Helmut Alt, Norbert Blum, Kurt Mehlhorn, and Markus Paul. Computing a maximum cardinality matching in a bipartite graph in time o(n  Home/Search   Document Not in Database   Summary   Related Articles   Check

....case running time is ( m c c n O # 0 1 which is bounded by ( m n O # . We want to mention that there are implementations of the basic scheme of Ford and Fulkerson which achieve a running time of ( m n O # by using an other strategy to find augmenting paths, see for example [7] or [1]. However, these algorithms are designed for computing maximum cardinality matchings, and if G contains a large but no p perfect matching then these algorithms may take a long time to discover this. Another reason for choosing the BFS strategy is that it can be adapted easily to the dynamic ....

Alt, H., Blum, N., Mehlhorn, K., Paul, M.: Computing a maximum cardinality matching in a bipartite graph in time ( ) n m n O log 5 . 1 . Inform.Processing Letters, 37(4):237-240, (1991).

....the total worst case running time is o(n (c 1 c o) m) which is bounded by O(n.m) We want to mention that there are implementations of the basic scheme of Ford and Fulkerson which achieve a running time of O(n. m) by using an other strategy to find augmenting paths, see for example [7] or [1]. However, these algorithms are designed for computing maximum cardinality matchings, and if G contains a large but no p perfect matching then these algorithms may take a long time to discover this. Another reason for choosing the BFS strategy is that it can be adapted easily to the dynamic ....

Alt, H., Blum, N., Mehlhorn, K., Paul, M.: Computing a maximum cardinality matching in a bipartite graph in time O(n 1.5 m/J--7 ). Inform. Processing Letters, 37(4):237-240, (1991).

....makespan for graphs in which the sum of the task lengths is bounded by a polynomial in n and the communication delays are bounded by a constant. Bodlaender and Fellows [3] showed that constructing a minimum length schedule for an arbitrary graph on k processors without communication delays is a W [2] hard problem (W [2] is the second class of the W hierarchy for parametrised problems, that was introduced by Downey and Fellows [8] Consequently, it is unlikely that an algorithm exists with an O(n ) running time that constructs minimum length schedules on k processors for some constant c ....

....in which the sum of the task lengths is bounded by a polynomial in n and the communication delays are bounded by a constant. Bodlaender and Fellows [3] showed that constructing a minimum length schedule for an arbitrary graph on k processors without communication delays is a W [2] hard problem (W [2] is the second class of the W hierarchy for parametrised problems, that was introduced by Downey and Fellows [8] Consequently, it is unlikely that an algorithm exists with an O(n ) running time that constructs minimum length schedules on k processors for some constant c independent of k. In ....

[Article contains additional citation context not shown here]

H. Alt, N. Blum, K. Mehlhorn and M. Paul. Computing a maximum cardinality matching in a bipartite graph in time O(n 1:5 m= log n). Information Processing Letters, 37(4):237--240, February 1991.

.... each of the n possible vertex degrees, the greedy procedure can be made to run in O(m n) time and storage for a given graph with n vertices and m edges [14] The O(m n) running time is asymptotically faster than the fastest known maximum matching algorithm for general graphs or bipartite graphs [1, 3, 5, 6, 7, 8, 9, 11]. The greedy procedure s success on some graphs, O(m n) time and storage requirements, low overhead, and simplicity motivate the investigation of its performance. The matching found by the greedy procedure may depend on how ties are broken. Let #(G) be the size of the smallest matching that can ....

H. Alt, N. Blum, K. Mehlhorn, and M. Paul. Computing a maximum cardinality matching in a bipartite graph in time O(n 1.5 m/ log n). Information Processing Letters, 37:237--240, February 1991.

....1;2;3 ; C 4 ) free graphs generalizing the result of Mosca. In the considered class itself, the stable set problem is obviously polynomially solvable due to a maximum matching algorithm that solves both the maximum matching and the maximum stable set problem for bipartite graphs. Due to result in [3], the both problems can be solved in general bipartite graphs in time n 1:5 q m= log n. We believe that the obtained characterization can be used to improve the power of the polynomial, when restricted to the class under consideration. A linear time algorithm for the maximum matching problem in ....

H. Alt, N. Blum, K. Mehlhorn and M. Paul, Computing a maximum cardinality matching in bipartite graphs in time n 1:5 q m= log n, Information processing Letters, 37 (1991) 237-240.

....done in time linear to the representation size [34, Thm. IV.6.1] i.e. in time O(n m) where n is the number of vertices and m is the number of edges of the graph. A maximal matching in a bipartite graph is obtainable in time O( p nm) 34, Thm. IV.9. 10] for more sophisticated algorithms, see [4, 19]) We make use of the following notation for a graph G = V; E) Writing G X means that we delete vertex X and all its incident edges from G. By NX we denote the set of neighbors of X in G. By X we denote the degree of vertex X, that is, jNXj. A graph is called r regular if every vertex has degree ....

H. Alt, N. Blum, K. Mehlhorn, and M. Paul. Computing a maximum cardinality matching in a bipartite graph in time o(n 1:5 q m= log n). Inform. Process. Lett., 37:237-240, 1991.

....the asymptotic bound O(q p 1 2 ) for Hopcroft and Karp s algorithm, 9] Hence we can state the following. Theorem 2 Let G be a bipartite graph with n = U(G) and # = E(G) If k = W U is fixed, then we can find a maximum matching of G in time O(# n 1 2 ) Alt et al. [2]) stated a matching algorithm with running time O(p 3 2 # q log p) which improves Hopcroft and Karp s algorithm for dense graphs. Consequently, applying the latter algorithm improves the running times of subsequently stated algorithms if formulas with dense formula graphs are considered. 5 ....

H. Alt, N. Blum, K. Mehlhorn, and M. Paul. Computing a maximum cardinality matching in a bipartite graph in time O(n 1.5 # m/ log n). Information Processing Letters, 37(4):237--240, 1991.

....of transitive orientation. The maximum clique and chromatic number problems on cocomparability graphs seem to be harder. To the best of our knowledge the complexity is dominated by finding a maximum matching in a bipartite graph. The time needed to solve this problem is almost O(n 2:5 ) see [ABMP]) and O(n 3 ) in the weighted case (see [PaSt] In Section 2 we give some definitions and replace graph terminology by order terminology that proves to be more convenient in designing our algorithms. We assume the vertices of the trapezoid graph to have some weights. To compute maximum ....

H. Alt, N. Blum, K. Mehlhorn and M. Paul, Computing a maximum cardinality matching in a bipartite graph in time O(n 1:5 (m= log n) 0:5 ), Inf. Proc. Letters 37 (1991), 237--240.

.... can be assigned to that leaf (i.e. if all the restrictions and the constraints that are on the path that starts at the root of the tree and ends to the leaf do not lead to a contradiction when they are applied to the variables of the tuple) A bipartite maximum cardinality matching algorithm [ALT, BLUM, MEHLHORN PAUL 91] and a pruning algorithm [COSTA 94] are then applied. The result of the pruning algorithm consist of 2 actions. The first action corresponds to enforce a given arc by stating all the constraints and restrictions that are on the path that lead to the leaf associated to the arc. The second action ....

H. Alt, N. Blum, K. Mehlhorn and M. Paul. Computing a maximum cardinality matching in a bipartite graph in time ( ) n m n log / 5 . 1 O . Information Processing Letters 37, 237-240, (1991).

....of transitive orientation. The maximum clique and chromatic number problems on cocomparability graphs seem to be harder. To the best of our knowledge the complexity is dominated by finding a maximum matching in a bipartite graph. The time needed to solve this problem is almost O(n 2:5 ) see [ABMP]) and O(n 3 ) in the weighted case (see [PaSt] In Section 2 we give some definitions and replace graph terminology by order terminology that proves to be more convenient in designing our algorithms. We assume the vertices of the trapezoid graph to have some weights. To compute maximum ....

H. Alt, N. Blum, K. Mehlhorn and M. Paul, Computing a maximum cardinality matching in a bipartite graph in time O(n

....be found with a depth rst search in O(n 2 ) time and at most n augmentations along alternating path are possible. With the Hopcroft and Karp algorithm [20] see also Tarjan s monograph [29] this can be improved to O(n 5=2 ) A small improvement to O(n 5=2 = p log n) has been obtained in [2]. If one only wants to compute the width of the poset rather than nd the corresponding antichain or chain cover, there is a randomized algorithm which has matrix multiplication rather than matching as its bottleneck step [7] A greedy chain decomposition of an order P = X; is a chain cover ....

H. Alt, N. Blum, K. Mehlhorn and M. Paul. Computing a maximum cardinality matching in a bipartite graph in time O(n 1:5 (m= log n) 0:5 ). Inf. Proc. Letters, 37, 237-240, 1991.

....in time linear to the representation size [30, Thm. IV.6.1] i.e. in time O(n m) where n is the number of vertices and m is the number of edges of the graph. A maximal matching in a bipartite graph is obtainable in time O( p nm) 30, Thm. IV.9. 10] for more sophisticated algorithms, see [2, 15]) We make use of the following notation for a graph G = V; E) Writing G X means that we delete vertex X and all its incident edges from G. By NX we denote the set of neighbors of X in G. By X we denote the degree of vertex X, that is, jNXj. A graph is called r regular if every vertex has ....

H. Alt, N. Blum, K. Mehlhorn, and M. Paul. Computing a maximum cardinality matching in a bipartite graph in time o(n 1:5 q m= log n). Information Processing Letters, 37:237-240, 1991.

.... to check, in polynomial time, the existence of a such subgraph in a given (general) digraph and find one, if it exists, see [31, 33, 34] using any polynomial maximum matching algorithm (for bipartite graphs) In particular, we can do it in time O(n 2:5 = p log n) applying the algorithm from [2]. J. Bang Jensen and G. Gutin [9] used this idea showing the following: Theorem 5.8 An extended LSD has a hamiltonian cycle if and only if it is strong and has a cycle factor. Given a spanning cycle subgraph of an extended LSD D, a hamiltonian cycle of D can be found in time O(n 2 ) where n is ....

H. Alt, N. Blum, K. Melhorn and M. Paul, Computing of maximum cardinality matching in a bipartite graph in time O(n 1:5 p m= log n). Inf. Proc. Letters 37 (1991) 237-240.

....all tables we use p to represent the number of processors used by parallel algorithms and U to denote the greatest absolute value among edge cost values. Sequential Algorithms Date Authors Ref. Complexity Primitive O(mn) 1973 Hopcroft and Karp [26] O(mn 1=2 ) 1991 Alt, Blum, Mehlhorn and Paul [3] O(n 3=2 p m= log n) Parallel Algorithms Date Authors Ref. Complexity Processors 1987 Kim and Chwa [27] O(n log n log log n) O(n 3 = log n) 1988 Goldberg, Plotkin and Vaidya [24] O(n 2=3 log 3 n) O(n 3 = log n) 1992 Goldberg, Plotkin, Shmoys and Tardos [23] O(m 1=2 log 3 n) O(m 3 ....

....path, the number of phases is reduced to O ( p n) Thus, Hopcroft 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 ....

H. Alt, N. Blum, K. Mehlhorn, and M. Paul. Computing a maximum cardinality matching in a bipartite graph in time o(n 1:5 p m= log n). Inf. Proc. Letters, 37:237--240, Feb 1991.

....setubal dcc.unicamp.br Abstract We present experimental results for four bipartite matching algorithms on 11 classes of graphs. The algorithms are depth first search (dfs) breadth first search (bfs) the push relabel algorithm [GT88b] and the algorithm by Alt, Blum, Mehlhorn, and Paul (abmp) [ABMP91]. dfs was thought to be a good choice for bipartite matching but our results show that, depending on the input graph, it can have very poor performance. bfs on the other hand has generally very good performance. The results also show that the abmp and push relabel implementations are similar in ....

.... From a computational complexity point of view, in the case of sparse graphs, the best sequential algorithm for finding a maximum matching is by Hopcroft and Karp [HK73] which achieves a worst case running time of O( p nm) For dense graphs the best algorithm is by Alt, Blum, Mehlhorn, and Paul [ABMP91], having a worst case bound of O(n 1:5 p m= log n) This work was supported in part by grants from Brazilian Research Agencies FAPESP and CNPq. Bipartite matching is an important problem from a practical point of view, since it has many applications [AMO93] Therefore it is important to ....

[Article contains additional citation context not shown here]

H. Alt, N. Blum, K. Mehlhorn, and M. Paul. Computing a maximum cardinality matching in a bipartite graph in time O(n 1:5 p m= log n). Inform. Process. Lett., 37:237--240, 1991.

.... check, in polynomial time, the existence of a such subgraph in a given (general) digraph and find one, if it exists, see [3, 53, 54, 55] using any polynomial maximum matching algorithm (for bipartite graphs) In particular, we can do it in time O(n 2:5 = p log n) applying the algorithm from [2]. Theorem 7.8 [14] An extended locally semicomplete digraph is hamiltonian if and only if it is strongly connected and has a cycle factor. Given a spanning cycle subgraph of an extended locally semicomplete digraph D, a hamiltonian cycle of D can be found in time O(n 2 ) where n is the number ....

H. Alt, N. Blum, K. Melhorn and M. Paul, Computing of maximum cardinality matching in a bipartite graph in time O(n 1:5 p m= log n). Inf. Proc. Letters 37 (1991) 237-240.

....the computational expenditure f keeps decreasing, which forces us to reconsider the analysis and rederive the bounds on T (n) For example, the complexity of bipartite matching was improved from O(rs 1.5 ) see Hopcroft and Karp [13] to O( r s) 1. 5 p rs log s) in 1990 by Alt et al. [2] and further to O(rs 1.5 log s) in 1991 by Feder and Motwani [8] r # s are the sizes of the two vertex sets) Since these new bounds do not satisfy the hypotheses of Theorems 1.1 and 1.2, one is forced to reanalyze the rooted subtree isomorphism algorithms from scratch. This rework can be ....

H. Alt, N. Blum, K. Mehlhorn, and M. Paul, Computing a maximum cardinality matching in a bipartite graph in time O(n 1.5 p m/logn), Inform. Process. Lett., 37 (1991), pp. 237-- 240.

.... each of the n possible vertex degrees, the greedy procedure can be made to run in O(m n) time and storage for a given graph with n vertices and m edges [14] The O(m n) running time is asymptotically faster than the fastest known maximum matching algorithm for general graphs or bipartite graphs [1, 3, 5, 6, 7, 8, 9, 11]. The greedy procedure s success on some graphs, O(m n) time and storage requirements, low overhead, and simplicity motivate the investigation of its performance. The matching found by the greedy procedure may depend on how ties are broken. Let fl(G) be the size of the smallest matching that can ....

H. Alt, N. Blum, K. Mehlhorn, and M. Paul. Computing a maximum cardinality matching in a bipartite graph in time O(n 1:5 p m= log n). Information Processing Letters, 37:237--240, February 1991.

....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 ....

....for all vertices of Y . See [14, 22] for more details. 3.6. Augment Relabel The augment relabel algorithm (see e.g. 1] uses depth first search in combination with the relabel operation to find augmenting paths. In the context of the bipartite matching problem, Alt, Blum, Melhorn and Paul [3] used a variant of the method in combination with certain word operations to obtain improved bounds for dense graphs. Our implementation of this method, ar, does not use the word operations. The ar algorithm can be thought of as a cross between the simple search and the push relabel algorithms, ....

[Article contains additional citation context not shown here]

H. Alt, N. Blum, K. Mehlhorn, and M. Paul. Computing a Maximum Cardinality Matching in a Bipartite Graph in time O(n 1:5 p m= log n). Information Processing Let., 37:237--240, 1991.

....of transitive orientation. The maximum clique and chromatic number problems on cocomparability graphs seem to be harder. To the best of our knowledge the complexity is dominated by finding a maximum matching in a bipartite graph. The time needed to solve this problem is almost O(n 2:5 ) see [ABMP]) and O(n 3 ) in the weighted case (see [PaSt] In Section 2 we give some definitions and replace graph terminology by order terminology that proves to be more convenient in designing our algorithms. We assume the vertices of the trapezoid graph to have some weights. To compute maximum ....

H. Alt, N. Blum, K. Mehlhorn and M. Paul, Computing a maximum cardinality matching in a bipartite graph in time O(n 1:5 (m= log n) 0:5 ), Inf. Proc. Letters 37 (1991), 237--240.

....makespan for graphs in which the sum of the task lengths is bounded by a polynomial in n and the communication delays are bounded by a constant. Bodlaender and Fellows [3] showed that constructing a minimum length schedule for an arbitrary graph on k processors without communication delays is a W [2] hard problem (W [2] is the second class of the W hierarchy for parametrised problems, that was introduced by Downey and Fellows [8] Consequently, it is unlikely that an algorithm exists with an O(n c ) running time that constructs minimum length schedules on k processors for some constant c ....

....in which the sum of the task lengths is bounded by a polynomial in n and the communication delays are bounded by a constant. Bodlaender and Fellows [3] showed that constructing a minimum length schedule for an arbitrary graph on k processors without communication delays is a W [2] hard problem (W [2] is the second class of the W hierarchy for parametrised problems, that was introduced by Downey and Fellows [8] Consequently, it is unlikely that an algorithm exists with an O(n c ) running time that constructs minimum length schedules on k processors for some constant c independent of k. In ....

[Article contains additional citation context not shown here]

H. Alt, N. Blum, K. Mehlhorn and M. Paul. Computing a maximum cardinality matching in a bipartite graph in time O(n 1:5 p m= log n). Information Processing Letters, 37(4):237--240, February 1991.

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Helmut Alt, Norbert Blum, Kurt Mehlhorn, and Markus Paul. Computing a maximum cardinality matching in a bipartite graph in time o(n

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H. Alt, N. Blum, K. Melhorn and M. Paul, Computing of maximum cardinality matching in a bipartite graph in time O(n 1.5 # m/ log n). Inf. Proc. Letters 37 (1991) 237-240.

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H. Alt, N. Blum, K. Melhorn and M. Paul, Computing of maximum cardinality matching in a bipartite graph in time O(n 1.5 # m/ log n). Inf. Proc. Letters 37 (1991) 237-240.

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H. Alt, N. Blum, K. Melhorn and M. Paul, Computing of maximum cardinality matching in a bipartite graph in time O(n 1:5 q m= log n). Inf. Proc. Letters 37 (1991) 237-240.

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