Abstract
Computing shortest paths in a directed graph has received considerable attention in the sequential RAM model of computation. However, developing a polylog-time parallel algorithm that is close to the sequential optimal in terms of the total work done remains an elusive goal. We present a first step in this direction by giving efficient parallel algorithms for shortest paths in planar layered digraphs.
We show that these graphs admit special kinds of separators calledone- way separators which allow the paths in the graph to cross it only once. We use these separators to give divide- and -conquer solutions to the problem of finding the shortest paths between any two vertices. We first give a simple algorithm that works in the CREW model and computes the shortest path between any two vertices in ann-node planar layered digraph in timeO(log2 n) usingn/logn processors. We then use results of Aggarwal and Park [1] and Atallah [4] to improve the time bound toO(log2 n) in the CREW model andO(logn log logn) in the CREW model. The processor bounds still remain asn/logn for the CREW model andn/log logn for the CRCW model.
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References
A. Aggarwal and J. Park, Notes on searching multidimensional monotone arrays,Proc. 29th Annual IEEE Symposium on Foundations of Computer Science, 1988, pp. 496–512.
N. Alon and Z. Galil, On the exponent of the all pairs shortest path problem,Proc. 32nd Annual IEEE Symposium on Foundations of Computer Science, 1991, pp. 569–575.
A. Apostolico, M. J. Atallah, L. Larmore, and H. S. Mcfaddin, Efficient parallel algorithms for string editing and related problems,SIAM Journal on Computing,19 (1990), 968–988.
M. J. Atallah, A faster parallel algorithm for a matrix searching problem,Proc. 2nd Scandinavian Workshop on Algorithm Theory, 1990, pp. 193–200.
R. Bellman, On a routing problem,Quarterly Journal of Applied Mathematics,16 (1958), 87–90.
E. Cohen, Efficient parallel shortest-paths in digraphs with a separator decomposition,Proc. 5th Annual Symposium on Parallel Algorithms and Architectures, 1993, pp. 57–67.
R. Cole and U. Vishkin, Optimal parallel algorithms for expression tree evaluation and list ranking,Proc. Third Aegean Workshop on Computing, 1988, pp. 91–100.
A. L. Deicher and S. R. Kosaraju, An NC algorithm for evaluating monotone planar circuits, Manuscript.
G. Di Battista and E. Nardelli, An algorithm for testing planarity of hierarchical graphs,Proc. Workshop WG86, Bernierd, June 1986 (1987), pp. 277–289.
E. W. Dijkstra, A note on two problems in connexion with graphs,Numerische Mathematik,1 (1959), 269–271.
G. N. Frederickson, Fast algorithms for shortest paths in planar graphs, with applications,SIAM Journal on Computing,16 (1987), 1004–1022.
M. L. Fredman and R. E. Tarjan, Fibonacci heaps and their uses in improved network optimization algorithms,Journal of the Association for Computing Machinery,34 (1987), 596–615.
H. Gazit and G. L. Miller, A parallel algorithm for finding a separator in planar graphs,Proc. 28th Annual IEEE Symposium on Foundations of Computer Science, 1987, pp. 238–248.
M. Gondran and M. Minoux, inGraphs and Algorithms, Wiley Interscience, New York, 1984.
Y. Han, V. Pan, and J. H. Reif, Efficient parallel algorithms for computing all pair shortest paths in directed graphs,Proc. Fourth Annual ACM Symposium on Parallel Algorithms and Architectures, 1992, pp. 353–362.
R. M. Karp and V. Ramachandran, A survey of parallel algorithms for shared memory machines, inHandbook of Theoretical Computer Science (J. van Leeuwen, ed.), North-Holland, Amsterdam, 1990, pp. 871–941.
P. N. Klein and S. Subramanian, A linear-processor polylog-time algorithm for shortest-paths in planar graphsProc. 1993 IEEE Symposium on Foundations of Computer Science, 1993, pp. 259–270
G. L. Miller, Finding small simple cycle separators for 2-connected planar graphs,Journal of Computer and System Sciences,32 (1986), 265–279.
G. L. Miller and W. Thurston, Separators in two and three dimensions,Proc. 22nd Annual ACM Symposium on Theory of Computing, 1990, pp. 300–309.
E. Lawler,Combinatorial Optimization: Networks and Matroids, Holt, Rinehart and Winston, New York, 1976.
A. Lempel, S. Even, and I. Cederbaum, An algorithm for planitary testing of graphs,Theory of Graphs, Proc. Internat. Symposium, 1966, pp. 215–232.
R. J. Lipton and R. E. Tarjan, Applications of a planar separator theorem,SIAM Journal on Computing,9 (1980), 615–627.
V. Pan and J. H. Reif, Fast and efficient solution of path algebra problems,Journal of Computer and System Sciences,38 (1989), 494–510.
V. Pan and J. H. Reif, The parallel computation of minimum cost paths in graphs by stream contraction,Information Processing Letters,40 (1991), 79–83.
G. Shannon and F. Wan, Subdividing Planar Graphs in Parallel, Technical Report, Department of Computer Science, Indiana Universty, 1991.
R. Tamassia and F. P. Preparata, Dynamic maintenance of planar digraphs, with applications,Algorithmica,5 (1990), 509–527.
R. Tamassia and J. S. Vitter, Parallel transitive closure and point location in planar structures,SIAM Journal on Computing,20 (1990), 708–725.
J. Valdes, R. E. Tarjan, and E. L. Lawler, The recognition of series parallel digraphs,SIAM Journal on Computing,11 (1982), 298–313.
S. Whitesides, Forms of hierarchy: a selected bibliography,General Systems,14 (1969), 3–15.
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Communicated by C. K. Wong.
Support for the first and third authors was provided in part by a National Science Foundation Presidential Young Investigator Award CCR-9047466 with matching funds from IBM, by NSF Research Grant CCR-9007851, by Army Research Office Grant DAAL03-91-G-0035, and by the Office of Naval Research and the Advanced Research Projects Agency under Contract N00014-91-J-4052, ARPA, Order 8225. Support for the second author was provided in part by NSF Research Grant CCR-9007851, by Army Research Office Grant DAAL03-91-G-0035, and by the Office of Naval Research and the Advanced Research Projects Agency under Contract N00014-91-J-4052 and ARPA Order 8225.
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Subramanian, S., Tamassia, R. & Vitter, J.S. An efficient parallel algorithm for shortest paths in planar layered digraphs. Algorithmica 14, 322–339 (1995). https://doi.org/10.1007/BF01294130
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DOI: https://doi.org/10.1007/BF01294130