On the multisearching problem for 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 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. 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' [??]Q in constant time. We present an improved algorithm for the multisearch problem, one that takes 0(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, with 0(1) memory registers per processor, and with one-port communication. Each register can store 0(log n) bits, so that a processor knows its ID

Truly efficient parallel algorithms: 1-optimal multisearch for an extension of the BSP model

AbstractIn this paper we design and analyse parallel algorithms with the goal to get exact bounds on their speed-ups on real machines. For this purpose we define an extension of Valiant's BSP model, BSP∗, that rewards blockwise communication, and use Valiant's notion of 1-optimality. Intuitively, a 1-optimal parallel algorithm for p processors achieves speed-up close to p. We consider the Multisearch Problem: Assume a strip in 2D to be partitioned into m segments. Given n query points in the strip, the task is to locate, for each query, its segment. For m ⩽n⩾ p we present a deterministic BSP∗ algorithm that is 1-optimal, if np⩾log2n. For m>n⩾p, we present a randomized BSP∗ algorithm that is l-optimal with high probability, if m⩽2p and n/p⩾log3n. Both results hold for a wide range of BSP∗ parameters where the range becomes larger with growing input size n. We further report on implementation work. Previous parallel algorithms for Multisearch were far away from being 1-optimal in our model and did not consider blockwise communication