Average-Case Communication-Optimal Parallel Parenthesis Matching

We provide the first non-trivial lower bound, p - 3/p ? n/p, where p is the number of the processors and n is the data size, on the average-case communication volume, ?, required to solve the parenthesis matching problem and present a parallel algorithm that takes linear (optimal) computation time and optimal expected message volume, ? + p.The kernel of the algorithm is to solve the all nearest smaller values problem. Provided n/p = ?(p), we present an algorithm that achieves optimal sequential computation time and uses only a constant number of communication phases, with the message volume in each phase bounded above by (n/p + p) in the worst case and p in the average case, assuming the input instances are uniformly distributed.

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

[2]  Clifford Stein,et al.  Introduction to Algorithms, 2nd edition. , 2001 .

[3]  Andrew Rau-Chaplin,et al.  Scalable parallel geometric algorithms for coarse grained multicomputers , 1993, SCG '93.

[4]  Allan Borodin,et al.  Routing, Merging, and Sorting on Parallel Models of Computation , 1985, J. Comput. Syst. Sci..

[5]  Uzi Vishkin,et al.  Finding all nearest neighbors for convex polygons in parallel: A new lower bound technique and a matching algorithm , 1990, Discret. Appl. Math..

[6]  Clyde P. Kruskal,et al.  Searching, Merging, and Sorting in Parallel Computation , 1983, IEEE Transactions on Computers.

[7]  Gad M. Landau,et al.  Optimal parallel suffix-prefix matching algorithm and applications , 1989, SPAA '89.

[8]  Uzi Vishkin,et al.  Optimal Doubly Logarithmic Parallel Algorithms Based on Finding All Nearest Smaller Values , 1993, J. Algorithms.

[9]  Gerhard Goos,et al.  Computer Science Today: Recent Trends and Developments , 1995 .

[10]  Zvi Galil,et al.  An Optimal O(log log n) Time Parallel String Matching Algorithm , 1990, SIAM J. Comput..

[11]  Selim G. Akl.  Parallel computation: models and methods , 1997 .

[12]  C. Greg Plaxton,et al.  All Nearest Smaller Values on the Hypercube , 1996, IEEE Trans. Parallel Distributed Syst..

[13]  Richard M. Karp,et al.  Parallel sorting with limited bandwidth , 1995, SPAA '95.

[14]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[15]  John H. Reif,et al.  Synthesis of Parallel Algorithms , 1993 .

[16]  Leslie G. Valiant,et al.  A bridging model for parallel computation , 1990, CACM.

[17]  Chun-Hsi Huang,et al.  Communication Efficient BSP Algorithm for All Nearest Smaller Values Problem , 2001, J. Parallel Distributed Comput..