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Minimum Cuts and Shortest NonSeparating Cycles via Homology Covers
 SYMPOSIUM ON DISCRETE ALGORITHMS
, 2011
"... Let G be a directed graph with weighted edges, embedded on a surface of genus g with b boundaries. We describe an algorithm to compute the shortest directed cycle in G in any given � 2homology class in 2 O(g+b) n log n time; this problem is NPhard even for undirected graphs. We also present two ap ..."
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Cited by 10 (4 self)
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Let G be a directed graph with weighted edges, embedded on a surface of genus g with b boundaries. We describe an algorithm to compute the shortest directed cycle in G in any given � 2homology class in 2 O(g+b) n log n time; this problem is NPhard even for undirected graphs. We also present two applications of our algorithm. The first is an algorithm to compute the shortest nonseparating directed cycle in G in 2 O(g) n log n time, improving the recent algorithm of Cabello et al. [SOCG 2010] for all g = o(log n). The second is a combinatorial algorithm to compute minimum (s, t)cuts in undirected surface graphs in 2 O(g) n log n time, improving an algorithm of Chambers et al. [SOCG 2009] for all positive g. Unlike earlier algorithms for surface graphs that construct and search finite portions of the universal cover, our algorithms use another canonical covering space, called the Z 2homology cover.
Shortest nontrivial cycles in directed surface graphs
 In Proc. 27th Ann. Symp. Comput. Geom
, 2011
"... Let G be a directed graph embedded on a surface of genus g. We describe an algorithm to compute the shortest nonseparating cycle in G in O(g 2 n log n) time, exactly matching the fastest algorithm known for undirected graphs. We also describe an algorithm to compute the shortest noncontractible cy ..."
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Cited by 5 (3 self)
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Let G be a directed graph embedded on a surface of genus g. We describe an algorithm to compute the shortest nonseparating cycle in G in O(g 2 n log n) time, exactly matching the fastest algorithm known for undirected graphs. We also describe an algorithm to compute the shortest noncontractible cycle in G in g O(g) n log n time, matching the fastest algorithm for undirected graphs of constant genus.
Faster shortest noncontractible cycles in directed surface graphs
 CoRR
"... Let G be a directed graph embedded on a surface of genus g with b boundary cycles. We describe an algorithm to compute the shortest noncontractible cycle in G in O((g 3 + g b)n log n) time. Our algorithm improves the previous best known time bound of (g + b) O(g+b) n log n for all positive g and b. ..."
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Cited by 2 (0 self)
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Let G be a directed graph embedded on a surface of genus g with b boundary cycles. We describe an algorithm to compute the shortest noncontractible cycle in G in O((g 3 + g b)n log n) time. Our algorithm improves the previous best known time bound of (g + b) O(g+b) n log n for all positive g and b. We also describe an algorithm to compute the shortest nonnullhomologous cycle in G in O((g 2 + g b)n log n) time, generalizing a known algorithm to compute the shortest nonseparating cycle.
Computing Replacement Paths in Surface Embedded Graphs
, 2011
"... Let s and t be vertices in a directed graph G with nonnegative edge weights. The replacement paths problem asks us to compute, for each edge e in G, the length of the shortest path from s to t that does not traverse e. We describe an algorithm that solves the replacement paths problem for directed ..."
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Let s and t be vertices in a directed graph G with nonnegative edge weights. The replacement paths problem asks us to compute, for each edge e in G, the length of the shortest path from s to t that does not traverse e. We describe an algorithm that solves the replacement paths problem for directed graphs embedded on a surface of any genus g in O(gn log n) time, generalizing a recent O(n log n)time algorithm of WulffNilsen for planar graphs [SODA 2010].
Combinatorial Optimization of Cycles and Bases
 PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS
"... We survey algorithms and hardness results for two important classes of topology optimization problems: computing minimumweight cycles in a given homotopy or homology class, and computing minimumweight cycle bases for the fundamental group or various homology groups. ..."
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We survey algorithms and hardness results for two important classes of topology optimization problems: computing minimumweight cycles in a given homotopy or homology class, and computing minimumweight cycle bases for the fundamental group or various homology groups.
Shortest Nontrivial Cycles in Directed and Undirected Surface Graphs
"... Let G be a graph embedded on a surface of genus g with b boundary cycles. We describe algorithms to compute multiple types of nontrivial cycles in G, using different techniques depending on whether or not G is an undirected graph. If G is undirected, then we give an algorithm to compute a shortest ..."
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Let G be a graph embedded on a surface of genus g with b boundary cycles. We describe algorithms to compute multiple types of nontrivial cycles in G, using different techniques depending on whether or not G is an undirected graph. If G is undirected, then we give an algorithm to compute a shortest nonseparating cycle in G in 2O(g) n log log n time. Similar algorithms are given to compute a shortest noncontractible or nonnullhomologous cycle in 2O(g+b) n log log n time. Our algorithms for undirected G combine an algorithm of Kutz with known techniques for efficiently enumerating homotopy classes of curves that may be shortest nontrivial cycles. Our main technical contributions in this work arise from assuming G is a directed graph with possibly asymmetric edge weights. For this case, we give an algorithm to compute a shortest noncontractible cycle in G in O((g 3 + g b)n log n) time. In order to achieve this time bound, we use a restriction of the infinite cyclic cover that may be useful in other contexts. We also describe an algorithm to compute a shortest nonnullhomologous cycle in G in O((g 2 + g b)n log n) time, extending a known algorithm of Erickson to compute a shortest nonseparating cycle. In both the undirected and directed cases, our algorithms improve the best time bounds known for many values of g and b. 1
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.