, Volume 54, Issue 3, pp 413-469
Date: 01 Dec 2007

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.

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.