B. Monien, How to find long paths efficiently, Annals of Discrete Mathematics 25 (1984), 239--254.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....k vertices from G which contain a subgraph isomorphic to H [4] As we shall see, the existential and constructive problems do not necessarily have the same complexity. Previous work on fixed subgraph isomorphism has concentrated on recognizing specific families of subgraphs, such as small paths [1, 3, 11], cycles [4, 10, 11, 15, 16] and cliques [12] A practical algorithm for subgraph isomorphism, without analysis, is presented by [19] In Section 2, we consider the complexity of recognizing paths of length k. This leads to a more general algorithm for fixed subgraph isomorphism, presented in ....

....which contain a subgraph isomorphic to H [4] As we shall see, the existential and constructive problems do not necessarily have the same complexity. Previous work on fixed subgraph isomorphism has concentrated on recognizing specific families of subgraphs, such as small paths [1, 3, 11] cycles [4, 10, 11, 15, 16], and cliques [12] A practical algorithm for subgraph isomorphism, without analysis, is presented by [19] In Section 2, we consider the complexity of recognizing paths of length k. This leads to a more general algorithm for fixed subgraph isomorphism, presented in Section 3, whose time ....

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B. Monien. How to find long paths efficiently. Annals of Discrete Mathematics, 25:239-- 254, 1985.

....can be found in O(V ) time. This in fact holds not only for planar graphs but for any non trivial minor closed family of graphs. Another contribution of this paper is an O(V ) algorithm for counting the number of C k s, for k 7, in a graph G = V; E) 2 Comparison with previous works Monien [Mon85] obtained, for any fixed k 3, an O(VE) algorithm for finding C k s in a directed or undirected graph G = V; E) In a previous work [AYZ94] we showed, using the color coding method, that a C k , for any fixed k 3, if one exists, can also be found in O(V ) expected time or in O(V log V ....

....or in O(V log V ) worst case time. Here we show that for any k 7 the number of C k s in a graph can be counted in O(V ) time. The counting method used here yields, in particular, a way of finding C k s for k 7, in O(V ) worst case time. 3 Finding cycles in sparse graphs Monien [Mon85] obtained his O(VE) algorithm by the use of representative collections . Such collections are also used by our algorithms. In the sequel, a p set is a set of size p. Definition 3.1 ( Mon85] Let F be a collection of p sets. A sub collection F F is q representative for F , if for every q set ....

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B. Monien. How to find long paths efficiently. Annals of Discrete Mathematics, 25:239--254, 1985. 12

....4.1, this information is represented by sets of colors. A naive approach would be to store directly the possible sets of nodes on the current paths. However, this requires storing O(jV j k ) of such sets instead of only O(2 k ) color sets. Prior to the color coding technique of [2] Monien [17] proposed an algorithm for finding a simple path of length k in time O(k jEj) He showed that for the dynamic programming step, much fewer sets of nodes have to be stored than with the naive approach. This algorithm can also be modified to solve the separation problem for node and edge search on ....

....: k, let F (v; i) denote the family of node sets S so that there is a path ending in v that is walked in i Gamma 1 steps and visits exactly the nodes in S (so jSj i) The families F (v; i) can be constructed inductively for successively larger values of i, starting with i = 1. Analogously to [17], we only store a certain representative subfamily F 0 (v; i) of F (v; i) that has the following property: For any set A of k Gamma i or fewer nodes and a set S in F (v; i) disjoint to A of maximum weight, there exists some S 0 in F 0 (v; i) that is also disjoint to A and has the same ....

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B. Monien (1985), How to find longs paths efficiently. In: Analysis and Design of Algorithms for Combinatorial Problems, eds. G. Ausiello and M. Lucertini. Annals of Discrete Mathematics 25, 239--254.

....= V; E) contain a directed cycle of an even length , for example, is not known to be in P, nor is it known to be NP complete (see [9] Though we do not shed any new light on the directed versions of the problems, we obtain surprisingly fast algorithms for some of the undirected versions. Monien [7] presented an O(VE) algorithm for finding all pairs of vertices that are connected by paths of length k Gamma 1, where k 2 is a fixed integer. Note that if k is part of the input, the problem is NP Hard) A simple consequence of his algorithm is an O(VE) algorithm for finding a cycle of length ....

....of the shortest path from almost the shortest possible. We also describe a simple O(M(V ) log V ) algorithm for finding a shortest odd cycle (SOLC for short) in an undirected graph G = V; E) and a simple O(VE) algorithm for finding a SOLC in a directed or undirected graph G = V; E) Monien [7] described an O(VE) algorithm for the undirected case. This paper is organized as follows. In Section 2 we present the algorithm for finding fixed length even cycles in undirected graphs. In Section 3 we investigate the combinatorial structure of SELCs. In Section 4 we describe the algorithm for ....

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B. Monien. How to find long paths efficiently. Annals of Discrete Mathematics, 25:239--254, 1985.

....is planar and under other restrictions. However, we are concerned only with the problem of finding simple cycles of a given length. The problem of finding given length simple cycles is one of the basic and natural algorithmic graph problems [Le 90] and was considered by many researchers, e.g. by [Mo 85] RL 85] Ri 86] YZ 94] Do 96] DW 96] Alon, Yuster and Zwick [AYZ 95] presented algorithms for this task which runs either in O(M(n) expected time or O(M(n) log n) worst case time, where M(n) n 2:376 is the complexity of matrix multiplication. Our algorithms for that problem, while ....

....the color coding method of [AYZ 95] with the idea of Grigoriev and Karpinski [GK 87] to utilize the nice properties of prime numbers. GK 87] make use of this properties to solve some matching problems efficiently in parallel (see also [KR 97] We improve an O(nm) worst case time bound of [Mo 85] in many cases) and upon the results of [AYZ 95] Furthermore, our results generalizes a result of [DW 96] We achieve a randomized algorithm which runs in O(maxfm;n log ng) expected time and a deterministic algorithm which runs in O(maxfm log n; n log ng) worst case time. We show that our ....

Monien, B., How to find long paths efficiently, Annals of Discrete Mathematics 25 (1985), pp. 239--254.

....of edge disjoint simple cycles (the points of the cycles being evident by context) of T , which are separating white and black regions. The problem of finding simple cycles is one of the most basic and natural algorithmic graph problems (see [Le 90] and was considered by many researchers e.g. Mo 85] AYZ 95] DW 97a] DW 97b] However, in the present paper we search for a special set of cycles namely the external contour. An (external) contour C (introduced in [DL 95] of a colored triangular embedding T consists of those cycles of S, which are not enclosed by another cycle of S. The ....

Monien, B., How to find long paths efficiently, Annals of Discrete Mathematics 25 (1985), pp. 239--254.

....Bonn, Germany, email: carsten cs.uni bonn.de Institut fur Informatik, Universitat Bonn, Romerstr. 164, D 53117 Bonn, Germany, email: wirtgen cs.uni bonn.de 1 Introduction The literature on detection and finding simple cycles of a given length has a long and varied history (see e.g. Mo 85] RL 85] Ri 86] YZ 94] Do 96] DW 96] DW 97a] DW 97b] However, there are only a few fast algorithms for counting the number of simple cycles of a given fixed length. AYZ 94] showed that for any k 7 the number of simple cycles of length k in a graph can be counted in O(M(n) time, ....

Monien, B., How to find long paths efficiently, Annals of Discrete Mathematics 25 (1985), pp. 239--254.

....problems when supplied with a meaningful, natural parameter yield parametric problems that are f.p. tractable. Examples: Given a graph and k pairs of nodes, are there node disjoint paths between all pairs of nodes [12] Given a graph and an integer k, is there a path of length k in the graph [10, 2] Both problems, and many others like them, have algorithms with running time f(k) Delta n c , where n is the input size and c a constant. In contrast, several other NP complete problems do not seem to be tractable when considered as parametric problems with the natural parameter; examples ....

....the query size and the number of variables as the parameter. Furthermore, we can evaluate such a query in f.p. polynomial time in the input and the output. A special case is the problem of finding simple paths of a specified length k in a graph. This problem was proved f.p. tractable by Monien [10], and an improved algorithm was given in [2] using an elegant color coding (hashing) technique. Our algorithm combines this technique with acyclic query processing techniques. The basic idea is to hash the domain D into a smaller domain (with size bounded by the number of variables) and use the ....

B. Monien, "How to Find Long Paths Efficiently", Ann. Disc. Math., pp. 239-254, 1985.

....graph G = V; E) that contains such a cycle, can be found in either 2 O(k) DeltaV E or 2 O(k) DeltaV expected time or in 2 O(k) DeltaV E log V or 2 O(k) DeltaV log V worst case time. This improves (in many cases) an O(k Delta VE) worst case bound obtained by Monien [Mon85] For k 7 we can count the number of cycles of length k in a graph G = V; E) in O(V ) worst case time. This uses different techniques and will appear elsewhere. In [YZ94] it is shown that for any even k, cycles of length k in undirected graphs that contain them can be found in O(V 2 ) ....

....This process is repeated an expected number of at most k =2 times. A very similar algorithm can be used to find directed simple cycles in directed graphs. 2 One can match the performance of the algorithm described in Theorem 2. 1 using a deterministic algorithm by combining techniques of Monien [Mon85] and Bodlaender [Bod93] The obtained algorithm works in O(k DeltaE) time for directed graphs or in O(k DeltaV ) time for undirected graphs. As this algorithm does not use the color coding method we omit its description. We note that although the O(V ) algorithm of Theorem 2.2 is extremely ....

B. Monien. How to find long paths efficiently. Annals of Discrete Mathematics, 25:239--254, 1985.

....also establishes that finding an approximate solution to Moderate Mean Cycle is at least as hard as finding good approximations to Hamiltonian paths, and our hardness results carry over to the new problem too. Before describing our results in greater detail we review some related work. Monien [21] presented an O(k nm) time algorithm that finds paths of length k in a Hamiltonian graph with n vertices and m edges. Our results are an improvement on this since in polynomialtime Monien s algorithm can only find paths of length O(log n= log log n) Furer and Raghavachari [15] present ....

B. Monien, How to find long paths efficiently, Annals of Discrete Mathematics, 25 (1984), pp. 239--254.

....k is among the most natural and easily stated algorithmic graph problems. If the cycle length k is part of the input then the problem is clearly NP complete as it includes in particular the Hamiltonian cycle problem. For every fixed k however, the problem can be solved in either O(VE) time (Monien [10]) or O(V log V ) 1] where 2:376 is the exponent of matrix multiplication. The main contribution of this paper is a collection of new bounds on the complexity of finding simple cycles of length exactly k, where k 3 is a fixed integer, in a directed or an undirected graph G = V; E) ....

....for planar graphs but for any non trivial minor closed family of graphs. Another contribution of this paper is an O(V ) algorithm for counting the number of C k s, for k 7, in a graph G = V; E) A preliminary version of this work appeared in [2] 2 Comparison with previous works Monien [10] obtained, for any fixed k 3, an O(VE) algorithm for finding C k s in a directed or undirected graph G = V; E) In a previous work [1] we showed, using the color coding method, that a C k , for any fixed k 3, if one exists, can also be found in O(V ) expected time or in O(V log V ) ....

[Article contains additional citation context not shown here]

B. Monien. How to find long paths efficiently. Annals of Discrete Mathematics, 25:239--254, 1985.

....In the proof of Prop. 4.1, this information is represented by sets of colors, instead of storing directly the possible sets of nodes on the current paths. However, this requires storing O(jV j k ) such sets instead of only O(2 k ) color sets. Prior to the color coding technique of [2] Monien [19] proposed an algorithm for finding a simple path of length k in time O(k jEj) He showed that for the dynamic programming step, much fewer sets of nodes have to be stored than with the naive approach. This algorithm can also be modified to solve the separation problem for node and edge search on ....

....: k, let F (v; i) denote the family of node sets S so that there is a path ending in v that is walked in i Gamma 1 steps and visits exactly the nodes in S (so jSj i) The families F (v; i) can be constructed inductively for successively larger values of i, starting with i = 1. Analogously to [19], we only store a certain representative subfamily F 0 (v; i) of F (v; i) that has the following property: For any set A of k Gamma i or fewer nodes and a set S in F (v; i) disjoint to A of maximum weight, there exists some S 0 in F 0 (v; i) that is also disjoint to A and has the same ....

[Article contains additional citation context not shown here]

B. Monien, How to find longs paths efficiently, in: G. Ausiello and M. Lucertini, eds., Analysis and Design of Algorithms for Combinatorial Problems, Annals of Discrete Mathematics, Vol. 25 (North-Holland, Amsterdam, 1985) 239--254.

....of 2 O(log 1 Gammaffl n) for any ffl 0, then NP has a quasi polynomial deterministic time simulation. The hardness results apply even to the special case where the input consists of bounded degree graphs. Before describing our results in greater detail we review some related work. Monien [20] showed that an O(k nm) time algorithm finds paths of length k in a Hamiltonian graph with n vertices and m edges. Our results are an improvement on this since in polynomial time Monien s algorithm can only find paths of length O(log n= log log n) Furer and Raghavachari [15] present ....

B. Monien, How to find long paths efficiently, Annals of Discrete Mathematics, 25 (1984), pp. 239--254.

....true if G is planar and under other restrictions. However, we are concerned only with the problem of finding simple cycles of a given length. The problem of finding given length simple cycles is one of the basic and natural algorithmic graph problems and was considered by many researchers, e. g by [16], 17] 18] Alon, Yuster and Zwick [1] presented a randomized sequential algorithm for this task which uses O(n) expected time. Our randomized parallel algorithm for that problem, while using ideas from [13] is largely a parallel implementation of the algorithm of [1] with results of [4] used ....

Burkhard Monien, How to find long paths efficiently, Annals of Discrete Mathematics 25 (1985), 239--254.

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B. Monien, How to find long paths efficiently, Annals of Discrete Mathematics 25 (1984), 239--254.

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