I/O-Efficient Planar Separators

Abstract

We present I/O-efficient algorithms for computing optimal separator partitions of planar graphs. Our main result shows that, given a planar graph G with N vertices and an integer $r > 0$, a vertex separator of size O$(N / \sqrt{r})$ that partitions G into O$(N / r)$ subgraphs of size at most r and boundary size O$(\sqrt{r})$ can be computed in O$(\operatorname{sort}(N))$ I/Os. This bound holds provided that $M \ge 56r \log^2 B$. Together with an I/O-efficient planar embedding algorithm presented in [N. Zeh, I/O-Efficient Algorithms for Shortest Path Related Problems, Ph.D. thesis, School of Computer Science, Carleton University, Ottawa, ON, Canada, 2002], this result is the basis for I/O-efficient solutions to many other fundamental problems on planar graphs, including breadth-first search and shortest paths [L. Arge, G. S. Brodal, and L. Toma, J. Algorithms, 53 (2004), pp. 186–206; L. Arge, L. Toma, and N. Zeh, I/O-efficient algorithms for planar digraphs, in Proceedings of the 15th ACM Symposium on Parallelism in Algorithms and Architectures, ACM, New York, 2003, pp. 85–93], depth-first search [L. Arge et al., J. Graph Algorithms Appl., 7 (2003), pp. 105–129; L. Arge and N. Zeh, I/O-efficient strong connectivity and depth-first search for directed planar graphs, in Proceedings of the 44th IEEE Symposium on Foundations of Computer Science, IEEE Press, Piscataway, NJ, 2003, pp. 261–270], strong connectivity [L. Arge and N. Zeh, I/O-efficient strong connectivity and depth-first search for directed planar graphs, in Proceedings of the 44th IEEE Symposium on Foundations of Computer Science, IEEE Press, Piscataway, NJ, 2003, pp. 261–270], and topological sorting [L. Arge and L. Toma, Simplified external memory algorithms for planar DAGs, in Proceedings of the 9th Scandinavian Workshop on Algorithm Theory, Lecture Notes in Comput. Sci. 3111, Springer-Verlag, Berlin, New York, 2004, pp. 493–503; L. Arge, L. Toma, and N. Zeh, I/O-efficient algorithms for planar digraphs, in Proceedings of the 15th ACM Symposium on Parallelism in Algorithms and Architectures, ACM, New York, 2003, pp. 85–93]. Our second result shows that, given I/O-efficient solutions to these problems, a general separator algorithm for graphs with costs and weights on their vertices [L. Aleksandrov et al., Partitioning planar graphs with costs and weights, in Proceedings of the 4th Workshop on Algorithm Engineering and Experiments, Lecture Notes in Comput. Sci. 2409, Springer-Verlag, Berlin, New York, 2002, pp. 98–107] can be made I/O-efficient. Many classical separator theorems are special cases of this result. In particular, our I/O-efficient version allows the computation of a separator as produced by our first separator algorithm, but without placing any constraints on r in relation to the memory size.