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Local Computation Algorithms for Graphs of Non-constant Degrees

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Abstract

In the model of local computation algorithms (LCAs), we aim to compute the queried part of the output by examining only a small (sublinear) portion of the input. Many recently developed LCAs on graph problems achieve time and space complexities with very low dependence on n, the number of vertices. Nonetheless, these complexities are generally at least exponential in d, the upper bound on the degree of the input graph. Instead, we consider the case where parameter d can be moderately dependent on n, and aim for complexities with subexponential dependence on d, while maintaining polylogarithmic dependence on n. We present:

  • a randomized LCA for computing maximal independent sets whose time and space complexities are quasi-polynomial in d and polylogarithmic in n;

  • for constant \(\varepsilon > 0\), a randomized LCA that provides a \((1-\varepsilon )\)-approximation to maximum matching with high probability, whose time and space complexities are polynomial in d and polylogarithmic in n.

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Notes

  1. Some recent work on LCAs uses the alternative term “probe” to refer to queries that \(\mathscr {A}\) makes to \(\mathscr {O}^G\), in order to distinguish them from queries that we ask \(\mathscr {A}\) to answer; the corresponding complexity is called the probe complexity. We choose not to adopt this notation since both types of queries will often be indistinguishable when we construct oracles for intermediate graphs during the execution of our algorithms. It will also contradict with the term “query tree,” which is unanimously used in other work on this method.

  2. Applying a similar reduction on the unmodified version gives an LCA with complexities \(2^{O(\log ^3 d + \log d \log \log n)}\).

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Acknowledgments

Levi was supported by National Science Foundation (NSF) Grants CCF-1217423 and CCF-1065125, and Israel Science Foundation (ISF) Grants 246/08 and 1536/14. Rubinfeld was supported by NSF Grants CCF-1217423, CCF-1065125, CCF-1420692, and ISF grant 1536/14. Yodpinyanee was supported by NSF Grants CCF-1217423, CCF-1065125, CCF-1420692, and the Development and Promotion of Science and Technology Talented Project scholarship, Royal Thai Government. We thank Dana Ron for her valuable contribution to this paper. We thank anonymous reviewers for their insightful comments on the preliminary version of this paper.

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Authors and Affiliations

  1. Ècole Normale Supèrieure and Universitè Paris Diderot, Paris, France

    Reut Levi

  2. CSAIL, MIT, Cambridge, MA, 02139, USA

    Ronitt Rubinfeld & Anak Yodpinyanee

  3. The Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, Israel

    Ronitt Rubinfeld

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  1. Reut Levi
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Levi, R., Rubinfeld, R. & Yodpinyanee, A. Local Computation Algorithms for Graphs of Non-constant Degrees. Algorithmica 77, 971–994 (2017). https://doi.org/10.1007/s00453-016-0126-y

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