Algorithms for Length Spectra of Combinatorial Tori
Consider a weighted, undirected graph cellularly embedded on a topological surface. The function assigning to each free homotopy class of closed curves the length of a shortest cycle within this homotopy class is called the marked length spectrum. The (unmarked) length spectrum is obtained by just listing the length values of the marked length spectrum in increasing order.
In this paper, we describe algorithms for computing the (un)marked length spectra of graphs embedded on the torus. More specifically, we preprocess a weighted graph of complexity n in time O(n² log log n) so that, given a cycle with 𝓁 edges representing a free homotopy class, the length of a shortest homotopic cycle can be computed in O(𝓁+log n) time. Moreover, given any positive integer k, the first k values of its unmarked length spectrum can be computed in time O(k log n).
Our algorithms are based on a correspondence between weighted graphs on the torus and polyhedral norms. In particular, we give a weight independent bound on the complexity of the unit ball of such norms. As an immediate consequence we can decide if two embedded weighted graphs have the same marked spectrum in polynomial time. We also consider the problem of comparing the unmarked spectra and provide a polynomial time algorithm in the unweighted case and a randomized polynomial time algorithm otherwise.
graphs on surfaces
length spectrum
polyhedral norm
Mathematics of computing~Geometric topology
Mathematics of computing~Graphs and surfaces
Mathematics of computing~Enumeration
26:1-26:16
Regular Paper
https://arxiv.org/abs/2303.08036
We would like to thank Marcos Cossarini who suggested to look at the norm point of view and Bruno Grenet who enlightened us about the Polynomial Identity Testing problem. We would also like to thank the referees for their helpful comments, in particular for suggesting the complexity improvement in the case of unweighted graphs.
Vincent
Delecroix
Vincent Delecroix
Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR 5800, F-33400 Talence, France
Matthijs
Ebbens
Matthijs Ebbens
Institut Fourier, CNRS, Université Grenoble Alpes, France
Francis
Lazarus
Francis Lazarus
G-SCOP/Institut Fourier, CNRS, Université Grenoble Alpes, France
This work is supported by the project TOFU of the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01) funded by the French program Investissement d’avenir.
Ivan
Yakovlev
Ivan Yakovlev
Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR 5800, F-33400 Talence, France
10.4230/LIPIcs.SoCG.2023.26
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Vincent Delecroix, Matthijs Ebbens, Francis Lazarus, and Ivan Yakovlev
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