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I/O-Efficient Algorithms for Graphs of Bounded Treewidth

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Abstract

We present an algorithm that takes \(\mathcal {O}(\mathrm {sort}(N))\) I/Os (sort(N)=Θ((N/(DB))log  M/B (N/B)) is the number of I/Os it takes to sort N data items) to compute a tree decomposition of width at most k, for any graph G of treewidth at most k and size N, where k is a constant. Given such a tree decomposition, we use a dynamic programming framework to solve a wide variety of problems on G in \(\mathcal {O}(N/(DB))\) I/Os, including the single-source shortest path problem and a number of problems that are NP-hard on general graphs. The tree decomposition can also be used to obtain an optimal separator decomposition of G. We use such a decomposition to perform depth-first search in G in  \(\mathcal {O}(N/(DB))\) I/Os.

As important tools that are used in the tree decomposition algorithm, we introduce flippable DAGs and present an algorithm that computes a perfect elimination ordering of a k-tree in \(\mathcal {O}(\mathrm {sort}(N))\) I/Os.

The second contribution of our paper, which is of independent interest, is a general and simple framework for obtaining I/O-efficient algorithms for a number of graph problems that can be solved using greedy algorithms in internal memory. We apply this framework in order to obtain an improved algorithm for finding a maximal matching and the first deterministic I/O-efficient algorithm for finding a maximal independent set of an arbitrary graph. Both algorithms take \(\mathcal {O}(\mathrm {sort}(|V|+|E|))\) I/Os. The maximal matching algorithm is used in the tree decomposition algorithm.

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Authors and Affiliations

  1. School of Computer Science, Carleton University, Ottawa, ON, K1S 5B6, Canada

    Anil Maheshwari

  2. Faculty of Computer Science, Dalhousie University, Halifax, NS, B3H 2Y5, Canada

    Norbert Zeh

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  1. Anil Maheshwari
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  2. Norbert Zeh
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Correspondence to Anil Maheshwari.

Additional information

An abstract of this paper was presented at the 12th Annual ACM-SIAM Symposium on Discrete Algorithms, Proceedings, pp. 89–90, 2001.

Research of A. Maheshwari supported by NSERC.

Part of this work was done while the second author was a Ph.D. student at the School of Computer Science of Carleton University.

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Maheshwari, A., Zeh, N. I/O-Efficient Algorithms for Graphs of Bounded Treewidth. Algorithmica 54, 413–469 (2009). https://doi.org/10.1007/s00453-007-9131-5

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