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Optimal Decremental Connectivity in Planar Graphs

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Abstract

We show an algorithm for dynamic maintenance of connectivity information in an undirected planar graph subject to edge deletions. Our algorithm may answer connectivity queries of the form “Are vertices u and v connected with a path?" in constant time. The queries can be intermixed with any sequence of edge deletions, and the algorithm handles all updates in a total of O(n) time. This results improves over a previously known O(n log n) time algorithm.

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Notes

  1. Throughout the paper we use n and m to denote, respectively, the number of vertices and the number of edges in the graph.

  2. Since the graph has constant degree, we may assure that both searches are synchronized in terms of number of vertices visited.

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Acknowledgements

This work is partly supported by the EU FET project MULTIPLEX no. 317532 and the Google Focused Award on “Algorithms for Large-scale Data Analysis”. Piotr Sankowski is supported by Polish National Science Centre grant UMO-2014/13/B/ST6/01811.

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Authors and Affiliations

  1. Sapienza University of Rome, Rome, Italy

    Jakub Łącki

  2. University of Warsaw, Warsaw, Poland

    Piotr Sankowski

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  1. Jakub Łącki
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  2. Piotr Sankowski
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Correspondence to Jakub Łącki.

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Łącki, J., Sankowski, P. Optimal Decremental Connectivity in Planar Graphs. Theory Comput Syst 61, 1037–1053 (2017). https://doi.org/10.1007/s00224-016-9709-x

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