On the Multisearching Problem of Hypercubes

We build on the work of Dehne and Rau-Chaplin and 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 aTe 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 SUQ, because every comparisonbased 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 0(1ogn(loglogn)3) time on a n-processor hypercube. This essentially replaces a logarithmic fador in the time complexities of previous schemes by a (log log n)3 fador. The hypercube model for which we claim our bounds is the standard one, with 0(1) memory registers per processor, and with oncport communication. Each register can store O(logn) bits, so that a processor knows its ID.