Efficient Deterministic Algorithms for Finding a Minimum Cycle Basis in Undirected Graphs

Amaldi, E.; Iuliano, Claudio; Rizzi, Roméo

doi:10.1007/978-3-642-13036-6_30

Cited by 19 publications

(33 citation statements)

References 15 publications

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“…Reconstruct MCB for G MCB, and then apply an independence test to extract an MCB. We will recall the basics of these concepts for completeness, in particular, the construction of Horton set [78] and isometric set [79], and the state-of-the-art method for independence tests [80], while refer interested reader to the review paper [64] for further reading.…”

Section: Computation Of Minimum Length Cycle Basismentioning

confidence: 99%

“…A consistent APSP can be computed by a lexicographic method [83], [64] (see Definition 3 and Lemma 4 in Appendix C). Given a consistent APSP, a cycle C is said to be isometric if for any two vertices u and v on C, the chosen shortest path P uv in APSP is contained in C [78], [79]. It can be further verified that a cycle C is isometric if and only if, for each vertex x ∈ C, there is a unique edge e uv , such that C = C(x, e uv ), with P xu and P xv being in the consistent APSP [79].…”

Section: B Superset Of Mcb: Isometric Circuitsmentioning

confidence: 99%

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Sparse Pose Graph Optimization in Cycle Space

2021

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The state-of-the-art modern pose-graph optimization (PGO) systems are vertex based. In this context, the number of variables might be high, albeit the number of cycles in the graph (loop closures) is relatively low. For sparse problems particularly, the cycle space has a significantly smaller dimension than the number of vertices. By exploiting this observation, in this article, we propose an alternative solution to PGO that directly exploits the cycle space. We characterize the topology of the graph as a cycle matrix, and reparameterize the problem using relative poses, which are further constrained by a cycle basis of the graph. We show that by using a minimum cycle basis, the cycle-based approach has superior convergence properties against its vertex-based counterpart, in terms of convergence speed and convergence to the global minimum. For sparse graphs, our cycle-based approach is also more time efficient than the vertex-based. As an additional contribution of this work, we present an effective algorithm to compute the minimum cycle basis. Albeit known in computer science, we believe that this algorithm is not familiar to the robotics community. All the claims are validated by experiments on both standard benchmarks and simulated datasets. To foster the reproduction of the results, we provide a complete open-source C++ implementation 1 of our approach. Index Terms-Minimum cycle basis, pose graph optimization (PGO), special Euclidean group (SE(3)), simultaneous localization and mapping (SLAM). I. INTRODUCTION P OSE graph optimization (PGO) is a fundamental problem, which arises in various research disciplines, such as simultaneous localization and mapping (SLAM) [1]-[5], structure from motion [6]-[8], calibration of multicamera rig [9], and sensor network localization [10], [11].A pose graph is a graph whose vertices encode positions and orientations of 3-D poses, and whose edges represent spatial constraints between the connected vertices. Taking a graph-based SLAM system as an example, the system processes the raw measurements to construct local maps. These local maps are Manuscript

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Section: Computation Of Minimum Length Cycle Basismentioning

confidence: 99%

Section: B Superset Of Mcb: Isometric Circuitsmentioning

confidence: 99%

Sparse Pose Graph Optimization in Cycle Space

2021

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“…The choice to raise this to a power less than unity gives preference to dealing with the edges of smaller tensors first, which works distinctly better in many cases than the opposing preference because it tends to spread the growth in index dimension more evenly through the network. The minimum cycle basis is computed following the approach of [47]. That is for each edge the corresponding Horton graph is constructed.…”

Section: A4 Cycle Elimination Heuristicmentioning

confidence: 99%

Automatic contraction of unstructured tensor networks

Jermyn

2020

SciPost Phys.

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The evaluation of partition functions is a central problem in statistical physics. For lattice systems and other discrete models the partition function may be expressed as the contraction of a tensor network. Unfortunately computing such contractions is difficult, and many methods to make this tractable require periodic or otherwise structured networks. Here I present a new algorithm for contracting unstructured tensor networks. This method makes no assumptions about the structure of the network and performs well in both structured and unstructured cases so long as the correlation structure is local. Contents arXiv:1709.03080v3 [physics.comp-ph] 7 Aug 2019 J = −2 J = −1 J = −0.5 J = 0 J = 0.5 J = 1 J .0 J = −1.0 J = −0.5 J = 0.0 J = 0.5 J = 1.0 J = 2.0

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“…The time needed to calculate the set of basic cycles is in turn bounded by

O ((K_{θ}^{2} \cdot K_{ψ}) /

(log

K_{ψ}

)) . Therefore, the time needed to calculate the set of basic cycles in advance (once per a given network topology) is expected to be relatively meaningless.…”

Section: The Basic Idea Of Our Approachmentioning

confidence: 99%

Toward fast calculation of communication paths for resilient routing

2017

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Utilization of alternate communication paths is a common technique to provide protection of transmission against failures of network nodes/links. However, a noticeable delay is encountered when calculating the relevant sets of disjoint paths using the available algorithms (e.g., using Bhandari's approach). This, in turn, may have a serious impact on the ability of a network to serve dynamic demands (i.e., characterized by a relatively short duration time). To provide a solution to this problem, in this article we introduce an approach to pre‐compute the sets of disjoint paths in advance to be able to start serving the demands shortly after their arrival. Our approach is based on the observation that the issue of establishing a set of node‐disjoint paths is equivalent to the problem of determining the cheapest cycle in the network topology traversing the end nodes of a demand. In particular, we propose a generalization of this scheme assuming that any pair of node‐disjoint paths can be obtained by means of merging a number of basic cycles defined for a network topology. A new method to calculate the cheapest end‐to‐end cycles based on the so called basic cycles is introduced, which, as verified for real network topologies, reduces up to 70% the time needed to establish node‐disjoint paths (compared with the results obtained for the reference Bhandari's scheme). © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 70(4), 308–326 2017

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Efficient Deterministic Algorithms for Finding a Minimum Cycle Basis in Undirected Graphs

Cited by 19 publications

References 15 publications

Sparse Pose Graph Optimization in Cycle Space

Sparse Pose Graph Optimization in Cycle Space

Automatic contraction of unstructured tensor networks

Toward fast calculation of communication paths for resilient routing

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