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Computing the stretch of an embedded graph
"... Let G be a graph embedded in an orientable surface Σ, possibly with edge weights, and denote by len(γ) the length (the number of edges or the sum of the edge weights) of a cycle γ in G. The stretch of a graph embedded on a surface is the minimum of len(α) · len(β) over all pairs of cycles α and β t ..."
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Let G be a graph embedded in an orientable surface Σ, possibly with edge weights, and denote by len(γ) the length (the number of edges or the sum of the edge weights) of a cycle γ in G. The stretch of a graph embedded on a surface is the minimum of len(α) · len(β) over all pairs of cycles α and β that cross exactly once. We provide an algorithm to compute the stretch of an embedded graph in time O(g 4 n log n) with high probability, or in time O(g 4 n log 2 n) in the worst case, where g is the genus of the surface Σ and n is the number of vertices in G.
Discrete Systolic Inequalities and Decompositions of Triangulated Surfaces
, 2014
"... How much cutting is needed to simplify the topology of a surface? We provide bounds for several instances of this question, for the minimum length of topologically nontrivial closed curves, pants decompositions, and cut graphs with a given combinatorial map in triangulated combinatorial surfaces (o ..."
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How much cutting is needed to simplify the topology of a surface? We provide bounds for several instances of this question, for the minimum length of topologically nontrivial closed curves, pants decompositions, and cut graphs with a given combinatorial map in triangulated combinatorial surfaces (or their dual crossmetric counterpart). Our work builds upon Riemannian systolic inequalities, which bound the minimum length of nontrivial closed curves in terms of the genus and the area of the surface. We first describe a systematic way to translate Riemannian systolic inequalities to a discrete setting, and viceversa. This implies a conjecture by Przytycka and Przytycki from 1993, a number of new systolic inequalities in the discrete setting, and the fact that a theorem of Hutchinson on the edgewidth of triangulated surfaces and Gromov’s systolic inequality for surfaces are essentially equivalent. We also discuss how these proofs generalize to higher dimensions. Then we focus on topological decompositions of surfaces. Relying on ideas of Buser, we prove the existence of pants decompositions of length O(g3/2n1/2) for any triangulated combinatorial surface of genus g with n triangles, and describe an O(gn)time algorithm to compute such a decomposition. Finally, we consider the problem of embedding a cut graph (or more generally a cellular graph) with a given combinatorial map on a given surface. Using random triangulations, we prove (essentially) that, for any choice of a combinatorial map, there are some surfaces on which any cellular embedding with that combinatorial map has length superlinear in the number of triangles of the triangulated combinatorial surface. There is also a similar result for graphs embedded on polyhedral triangulations.
COMBINATORIAL OPTIMIZATION ON EMBEDDED CURVES
, 2012
"... We describe several algorithms for classifying, comparing and optimizing curves on surfaces. We give algorithms to compute the minimum member of a given homology class, particularly computing the maximum flow and minimum cuts, in surface embedded graphs. We describe approximation algorithms to compu ..."
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We describe several algorithms for classifying, comparing and optimizing curves on surfaces. We give algorithms to compute the minimum member of a given homology class, particularly computing the maximum flow and minimum cuts, in surface embedded graphs. We describe approximation algorithms to compute certain similarity measures for embedded curves on a surface. Finally, we present algorithms to solve computational problems for compactly presented curves. We describe the first algorithms to compute the shortest representative of a Z2homology class. Given a directed graph embedded on a surface of genus g with b boundary cycles, we can compute the shortest single cycle Z2homologous to a given even subgraph in 2O(g+b)n log n time. As a consequence we obtain an algorithm to compute the shortest directed nonseparating cycle in 2O(g)n log n time, which improves the previous best algorithm by a factor of O( p n) if the genus is