow and testing graph connectivity", SIAM J. Comput., 4 (1975), pp.507-518.

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