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An Efficient Oracle for Counting Shortest Paths in Planar Graphs

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Algorithmic Aspects in Information and Management (AAIM 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 13153))

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Abstract

We propose an \(O(\sqrt{n})\) query time and \(O(n^{1.5})\) size oracle which, given a pair of vertices u and v in a planar graph G of n vertices, answers the number of shortest paths from u to v. Our oracle can answer a query whether there is a unique shortest path from u to v in \(O(\log n)\) time. Bezáková and Searns [ISAAC 2018] give an \(O(\sqrt{n})\) query time and \(O(n^{1.5})\) size oracle for counting shortest paths in planar graphs. Applying this oracle directly, it takes \(O(\sqrt{n})\) time to answer whether there is a unique shortest path from u to v. A key component in our oracle is to apply Voronoi diagrams on planar graphs to speed up the query time. Computational studies show that our oracle is faster to answer queries than the oracle of Bezáková and Searns for large graphs. Applying Voronoi diagrams on planar graphs, significant theoretical improvements have been made for distance oracles. Our studies confirm that Voronoi diagrams are efficient data structures for distance oracles in practice.

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Notes

  1. 1.

    It is assumed that every arithmetic operation takes O(1) time. This paper also follows this assumption.

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Acknowledgement

The authors thank anonymous reviewers for their constructive comments. The research was partially supported by NSERC discovery grant.

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Authors and Affiliations

  1. School of Computing Science, Simon Fraser University, Burnaby, Canada

    Ye Gong & Qian-Ping Gu

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  1. Ye Gong
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  2. Qian-Ping Gu
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Correspondence to Qian-Ping Gu .

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Editors and Affiliations

  1. The University of Texas at Dallas, Richardson, TX, USA

    Weili Wu

  2. Harbin Institute of Technology, Shenzhen, China

    Hongwei Du

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Gong, Y., Gu, QP. (2021). An Efficient Oracle for Counting Shortest Paths in Planar Graphs. In: Wu, W., Du, H. (eds) Algorithmic Aspects in Information and Management. AAIM 2021. Lecture Notes in Computer Science(), vol 13153. Springer, Cham. https://doi.org/10.1007/978-3-030-93176-6_35

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  • DOI: https://doi.org/10.1007/978-3-030-93176-6_35

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-93175-9

  • Online ISBN: 978-3-030-93176-6

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