Parallel and Pipelined Parallel Consecutive Sums on a Hypercube with Application to Ray Casting

Abstract

Communication penalty for parallel computation is related to message startup time and speed of data transmission between the host and processing elements (PEs). We propose two algorithms in this paper to show that the first factor can be alleviated by reducing the number of messages and the second by making the host-PE communication concurrent with computation on the PE array. The algorithms perform 2n consecutive sums of 2n numbers each on a hypercube of degree n. The first algorithm leaves one sum on each processor. It takes n steps to complete the sums and reduces the number of messages generated by a PE from 2n to n. The second algorithm sends all the sums back to the host as the sums are generated one by one. It takes 2n+n−1 steps to complete the sums in a pipeline so that one sum is completed every step after the initial (n−1) steps. We apply our second algorithm to the front-to-back composition for ray casting. For large number of rays, the efficiency and speedup of our algorithm are close to theoretically optimal values.