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Compact I/O-Efficient Representation of Separable Graphs and Optimal Tree Layouts

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Theory and Applications of Models of Computation (TAMC 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11436))

Abstract

Compact and I/O-efficient data representations play an important role in efficient algorithm design, as memory bandwidth and latency can present a significant performance bottleneck, slowing the computation by orders of magnitude. While this problem is very well explored in e.g. uniform numerical data processing, structural data applications (e.g. on huge graphs) require different algorithm-dependent approaches. Separable graph classes (i.e. graph classes with balanced separators of size \(\mathcal {O}(n^c)\) with \(c<1\)) include planar graphs, bounded genus graphs, and minor-free graphs.

In this article we present two generalizations of the separator theorem, to partitions with small regions only on average and to weighted graphs. Then we propose I/O-efficient succinct representation and memory layout for random walks in (weighted) separable graphs in the pointer machine model, including an efficient algorithm to compute them. Finally, we present a worst-case I/O-optimal tree layout algorithm for root-leaf path traversal, show an additive (+1)-approximation of optimal compact layout and contrast this with NP-completeness proof of finding an optimal compact layout.

The work was supported by the Czech Science Foundation (GACR) project 17-10090Y “Network optimization”.

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Notes

  1. 1.

    A succinct (resp. compact) data representation uses \(H+o(H)\) (resp. \(\mathcal {O}(H)\)) bits where H is the class information entropy.

  2. 2.

    Note that \(\varOmega (K/\log B)\) blocks may be required even for trees. Standard graph representation would access \(\mathcal {O}(K)\) blocks.

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Authors and Affiliations

  1. Department of Applied Mathematics, Charles University, Malostranské nám. 25, 11800, Prague, Czech Republic

    Tomáš Gavenčiak & Jakub Tětek

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  1. Tomáš Gavenčiak
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Correspondence to Jakub Tětek .

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  1. Anna University, Chennai, India

    T.V. Gopal

  2. Waseda University, Kitakyushu, Japan

    Junzo Watada

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Gavenčiak, T., Tětek, J. (2019). Compact I/O-Efficient Representation of Separable Graphs and Optimal Tree Layouts. In: Gopal, T., Watada, J. (eds) Theory and Applications of Models of Computation. TAMC 2019. Lecture Notes in Computer Science(), vol 11436. Springer, Cham. https://doi.org/10.1007/978-3-030-14812-6_14

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  • DOI: https://doi.org/10.1007/978-3-030-14812-6_14

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