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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 7 (5 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.
DETERMINISTICALLY ISOLATING A PERFECT MATCHING IN BIPARTITE PLANAR GRAPHS
"... Abstract. We present a deterministic way of assigning small (log bit) weights to the edges of a bipartite planar graph so that the minimum weight perfect matching becomes unique. The isolation lemma as described in [MVV87] achieves the same for general graphs using a randomized weighting scheme, whe ..."
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Cited by 3 (3 self)
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Abstract. We present a deterministic way of assigning small (log bit) weights to the edges of a bipartite planar graph so that the minimum weight perfect matching becomes unique. The isolation lemma as described in [MVV87] achieves the same for general graphs using a randomized weighting scheme, whereas we can do it deterministically when restricted to bipartite planar graphs. As a consequence, we reduce both decision and construction versions of the matching problem to testing whether a matrix is singular, under the promise that its determinant is 0 or 1, thus obtaining a highly parallel SPL algorithm for bipartite planar graphs. This improves the earlier known bounds of nonuniform SPL by [ARZ99] and NC 2 by [MN95, MV00]. It also rekindles the hope of obtaining a deterministic parallel algorithm for constructing a perfect matching in nonbipartite planar graphs, which has been open for a long time. Our techniques are elementary and simple. 1.
www.stacsconf.org DETERMINISTICALLY ISOLATING A PERFECT MATCHING IN BIPARTITE PLANAR GRAPHS
"... Abstract. We present a deterministic way of assigning small (log bit) weights to the edges of a bipartite planar graph so that the minimum weight perfect matching becomes unique. The isolation lemma as described in [MVV87] achieves the same for general graphs using a randomized weighting scheme, whe ..."
Abstract
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Abstract. We present a deterministic way of assigning small (log bit) weights to the edges of a bipartite planar graph so that the minimum weight perfect matching becomes unique. The isolation lemma as described in [MVV87] achieves the same for general graphs using a randomized weighting scheme, whereas we can do it deterministically when restricted to bipartite planar graphs. As a consequence, we reduce both decision and construction versions of the matching problem to testing whether a matrix is singular, under the promise that its determinant is 0 or 1, thus obtaining a highly parallel SPL algorithm for bipartite planar graphs. This improves the earlier known bounds of nonuniform SPL by [ARZ99] and NC 2 by [MN95, MV00]. It also rekindles the hope of obtaining a deterministic parallel algorithm for constructing a perfect matching in nonbipartite planar graphs, which has been open for a long time. Our techniques are elementary and simple. 1.
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.
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 ..."
Abstract
 Add to MetaCart
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(g 2 n log n) time.