Efficient Vertex-Label Distance Oracles for Planar Graphs

Efficient Vertex-Label Distance Oracles for Planar Graphs We consider distance queries in vertex-labeled planar graphs. For any fixed 0 < πœ– ≤ 1/2 we show how to preprocess a directed planar graph with vertex labels and arc lengths into a data structure that answers queries of the following form. Given a vertex u and a label λ return a (1 + πœ–)-approximation of the distance from u to its closest vertex with label λ. For a directed planar graph with n vertices, such that the ratio of the largest to smallest arc length is bounded by N, the preprocessing time is O(πœ– − 2 n lg 3n lg(n N)), the data structure size is O(πœ– − 1 n lg n lg(n N)), and the query time is O(lg lg n lg lg(n N) + πœ– − 1). We also point out that a vertex label distance oracle for undirected planar graphs suggested in an earlier version of this paper is incorrect. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Theory of Computing Systems Springer Journals

Efficient Vertex-Label Distance Oracles for Planar Graphs

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Publisher
Springer US
Copyright
Copyright Β© 2017 by Springer Science+Business Media, LLC, part of Springer Nature
Subject
Computer Science; Theory of Computation
ISSN
1432-4350
eISSN
1433-0490
D.O.I.
10.1007/s00224-017-9827-0
Publisher site
See Article on Publisher Site

Abstract

We consider distance queries in vertex-labeled planar graphs. For any fixed 0 < πœ– ≤ 1/2 we show how to preprocess a directed planar graph with vertex labels and arc lengths into a data structure that answers queries of the following form. Given a vertex u and a label λ return a (1 + πœ–)-approximation of the distance from u to its closest vertex with label λ. For a directed planar graph with n vertices, such that the ratio of the largest to smallest arc length is bounded by N, the preprocessing time is O(πœ– − 2 n lg 3n lg(n N)), the data structure size is O(πœ– − 1 n lg n lg(n N)), and the query time is O(lg lg n lg lg(n N) + πœ– − 1). We also point out that a vertex label distance oracle for undirected planar graphs suggested in an earlier version of this paper is incorrect.

Journal

Theory of Computing SystemsSpringer Journals

Published: Dec 12, 2017

References

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