On the Multisearching Problem for Hypercubes

In this paper we give improved bounds for the multisearch problem on a hypercube. This is a parallel search problem where the elements in the 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′. This problem is fundamental in computational geometry, for example it models planar point location in a slab. More precisely, we are given on a n-processor hypercube a sorted n-element sequence S, and a set Q of n queries, and we need to find for each query q e Q its location in the sorted S. Note that one cannot solve this problem by sorting S ∪ Q, because every comparison-based parallel sorting algorithm needs to compare a pair q,q′ e Q in constant time. We present an improved algorithm for the multisearch problem, one that takes O(log n(log log n)3) time on a n- processor hypercube. This essentially replaces a logarithmic factor in the time complexities of previous schemes by a (log log n)3 factor. 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]  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).

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

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

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

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

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

[7]  Frank Thomson Leighton.  Introduction to parallel algorithms and architectures: arrays , 1992 .

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

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

[10]  C. Greg Plaxton,et al.  Deterministic sorting in nearly logarithmic time on the hypercube and related computers , 1990, Symposium on the Theory of Computing.

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

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

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