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# Faster Shortest-Path Algorithms for Planar Graphs

09/1996;
Source: CiteSeer

ABSTRACT We give a linear-time algorithm for single-source shortest paths in planar graphs with nonnegative edge-lengths. Our algorithm also yields a linear-time algorithm for maximum flow in a planar graph with the source and sink on the same face. For the case where negative edge-lengths are allowed, we give an algorithm requiring O(n 4=3 log(nL)) time, where L is the absolute value of the most negative length. This algorithm can be used to obtain similar bounds for computing a feasible flow in a planar network, for finding a perfect matching in a planar bipartite graph, and for finding a maximum flow in a planar graph when the source and sink are not on the same face. We also give parallel and dynamic versions of these algorithms. 1 Introduction Computing shortest paths is a fundamental and ubiquitous problem in network analysis. Aside from the importance of this problem in its own right, often the problem arises in the solution of other problems (e.g. network flow and matching). In thi...

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##### Article:Shortest Non-trivial Cycles in Directed and Undirected Surface Graphs
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ABSTRACT: Let G be a graph embedded on a surface of genus g with b boundary cycles. We describe algorithms to compute multiple types of non-trivial 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 non-separating cycle in 2^O(g) n log log n time. Similar algorithms are given to compute a shortest non-contractible or non-null-homologous cycle in 2^O(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 non-trivial 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 non-contractible 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 non-null-homologous cycle in G in O((g^2 + g b)n log n) time, extending a known algorithm of Erickson to compute a shortest non-separating cycle. In both the undirected and directed cases, our algorithms improve the best time bounds known for many values of g and b.
11/2011;
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##### Article:Approximating the Diameter of Planar Graphs in Near Linear Time
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ABSTRACT: We present a $(1+\epsilon)$-approximation algorithm running in $O(f(\epsilon)\cdot n \log^4 n)$ time for finding the diameter of an undirected planar graph with non-negative edge lengths.
12/2011;
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##### Conference Proceeding:Global minimum cuts in surface embedded graphs.
Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23-25, 2011; 01/2012

### Keywords

1 Introduction

absolute value

algorithms

dynamic versions

feasible flow

linear-time algorithm

maximum flow

negative edge-lengths

negative length

network analysis

network flow

nonnegative edge-lengths

planar bipartite graph

planar graph

planar graphs

planar network

shortest paths

similar bounds

single-source shortest paths

ubiquitous problem