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Saving Probe Bits by Cube Domination

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Graph-Theoretic Concepts in Computer Science (WG 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11159))

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Abstract

We consider the problem of storing a single element from an m-element set as a binary string of optimal length, and comparing any queried string to the stored string without reading all bits. This is the one-element version of the problem of membership testing in the bit probe model, and solutions can serve as building blocks of general membership testers. Our principal contribution is the equivalence of saving probe bits with some generalized notion of domination in hypercubes. This domination variant requires that every vertex outside the dominating set belongs to a sub-hypercube, of fixed dimension, in which all other vertices belong to in the dominating set. This fixed dimension equals the number of saved probe bits. We give specific constructions showing that up to three probe bits can be ignored when m is far enough from the next larger power of 2. The main technical idea is to use low-dimensional (grid) relaxations of the problem. The design of optimal schemes remains an open problem, however one has to notice that even usual domination in hypercubes is far from being completely understood.

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Acknowledgment

Special thanks go to the anonymous reviewer who pointed out additional references around Theorem 1.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Chalmers University, 41296, Göteborg, Sweden

    Peter Damaschke

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  1. Peter Damaschke
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Correspondence to Peter Damaschke .

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Editors and Affiliations

  1. Universität Rostock, Rostock, Germany

    Andreas Brandstädt

  2. Brandenburgische Technische Universität Cottbus-Senftenberg, Cottbus, Germany

    Ekkehard Köhler

  3. Brandenburgische Technische Universität Cottbus-Senftenberg, Cottbus, Germany

    Klaus Meer

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Damaschke, P. (2018). Saving Probe Bits by Cube Domination. In: Brandstädt, A., Köhler, E., Meer, K. (eds) Graph-Theoretic Concepts in Computer Science. WG 2018. Lecture Notes in Computer Science(), vol 11159. Springer, Cham. https://doi.org/10.1007/978-3-030-00256-5_12

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  • DOI: https://doi.org/10.1007/978-3-030-00256-5_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-00255-8

  • Online ISBN: 978-3-030-00256-5

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