Linear algorithms for analysis of minimum spanning and shortest path trees in planar graphs (1994) [15 citations — 0 self]
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@ARTICLE{Booth94linearalgorithms,author = {Heather Booth and Jeffery Westbrook},
title = {Linear algorithms for analysis of minimum spanning and shortest path trees in planar graphs},
journal = {Algorithmica},
year = {1994},
volume = {11},
pages = {341--352}
}
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Abstract:
We give a linear time and space algorithm for analyzing trees in planar graphs. The algorithm can be used to analyze the sensitivity of a minimum spanning tree to changes in edge costs, to find its replacement edges, and to verify its minimality. It can also be used to analyze the sensitivity of a singlesource shortest path tree to changes in edge costs, and to analyze the sensitivity of a minimum cost network flow. The algorithm is simple and practical. It uses the properties of a planar embedding, combined with a heap-ordered queue data structure. Let G = (V; E) be a planar graph, either directed or undirected, with n vertices and m = O(n) edges. Each edge e 2 E has a real-valued cost cost(e). A minimum spanning tree of a connected, undirected planar graph G is a spanning tree of minimum total edge cost. If G is directed and r is a vertex from which all other vertices are reachable, then a shortest path tree from r is a spanning tree that contains a minimum-cost path from r to every other vertex. We consider the following problems: ffl Finding the replacement edges of a minimum spanning tree, and verifying its minimality.
