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34
Tightening NonSimple Paths and Cycles on Surfaces
 SUBMITTED TO SIAM JOURNAL ON COMPUTING
"... We describe algorithms to compute the shortest path homotopic to a given path, or the shortest cycle freely homotopic to a given cycle, on an orientable combinatorial surface. Unlike earlier results, our algorithms do not require the input path or cycle to be simple. Given a surface with complexity ..."
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Cited by 28 (9 self)
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We describe algorithms to compute the shortest path homotopic to a given path, or the shortest cycle freely homotopic to a given cycle, on an orientable combinatorial surface. Unlike earlier results, our algorithms do not require the input path or cycle to be simple. Given a surface with complexity n, genus g ≥ 2, and no boundary, we construct in O(gn log n) time a tight octagonal decomposition of the surface—a set of simple cycles, each as short as possible in its free homotopy class, that decompose the surface into a complex of octagons meeting four at a vertex. After the surface is preprocessed, we can compute the shortest path homotopic to a given path of complexity k in O(gnk) time, or the shortest cycle homotopic to a given cycle of complexity k in O(gnk log(nk)) time. A similar algorithm computes shortest homotopic curves on surfaces with boundary or with genus 1. We also prove that the recent algorithms of Colin de Verdière and Lazarus for shortening embedded graphs and sets of cycles have running times polynomial in the complexity of the surface and the input curves, regardless of the surface geometry.
Minimum Cuts and Shortest Homologous Cycles
 SYMPOSIUM ON COMPUTATIONAL GEOMETRY
, 2009
"... We describe the first algorithms to compute minimum cuts in surfaceembedded graphs in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s, t)cut in g O(g) n log n time. Except for the spec ..."
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Cited by 20 (7 self)
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We describe the first algorithms to compute minimum cuts in surfaceembedded graphs in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s, t)cut in g O(g) n log n time. Except for the special case of planar graphs, for which O(n log n)time algorithms have been known for more than 20 years, the best previous time bounds for finding minimum cuts in embedded graphs follow from algorithms for general sparse graphs. A slight generalization of our minimumcut algorithm computes a minimumcost subgraph in every Z2homology class. We also prove that finding a minimumcost subgraph homologous to a single input cycle is NPhard.
Many distances in planar graphs
 In SODA ’06: Proc. 17th Symp. Discrete algorithms
, 2006
"... We show how to compute in O(n 4/3 log 1/3 n+n 2/3 k 2/3 logn) time the distance between k given pairs of vertices of a planar graph G with n vertices. This improves previous results whenever (n/logn) 5/6 ≤ k ≤ n 2 /log 6 n. As an application, we speed up previous algorithms for computing the dilatio ..."
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Cited by 18 (5 self)
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We show how to compute in O(n 4/3 log 1/3 n+n 2/3 k 2/3 logn) time the distance between k given pairs of vertices of a planar graph G with n vertices. This improves previous results whenever (n/logn) 5/6 ≤ k ≤ n 2 /log 6 n. As an application, we speed up previous algorithms for computing the dilation of geometric planar graphs. 1
Homology flows, cohomology cuts
 ACM SYMPOSIUM ON THEORY OF COMPUTING
, 2009
"... We describe the first algorithms to compute maximum flows in surfaceembedded graphs in nearlinear time. Specifically, given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, we can compute a maximum (s, t)flow in O(g 7 n log 2 n log 2 C) time fo ..."
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Cited by 16 (6 self)
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We describe the first algorithms to compute maximum flows in surfaceembedded graphs in nearlinear time. Specifically, given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, we can compute a maximum (s, t)flow in O(g 7 n log 2 n log 2 C) time for integer capacities that sum to C, or in (g log n) O(g) n time for real capacities. Except for the special case of planar graphs, for which an O(n log n)time algorithm has been known for 20 years, the best previous time bounds for maximum flows in surfaceembedded graphs follow from algorithms for general sparse graphs. Our key insight is to optimize the relative homology class of the flow, rather than directly optimizing the flow itself. A dual formulation of our algorithm computes the minimumcost cycle or circulation in a given (real or integer) homology class.
Finding one tight cycle
 Proc. 19th Ann. ACMSIAM Symp. Discrete Algorithms
"... A cycle on a combinatorial surface is tight if it as short as possible in its (free) homotopy class. We describe an algorithm to compute a single tight, noncontractible, essentially simple cycle on a given orientable combinatorial surface in O(n log n) time. The only method previously known for thi ..."
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Cited by 15 (12 self)
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A cycle on a combinatorial surface is tight if it as short as possible in its (free) homotopy class. We describe an algorithm to compute a single tight, noncontractible, essentially simple cycle on a given orientable combinatorial surface in O(n log n) time. The only method previously known for this problem was to compute the globally shortest noncontractible or nonseparating cycle in O(min{g 3, n} nlog n) time, where g is the genus of the surface. As a consequence, we can compute the shortest cycle freely homotopic to a chosen boundary cycle in O(n log n) time, a tight octagonal decomposition in O(gn log n) time, and a shortest contractible cycle enclosing a nonempty set of faces in O(nlog 2 n) time.
Polynomialtime approximation schemes for subsetconnectivity problems in boundedgenus graphs
 In Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science
, 2009
"... Abstract. We present the first polynomialtime approximation schemes (PTASes) for the following subsetconnectivity problems in edgeweighted graphs of bounded genus: Steiner tree, lowconnectivity survivablenetwork design, and subset TSP. The schemes run in O(n log n) time for graphs embedded on b ..."
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Cited by 11 (2 self)
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Abstract. We present the first polynomialtime approximation schemes (PTASes) for the following subsetconnectivity problems in edgeweighted graphs of bounded genus: Steiner tree, lowconnectivity survivablenetwork design, and subset TSP. The schemes run in O(n log n) time for graphs embedded on both orientable and nonorientable surfaces. This work generalizes the PTAS frameworks of Borradaile, Klein, and Mathieu [BMK07, Kle06] from planar graphs to boundedgenus graphs: any future problems shown to admit the required structure theorem for planar graphs will similarly extend to boundedgenus graphs. 1.
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.
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 10 (4 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.
Computing the shortest essential cycle
, 2008
"... An essential cycle on a surface is a simple cycle that cannot be continuously deformed to a point or a single boundary. We describe algorithms to compute the shortest essential cycle in an orientable combinatorial surface in O(n 2 log n) time, or in O(n log n) time when both the genus and number of ..."
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Cited by 8 (4 self)
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An essential cycle on a surface is a simple cycle that cannot be continuously deformed to a point or a single boundary. We describe algorithms to compute the shortest essential cycle in an orientable combinatorial surface in O(n 2 log n) time, or in O(n log n) time when both the genus and number of boundaries are fixed. Our result corrects an error in a paper of Erickson and HarPeled.
Maximum Flows and Parametric Shortest Paths in Planar Graphs
"... We observe that the classical maximum flow problem in any directed planar graph G can be reformulated as a parametric shortest path problem in the oriented dual graph G ∗. This reformulation immediately suggests an algorithm to compute maximum flows, which runs in O(n log n) time. As we continuously ..."
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Cited by 7 (1 self)
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We observe that the classical maximum flow problem in any directed planar graph G can be reformulated as a parametric shortest path problem in the oriented dual graph G ∗. This reformulation immediately suggests an algorithm to compute maximum flows, which runs in O(n log n) time. As we continuously increase the parameter, each change in the shortest path tree can be effected in O(log n) time using standard dynamic tree data structures, and the special structure of the parametrization implies that each directed edge enters the evolving shortest path tree at most once. The resulting maximumflow algorithm is identical to the recent algorithm of Borradaile and Klein [J. ACM 2009], but our new formulation allows a simpler presentation and analysis. On the other hand, we demonstrate that for a similarly structured parametric shortest path problem on the torus, the shortest path tree can change Ω(n²) times in the worst case, suggesting that a different method may be required to efficiently compute maximum flows in highergenus graphs.