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Approximate Shortest Paths Avoiding a Failed Vertex: Near Optimal Data Structures for Undirected Unweighted Graphs

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Abstract

Let G=(V,E) be an undirected unweighted graph. A path between any two vertices u,v∈V is said to be t-approximate shortest path if its length is at most t times the length of the shortest path between u and v. We address the problem of building a compact data structure which can efficiently answer the following query for any u,v,x∈V and t>1:

Report t-approximate shortest path between u and v when vertex x fails.

We present data structures for the single source as well as all-pairs versions of this problem. The query time guaranteed by our data structures is optimal up to a constant factor. Moreover, the size of each of them nearly matches the size of the corresponding data structure with no failures.

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Acknowledgements

Part of the work was done while the authors were at Max-Planck Institute for Computer Science, Saarbruecken, Germany for the period May–July 2009. The authors are also grateful to anonymous referees for providing useful comments which led to improving the readability of the paper.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, IIT Kanpur, Kanpur, 208016, India

    Surender Baswana

  2. Oracle India Pvt. Ltd., Bangalore, 560029, India

    Neelesh Khanna

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  1. Surender Baswana
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  2. Neelesh Khanna
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Correspondence to Surender Baswana.

Additional information

The results of the preliminary version of this article appeared in the proceedings of 27th International Symposium on Theoretical Aspects of Computer Science (STACS) held at Nancy, France during 6–8 March 2010.

This work was supported by Research I Foundation, CSE, IIT Kanpur and by Indo-German Max Planck Center for Computer Science (IMPECS).

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Baswana, S., Khanna, N. Approximate Shortest Paths Avoiding a Failed Vertex: Near Optimal Data Structures for Undirected Unweighted Graphs. Algorithmica 66, 18–50 (2013). https://doi.org/10.1007/s00453-012-9621-y

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  • DOI: https://doi.org/10.1007/s00453-012-9621-y

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