Abstract
For many types of graphs, including directed acyclic graphs, undirected graphs, tournament graphs, and graphs with bounded independence number, the shortest path problem is NL-complete. The longest path problem is even NP-complete for many types of graphs, including undirected K 5-minor-free graphs and planar graphs. In the present paper we present logspace algorithms for computing shortest and longest paths in series-parallel graphs where the edges can be directed arbitrarily. The class of series-parallel graphs that we study can be characterized alternatively as the class of K 4-minor-free graphs and also as the class of graphs of tree-width 2. It is well-known that for graphs of bounded tree-width many intractable problems can be solved efficiently, but previous work was focused on finding algorithms with low parallel or sequential time complexity. In contrast, our results concern the space complexity of shortest and longest path problems. In particular, our results imply that for directed graphs of tree-width 2 these problems are L-complete.
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References
Allender, E., Barrington, D.A.M., Chakraborty, T., Datta, S., Roy, S.: Grid graph reachability problems. In: 21th Annual IEEE Conference on Computational Complexity (CCC), pp. 299–313 (2006)
Ben-Or, M., Cleve, R.: Computing algebraic formulas using a constant number of registers. SIAM J. Comput. 21, 54–58 (1992)
Bodlaender, H.L.: NC-algorithms for graphs with small treewidth. In: 14th International Workshop on Graph-Theoretic Concepts in Computer Science (WG), pp. 1–10.
Bodlaender, H.L.: Treewidth: Characterizations, applications, and computations. In: 32nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG), pp. 1–14.
Bodlaender, H.L., de Fluiter, B.A.: Parallel algorithms for series parallel graphs and graphs with treewidth two. Algorithmica 29(4), 534–559 (2001)
Bodlaender, H.L., Hagerup, T.: Parallel algorithms with optimal speedup for bounded treewidth. SIAM J. Comput. 27, 1725–1746 (1998)
Borodin, A., Cook, S.A., Dymond, P.W., Ruzzo, W.L., Tompa, M.: Two applications of inductive counting for complementation problems. SIAM J. on Computing 18(3), 559–578 (1989)
Bourke, C., Tewari, R., Vinodchandran, N.V.: Directed planar reachability is in unambiguous log-space. In: 22th Annual IEEE Conference on Computational Complexity (CCC), pp. 217–221 (2007)
Buss, S., Cook, S., Gupta, A., Ramachandran, V.: An optimal parallel algorithm for formula evaluation. SIAM J. Comput. 21, 755–780 (1992)
Chaudhuri, S., Zaroliagis, C.D.: Shortest paths in digraphs of small treewidth. Part II: Optimal parallel algorithms. Theoretical Comput. Sci. 203, 205–223 (1998)
Chaudhuri, S., Zaroliagis, C.D.: Shortest paths in digraphs of small treewidth. Part I: Sequential algorithms. Algorithmica 27(3), 212–226 (2000)
Chiu, A., Davida, G., Litow, B.: Division in logspace-uniform NC1. Theoretical Informatics and Applications 35, 259–275 (2001)
Duffin, R.: Topology of series-parallel networks. J. Math. Analysis and Applications 10, 303–318 (1965)
Eppstein, D.: Parallel recognition of series-parallel graphs. Inf. and Comp. 98, 41–55 (1992)
He, X., Yesha, Y.: Parallel recognition and decomposition of two terminal series parallel graphs. Inf. and Comp. 75, 15–38 (1987)
Hesse, W.: Division is in uniform TC0. In: 28th International Colloquium on Automata, Languages and Programming (ICALP), pp. 104–114
Hohberg, W., Reischuk, R.: A framework to design algorithms for optimization problems on graphs. Technical Report ITI, Technical University Darmstadt (1990)
Jakoby, A., Liśkiewicz, M.: Paths problems in symmetric logarithmic space. In: 29th International Colloquium on Automata, Languages and Programming (ICALP), pp. 269–280.
Jakoby, A., Liśkiewicz, M., Reischuk, R.: Space efficient algorithms for series-parallel graphs. J. of Algorithms 60, 85–114 (2006)
Lagergren, J.: Efficient parallel algorithms for graphs of bounded tree-width. J. of Algorithms 20, 20–44 (1996)
Nickelsen, A., Tantau, T.: The complexity of finding paths in graphs with bounded independence number. SIAM J. Comput. 34(5), 1176–1195 (2005)
Reingold, O.: Undirected s-t-connectivity in log-space. In: 37th ACM Symposium on Theory of Computing (STOC), pp. 376–385 (2005)
Toda, S.: Counting problems computationally equivalent to computing the determinant. Technical Report CSIM 91-07, Dept. Comp. Sci. and Inform. Math., Univ. Elect.-Comm., Chofu-shi, Tokyo 182, Japan (1991)
Valdes, J., Tarjan, R., Lawlers, E.: The recognition of series parallel digraphs. SIAM J. Comput. 11, 298–313 (1982)
Wanke, E.: Bounded tree-width and LOGCFL. J. of Algorithms 16, 470–491 (1994)
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Jakoby, A., Tantau, T. (2007). Logspace Algorithms for Computing Shortest and Longest Paths in Series-Parallel Graphs. In: Arvind, V., Prasad, S. (eds) FSTTCS 2007: Foundations of Software Technology and Theoretical Computer Science. FSTTCS 2007. Lecture Notes in Computer Science, vol 4855. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77050-3_18
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DOI: https://doi.org/10.1007/978-3-540-77050-3_18
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