Implementing minimum cycle basis algorithms

Mehlhorn, Kurt; Michail, Dimitrios

doi:10.1145/1187436.1216582

Cited by 25 publications

(18 citation statements)

References 5 publications

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“…The cyclic structural building blocks, i.e. the n-cycles, are found from the minimal cycle basis of the contact network (Horton, 1987;Mehlhorn and Michail, 2006). The identification of the linear structural building blocks of force chains require additional information, namely, the contact forces as well as their orientation.…”

Section: Materials Characterisation Using Complex Networkmentioning

confidence: 99%

“…There are a number of established algorithms for obtaining a minimal cycle basis for a complex network. We use the algorithm of Horton (1987) in conjunction with a faster variant presented by Mehlhorn and Michail (2006). We can treat ncycles as structures in both the abstract domain of the complex network or in the physical domain.…”

Section: Appendix a Detection Of Linear And Cyclic Building Blocksmentioning

confidence: 99%

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Force chain and contact cycle evolution in a dense granular material under shallow penetration

Tordesillas

Steer

Walker

2014

Nonlin. Processes Geophys.

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Abstract. The mechanical response of a dense granular material submitted to indentation by a rigid flat punch is examined. The resultant deformation is viewed as a process of self-organisation. Four aspects of the mechanical response (i.e. indentation resistance, failure, Reynolds' dilatancy, the undeforming "dead zone") are explored with respect to the linear and cyclic structural building blocks of granular media self-organisation: force chains and contact network cycles. Formation and breaking of 3-cycle contacts preferentially occur around and close to the punch uncovering a "dilation zone". This zone encapsulates (i) most of the indentation resistance and is populated by force chains consisting of six or more particles, (ii) all buckling force chains, and (iii) a central, near-triangular, undeforming cluster of grains beneath the punch face. Force chain buckling is confined to the zone's outer regions, beneath the corners and to the sides of the punch where surface material heave forms. Grain rearrangements here involve the creation of 6-, 7-, and 8-cycles -in contrast with Reynolds' postulated cubic packing rearrangements (i.e. 3-cycles opening up to form 4-cycles). In between these intensely dilatant regions lies a compacted triangular grain cluster which moves in near-rigid body with the punch when jammed, but this dead zone unjams and deforms in the failure regimes when adjacent force chains buckle. The long force chains preferentially percolate from the punch face, through the dead zone, fanning downwards and outwards into the material.

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Section: Materials Characterisation Using Complex Networkmentioning

confidence: 99%

Section: Appendix a Detection Of Linear And Cyclic Building Blocksmentioning

confidence: 99%

Force chain and contact cycle evolution in a dense granular material under shallow penetration

Tordesillas

Steer

Walker

2014

Nonlin. Processes Geophys.

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“…We restrict attention to computing minimum undirected bases. Implementations of cycle basis algorithms are described in Gleiss (2001a);Huber (2003); Bauer (2004); Mehlhorn and Michail (2006).…”

Section: Algorithm Engineeringmentioning

confidence: 99%

“…Experiments in Mehlhorn and Michail (2006) with random graphs suggest, that the use of fast matrix multiplication is not necessary even for medium to large instances. The reason is that the cycles computation part of the algorithm dominates the running time, although in theory it is the other way around.…”

Section: Algorithm Engineeringmentioning

confidence: 99%

Cycle bases in graphs characterization, algorithms, complexity, and applications

Kavitha

Liebchen

Mehlhorn

et al. 2009

Computer Science Review

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146

140

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Cycles in graphs play an important role in many applications, e.g., analysis of electrical networks, analysis of chemical and biological pathways, periodic scheduling, and graph drawing. From a mathematical point of view, cycles in graphs have a rich structure. Cycle bases are a compact description of the set of all cycles of a graph. In this paper, we survey results on cycle bases and prove new ones. We introduce different kinds of cycle bases, characterize them in terms of their cycle matrix, and prove structural results about them, in particular, a-priori length bounds. We give polynomial algorithms for the minimum cycle basis problem for some of the classes and prove APX -hardness for others. We also discuss three applications and show that they require different kinds of cycle bases.

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“…A minimum cycle basis (MCB) is a cycle basis of minimum weight; in general, it is not induced by a spanning tree. MCBs can be computed efficiently: the currently best algorithm (Kavitha et al, 2004) runs in time O(m 2 n) using O(m 2 + mn) space; the practical performance seems to be much better (Mehlhorn and Michail, 2006).…”

Section: Introductionmentioning

confidence: 99%