Abstract
We present an algorithm that takes $\mathcal {O}(\mathrm {sort}(N))$I/Os (sort(N)=Î((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.
Published Version
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