NDLI: Multiple-Source Shortest Paths in Embedded Graphs

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Erickson, Jeff Cabello, Sergio Chambers, Erin W.
Copyright Year	2013
Abstract	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 a randomized algorithm to preprocess the graph in $O(gn \log n)$ time with high probability, so that the shortest-path 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 [Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, 2005] for multiple-source shortest paths in planar graphs. Intuitively, our preprocessing algorithm maintains a shortest-path 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 with high probability. Our high-probability time bounds hold in the worst case for generic edge weights or with an additional $O(\log n)$ factor for arbitrary edge weights.
Starting Page	1542
Ending Page	1571
Page Count	30
File Format	PDF
ISSN	00975397
DOI	10.1137/120864271
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	4
Volume Number	42
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2013-07-18
Access Restriction	Subscribed
Subject Keyword	Planar graphs; geometric and topological aspects of graph theory Nonnumerical algorithms computational topology topological graph theory parametric shortest paths Graph theory Graph algorithms dynamic data structures Relations with graph theory
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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