An Improved Hypercube Bound for Multisearching and Its Applications

We give a result that implies an improvement by a factor of log log n in the hypercube bounds for the geometric problems of batched planar point location, trapezoidal decomposition, and polygon triangulation. The improvements are achieved through a better solution to the multisearch problem on a hypercube, a parallel search problem where the elements in the data structure S to be searched are totally ordered, but where it is not possible to compare in constant time any two given queries q and q′. Whereas the previous best solution to this problem took O(log n(log log n)3) time on an n-processor hypercube, the solution given here takes O(log n(log log n)2) time on an n-processor hypercube. The hypercube model for which we claim our bounds is the standard one, SIMD, with O(1) memory registers per processor, and with one-port communication. Each register can store O(log n) bits, so that a processor knows its ID.

[1]  F. Leighton,et al.  Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes , 1991 .

[2]  C. Greg Plaxton,et al.  Deterministic sorting in nearly logarithmic time on the hypercube and related computers , 1990, STOC '90.

[3]  John H. Reif,et al.  Randomized Algorithms for Binary Search and Load Balancing on Fixed Connection Networks with Geometric Applications , 1994, SIAM J. Comput..

[4]  Leonidas J. Guibas,et al.  Parallel computational geometry , 1988, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[5]  Andrew Rau-Chaplin,et al.  Implementing Data Structures on a Hypercube Multiprocessor, and Applications in Parallel Computational Geometry , 1989, J. Parallel Distributed Comput..

[6]  Mikhail J. Atallah,et al.  On the Multisearching Problem for Hypercubes , 1995, Comput. Geom..

[7]  D. T. Lee,et al.  Parallel Batched Planar Point Location on the CCC , 1989, Inf. Process. Lett..

[8]  Richard Cole,et al.  Cascading divide-and-conquer: A technique for designing parallel algorithms , 1989, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[9]  Kenneth E. Batcher,et al.  Sorting networks and their applications , 1968, AFIPS Spring Joint Computing Conference.

[10]  Afonso Ferreira,et al.  Parallel Fractional Cascading on Hypercube Multiprocessors , 1992, Comput. Geom..

[11]  Sartaj Sahni,et al.  A Self-Routing Benes Network and Parallel Permutation Algorithms , 1981, IEEE Transactions on Computers.

[12]  Sartaj Sahni,et al.  Data broadcasting in SIMD computers , 1981, IEEE Transactions on Computers.