Parallel Numerical Interpolation on Necklace Hypercubes

Abstract

The necklace hypercube has been recently proposed as an attractive topology for multicomputers and was shown to have many desirable properties such as well-scalability and suitability for VLSI implementation. This paper introduces a parallel algorithm for computing an N-point Lagrange interpolation on a necklace hypercube multiprocessor. This algorithm consists of 3 phases: initialization, main and final. There is no computation in the initialization phase. The main phase consists of lceilE/2rceil steps (with E being the number of edges of the network), each consisting of 4 multiplications and 4 subtractions, and an additional step including 1 division and 1 multiplication. Communication in the main phase is based on an all-to-all broadcast algorithm using some Eulerian rings embedded in the host necklace hypercube. The final phase is carried out in three sub-phases. There are lceilk/2rceil steps in the first sub-phase where k is the size of necklace. Each of sub-phases two and three contains n steps. Our study reveals that when implementation cost in taken into account, there is no speedup difference between low-dimensional and high-dimensional necklace networks