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Succinct Encodings for Families of Interval Graphs

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Abstract

We consider the problem of designing succinct data structures for interval graphs with n vertices while supporting degree, adjacency, neighborhood and shortest path queries in optimal time. Towards showing succinctness, we first show that at least \(n\log _2{n} - 2n\log _2\log _2 n - O(n)\) bits are necessary to represent any unlabeled interval graph G with n vertices, answering an open problem of Yang and Pippenger (Proc Am Math Soc Ser B 4(1):1–3, 2017). This is augmented by a data structure of size \(n\log _2{n} +O(n)\) bits while supporting not only the above queries optimally but also capable of executing various combinatorial algorithms (like proper coloring, maximum independent set etc.) on interval graphs efficiently. Finally, we extend our ideas to other variants of interval graphs, for example, proper/unit interval graphs, k-improper interval graphs, and circular-arc graphs, and design succinct data structures for these graph classes as well along with supporting queries on them efficiently.

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Notes

  1. Throughout the paper, we use \(\log\) to denote the logarithm to the base 2.

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Acknowledgements

Hüseyin Acan: Research supported by a National Science Foundation Fellowship (Award No. 1502650). Sankardeep Chakraborty: This work is supported by JSPS KAKENHI Grant Number 18H05291. Seungbum Jo: Work done while the author was at Université libre de Bruxelles. Research supported by Fonds de la Recherche Scientifique-FNRS under Grant No MISU F 6001 1, and by the research grant of the Chungbuk National University in 2019. A preliminary version of these results have appeared in the 16th International Symposium on Algorithms and Data Structures (WADS, 2019) [1].

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Authors and Affiliations

  1. Drexel University, Philadelphia, USA

    Hüseyin Acan

  2. National Institute of Informatics, Tokyo, Japan

    Sankardeep Chakraborty

  3. Chungbuk National University, Cheongju, South Korea

    Seungbum Jo

  4. Seoul National University, Seoul, South Korea

    Srinivasa Rao Satti

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  1. Hüseyin Acan
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  2. Sankardeep Chakraborty
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  3. Seungbum Jo
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  4. Srinivasa Rao Satti
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Correspondence to Seungbum Jo.

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Acan, H., Chakraborty, S., Jo, S. et al. Succinct Encodings for Families of Interval Graphs. Algorithmica 83, 776–794 (2021). https://doi.org/10.1007/s00453-020-00710-w

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