I/O-Efficient Planar Separators
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 mos...
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| Published in: | SIAM journal on computing Vol. 38; no. 3; pp. 767 - 801 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Philadelphia
Society for Industrial and Applied Mathematics
01.01.2008
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| Subjects: | |
| ISSN: | 0097-5397, 1095-7111 |
| Online Access: | Get full text |
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| Summary: | 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. |
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| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 content type line 14 |
| ISSN: | 0097-5397 1095-7111 |
| DOI: | 10.1137/S0097539705446925 |