Efficient Parallel Algorithms for Planar st-Graphs

In this paper, we present efficient parallel algorithms for solving several fundamental problems on planar st-graphs. The problems we consider include all-pairs shortest paths in weighted planar st-graphs, single-source shortest paths in weighted planar layered digraphs (which can be reduced to single-source shortest paths in certain special planar st-graphs), and depth-first search in planar st-graphs. The parallel algorithms we obtain are considerable improvement over the previously best known solutions.

[1]  Ming-Yang Kao,et al.  Towards overcoming the transitive-closure bottleneck: efficient parallel algorithms for planar digraphs , 1990, STOC '90.

[2]  Gary L. Miller,et al.  Finding Small Simple Cycle Separators for 2-Connected Planar Graphs , 1986, J. Comput. Syst. Sci..

[3]  Joseph JáJá,et al.  An Introduction to Parallel Algorithms , 1992 .

[4]  Gary L. Miller,et al.  A parallel algorithm for finding a separator in planar graphs , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[5]  Justin R. Smith.  Parallel Algorithms for Depth-First Searches I. Planar Graphs , 1986, SIAM J. Comput..

[6]  Ming-Yang Kao,et al.  All Graphs have Cycle Separators and Planar Directed Depth-First Search is in DNC , 1988, AWOC.

[7]  Robert E. Tarjan,et al.  Fibonacci heaps and their uses in improved network optimization algorithms , 1984, JACM.

[8]  Robert E. Tarjan,et al.  Applications of a planar separator theorem , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[9]  Victor Y. Pan,et al.  Fast and Efficient Solution of Path Algebra Problems , 1989, J. Comput. Syst. Sci..

[10]  Torben Hagerup,et al.  Planar Depth-First Search in O(log n) Parallel Time , 1990, SIAM J. Comput..

[11]  Jon Freeman,et al.  Parallel Algorithms for Depth-First Search , 1991 .

[12]  Xin He,et al.  Efficient Parallel Algorithms for Series Parallel Graphs , 1991, J. Algorithms.

[13]  Joseph JaJa,et al.  Parallel algorithms for planar graph isomorphism and related problems , 1988 .

[14]  John H. Reif,et al.  Depth-First Search is Inherently Sequential , 1985, Inf. Process. Lett..

[15]  Yijie Han,et al.  Efficient parallel algorithms for computing all pair shortest paths in directed graphs , 1992, SPAA '92.

[16]  Richard P. Brent,et al.  The Parallel Evaluation of General Arithmetic Expressions , 1974, JACM.

[17]  Enrico Nardelli,et al.  An Algorithm for Testing Planarity of Hierarchical Graphs , 1987, WG.

[18]  Gary L. Miller,et al.  Finding small simple cycle separators for 2-connected planar graphs. , 1984, STOC '84.

[19]  Mikhail J. Atallah,et al.  An efficient parallel algorithm for the row minima of a totally monotone matrix , 1991, SODA '91.

[20]  R. Tarjan,et al.  A Separator Theorem for Planar Graphs , 1977 .

[21]  Alok Aggarwal,et al.  Notes on searching in multidimensional monotone arrays , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[22]  Philip N. Klein.  Efficient Parallel Algorithms for Chordal Graphs , 1996, SIAM J. Comput..

[23]  Gregory E. Shannon.  A Linear-Processor Algorithm for Depth-First Search in Planar Graphs , 1988, Inf. Process. Lett..

[24]  Xin He,et al.  A Nearly Optimal Parallel Algorithm for Constructing Depth First Spanning Trees in Planar Graphs , 1988, SIAM J. Comput..

[25]  Philip N. Klein,et al.  A linear-processor polylog-time algorithm for shortest paths in planar graphs , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[26]  Mikhail J. Atallah,et al.  Efficient Parallel Algorithms for String Editing and Related Problems , 1990, SIAM J. Comput..

[27]  Jeffrey Scott Vitter,et al.  Parallel Transitive Closure and Point Location in Planar Structures , 1991, SIAM J. Comput..

[28]  S. Sitharama Iyengar,et al.  Introduction to parallel algorithms , 1998, Wiley series on parallel and distributed computing.

[29]  Edith Cohen.  Efficient parallel shortest-paths in digraphs with a separator decomposition , 1993, SPAA '93.

[30]  Victor Y. Pan,et al.  The Parallel Computation of Minimum Cost Paths in Graphs by Stream Contraction , 1991, Inf. Process. Lett..

[31]  Eugene L. Lawler,et al.  The recognition of Series Parallel digraphs , 1979, SIAM J. Comput..

[32]  Ming-Yang Kao,et al.  Planar Strong Connectivity Helps in Parallel Depth-First Search , 1995, SIAM J. Comput..