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′. 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 Q its location in the sorted S. 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 problem is fundamental in computational geometry, for example it models planar point location in a slab. We give as application a trapezoidal decomposition algorithm with the same time complexity on a n log 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.