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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 ..."
Abstract

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.
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.
A PrimalDual Clustering Technique with Applications in Network Design
, 2011
"... Network design problems deal with settings where the goal is to design a network (i.e., find a subgraph of a given graph) that satisfies certain connectivity requirements. Each requirement is in the form of connecting (or, more generally, providing large connectivity between) a pair of vertices of t ..."
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Network design problems deal with settings where the goal is to design a network (i.e., find a subgraph of a given graph) that satisfies certain connectivity requirements. Each requirement is in the form of connecting (or, more generally, providing large connectivity between) a pair of vertices of the graph. The goal is to find a network of minimum length, and in some cases requirements can be compromised after paying their “penalties. ” These are usually called prizecollecting Steiner network problems. In practical scenarios of physical networking, with cable or fiber embedded in the ground, crossings are rare or nonexistent. Hence, planar instances of network design problems are a natural subclass of interest. We can usually take advantage of this structure to find better performance guarantees. In this thesis, we develop a primaldual clustering technique called “prizecollecting clustering, ” which is used to give improved approximation algorithms for several planar and nonplanar network design problems. The technique is based on a famous moat growing procedure due to Agrawal, Klein, Ravi [AKR95] and Goemans and Williamson [GW95]. It provides a paradigm for clustering the vertices of a graph
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.