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Finding shortest nontrivial cycles in directed graphs on surfaces
 In These Proceedings
, 2010
"... Let D be a weighted directed graph cellularly embedded in a surface of genus g, orientable or not, possibly with boundary. We describe algorithms to compute a shortest noncontractible and a shortest surface nonseparating cycle in D. This generalizes previous results that only dealt with undirected ..."
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Cited by 14 (3 self)
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Let D be a weighted directed graph cellularly embedded in a surface of genus g, orientable or not, possibly with boundary. We describe algorithms to compute a shortest noncontractible and a shortest surface nonseparating cycle in D. This generalizes previous results that only dealt with undirected graphs. Our first algorithm computes such cycles in O(n 2 log n) time, where n is the total number of vertices and edges of D, thus matching the complexity of the best known algorithm in the undirected case. It revisits and extends Thomassen’s 3path condition; the technique applies to other families of cycles as well. We also give an algorithm with subquadratic complexity in the complexity of the input graph, if g is fixed. Specifically, we can solve the problem in O ( √ g n 3/2 log n) time, using a divideandconquer technique that simplifies the graph while preserving the topological properties of its cycles. A variant runs in O(ng log g + nlog 2 n) for graphs of bounded treewidth.
Multiplesource shortest paths in embedded graphs
, 2012
"... Let G be a directed graph with n vertices and nonnegative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe an algorithm to preprocess the graph in O(gn log n) time, so that the shortestpath distance from any vertex on the boundary of ..."
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Cited by 12 (6 self)
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Let G be a directed graph with n vertices and nonnegative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe an algorithm to preprocess the graph in O(gn log n) time, so that the shortestpath distance from any vertex on the boundary of f to any other vertex in G can be retrieved in O(log n) time. Our result directly generalizes the O(n log n)time algorithm of Klein [Multiplesource shortest paths in planar graphs. In Proc. 16th Ann. ACMSIAM Symp. Discrete Algorithms, 2005] for multiplesource shortest paths in planar graphs. Intuitively, our preprocessing algorithm maintains a shortestpath tree as its source point moves continuously around the boundary of f. As an application of our algorithm, we describe algorithms to compute a shortest noncontractible or nonseparating cycle in embedded, undirected graphs in O(g² n log n) time.
Global Minimum Cuts in Surface Embedded Graphs
"... We give a deterministic algorithm to find the minimum cut in a surfaceembedded graph in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, our algorithm computes the minimum cut in g O(g) n log log n time, matching the running time of the fastest algorithm kno ..."
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Cited by 8 (5 self)
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We give a deterministic algorithm to find the minimum cut in a surfaceembedded graph in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, our algorithm computes the minimum cut in g O(g) n log log n time, matching the running time of the fastest algorithm known for planar graphs, due to Ł ˛acki and Sankowski, for any constant g. Indeed, our algorithm calls Ł ˛acki and Sankowski’s recent O(n log log n) time planar algorithm as a subroutine. Previously, the best time bounds known for this problem followed from two algorithms for general sparse graphs: a randomized algorithm of Karger that runs in O(n log³ n) time and succeeds with high probability, and a deterministic algorithm of Nagamochi and Ibaraki that runs in O(n² log n) time. We can also achieve a deterministic g O(g) n² log log n time bound by repeatedly applying the best known algorithm for minimum (s, t)cuts in surface graphs. The bulk of our work focuses on the case where the dual of the minimum cut splits the underlying surface into multiple components with positive genus.
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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Cited by 5 (3 self)
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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
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.
FAST ALGORITHMS FOR SURFACE EMBEDDED GRAPHS VIA HOMOLOGY
, 2013
"... We describe several results on combinatorial optimization problems for graphs where the input comes with an embedding on an orientable surface of small genus. While the specific techniques used differ between problems, all the algorithms we describe share one common feature in that they rely on the ..."
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We describe several results on combinatorial optimization problems for graphs where the input comes with an embedding on an orientable surface of small genus. While the specific techniques used differ between problems, all the algorithms we describe share one common feature in that they rely on the algebraic topology construct of homology. We describe algorithms to compute global minimum cuts and count minimum s, tcuts. We describe new algorithms to compute short cycles that are topologically nontrivial. Finally, we describe ongoing work in designing a new algorithm for computing maximum s, tflows in surface embedded graphs. We begin by describing an algorithm to compute global minimum cuts in edge weighted genus g graphs in gO(g)n log log n time. When the genus is a constant, our algorithm’s running time matches the best time bound known for planar graphs due to La̧cki and Sankowski. In our algorithm, we reduce to the problem of finding a minimum weight separating subgraph in the dual
A Polynomialtime Bicriteria Approximation Scheme for Planar Bisection
, 2015
"... Given an undirected graph with edge costs and node weights, the minimum bisection problem asks for a partition of the nodes into two parts of equal weight such that the sum of edge costs between the parts is minimized. We give a polynomial time bicriteria approximation scheme for bisection on planar ..."
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Given an undirected graph with edge costs and node weights, the minimum bisection problem asks for a partition of the nodes into two parts of equal weight such that the sum of edge costs between the parts is minimized. We give a polynomial time bicriteria approximation scheme for bisection on planar graphs. Specifically, let W be the total weight of all nodes in a planar graph G. For any constant ε> 0, our algorithm outputs a bipartition of the nodes such that each part weighs at most W/2+ε and the total cost of edges crossing the partition is at most (1+ε) times the total cost of the optimal bisection. The previously best known approximation for planar minimum bisection, even with unit node weights, was O(log n). Our algorithm actually solves a more general problem where the input may include a target weight for the smaller side of the bipartition.
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