Implementing Minimum Cycle Basis Algorithms

Mehlhorn, Kurt; Michail, Dimitrios

doi:10.1007/11427186_5

Cited by 21 publications

(32 citation statements)

References 9 publications

(21 reference statements)

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“…The proof is almost identical with the proof of de Pina's [1995] original algorithm, the only difference being that we search for cycles only in set A. A similar proof using A = H can be found in Mehlhorn and Michail [2006]. Let B = {B 1 , .…”

Section: The Algorithmmentioning

confidence: 70%

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Minimum cycle bases

Mehlhorn

Michail

2009

ACM Trans. Algorithms

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We consider the problem of computing exact or approximate minimum cycle bases of an undirected (or directed) graph G with m edges, n vertices and nonnegative edge weights. In this problem, a {0, 1} ({−1, 0, 1}) incidence vector is associated with each cycle and the vector space over F 2 (Q) generated by these vectors is the cycle space of G. A set of cycles is called a cycle basis of G if it forms a basis for its cycle space. A cycle basis where the sum of the weights of the cycles is minimum is called a minimum cycle basis of G. Cycle bases of low weight are useful in a number of contexts, for example, the analysis of electrical networks, structural engineering, chemistry, and surface reconstruction.There exists a set of (mn) cycles which is guaranteed to contain a minimum cycle basis. A minimum basis can be extracted by Gaussian elimination. The resulting algorithm [Horton 1987] was the first polynomial-time algorithm. Faster and more complicated algorithms have been found since then.We present a very simple method for extracting a minimum cycle basis from the candidate set with running time O(m 2 n), which improves the running time for sparse graphs. Furthermore, in the undirected case by using bit-packing we improve the running time also in the case of dense graphs. For undirected graphs we derive an O(m 2 n/ log n + n 2 m) algorithm. For directed graphs we get an O(m 3 n) deterministic and an O(m 2 n) randomized algorithm.Our results improve the running times of both exact and approximate algorithms. Finally, we derive a smaller candidate set with size in (m) ∩ O(mn).

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Section: The Algorithmmentioning

confidence: 70%

“…The technique that we use was originally introduced in Mehlhorn and Michail [2006] where an O(m 2 n 2 )-time algorithm was presented. The algorithm tried to combine the two main different approaches for an MCB computation.…”

Section: Introductionmentioning

confidence: 99%

Minimum cycle bases

Mehlhorn

Michail

2009

ACM Trans. Algorithms

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“…To improve the running time we also used the following heuristics that are suggested in Mehlhorn and Michail [2006].…”

Section: Implementation Heuristicsmentioning

confidence: 99%

An improved heuristic for computing short integral cycle bases

Kavitha

Krishna

2009

ACM J. Exp. Algorithmics

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In this article, we consider the problem of constructing low-weight integral cycle bases in a directed graph G = (V , E) with nonnegative edge weights. It has been shown that low-weight integral cycle bases have applications in the periodic event scheduling problem and cyclic time tabling. However, no polynomial time algorithm is known for computing a minimum weight integral cycle basis in a given directed graph.We give a necessary condition that has to be satisfied by any minimum weight integral cycle basis and guided by this necessary condition, we propose a new heuristic for constructing a low weight integral cycle basis. To the best of our knowledge, this is the first algorithm to construct integral cycle bases that are not necessarily fundamental. We implemented our heuristic and tested it on random graphs and compared the weight of the integral cycle basis computed by our algorithm with the weight of a minimum cycle basis and the weights of integral cycle bases (these are, in fact, fundamental cycle bases) computed by already existing and new heuristics for this problem. Empirical results suggest that our heuristic performs better than the existing heuristics for this problem and in fact, it computes a minimum weight integral cycle basis on these graphs. (In the above comparison, when the weight of the integral cycle basis computed and the weight of a minimum cycle basis turn out to be equal, it immediately implies that we have in fact found a minimum integral cycle basis.) The time needed for our heuristic is typically a little higher compared to the time taken by the other heuristics that compute fundamental cycle bases.

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“…Mehlhorn and Michail [22] also addressed the implementation issues of the above algorithms. The best currently known randomized algorithm, due to Amaldi et al [1],…”

Section: Introductionmentioning

confidence: 96%