On External-Memory MST, SSSP, and Multi-way Planar Graph Separation

Recently external memory graph problems have received considerable attention because massive graphs arise naturally in many applications involving massive data sets. Even though a large number of I/O-efficient graph algorithms have been developed, a number of fundamental problems still remain open. The results in this paper fall in two main classes. First we develop an improved algorithm for the problem of computing a minimum spanning tree (MST) of a general undirected graph. Second we show that on planar undirected graphs the problems of computing a multi-way graph separation and single source shortest paths (SSSP) can be reduced I/O-efficiently to planar breadth-first search (BFS). Since BFS can be trivially reduced to SSSP by assigning all edges weight one, it follows that in external memory planar BFS, SSSP, and multi-way separation are equivalent. That is, if any of these problems can be solved I/O-efficiently, then all of them can be solved I/O-efficiently in the same bound. Our planar graph results have subsequently been used to obtain I/O-efficient algorithms for all fundamental problems on planar undirected graphs. ✩ An extended abstract version of this paper was presented at the Seventh Scandinavian Workshop on Algorithm Theory (SWAT 2000) [Lecture Notes in Comput. Sci., vol 1851, 2000, pp. 433–447]. * Corresponding author. E-mail addresses: [email protected] (L. Arge), [email protected] (G.S. Brodal), [email protected] (L. Toma). 1 Supported in part by the National Science Foundation through ESS grant EIA-9870734, RI grant EIA9972879, CAREER grant CCR-9984099, ITR grant EIA-0112849, and U.S.–Germany Cooperative Research Program grant INT-0129182. 2 Supported in part by the IST Programme of the EU under contract number IST-1999-14186 (ALCOM-FT) and by the Carlsberg foundation under contract number ANS-0257/20. 0196-6774/$ – see front matter  2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgor.2004.04.001 L. Arge et al. / Journal of Algorithms 53 (2004) 186–206 187  2004 Elsevier Inc. All rights reserved.