Optimal Schedules for Parallel Prefix Computation with Bounded Resources

Abstract

Given x 1 ; . . . ; xN , parallel prefix computes x 1 ffi x 2 ffi . . . ffi x k , for 1 k N , with associative operation ffi. We show optimal schedules for parallel prefix computation with a fixed number of resources p 2 for a prefix of size N p(p + 1)=2 . The time of the optimal schedules with p resources is d2N=(p + 1)e for N p(p + 1)=2, which we prove to be the strict lower bound(i.e., which is what can be achieved maximally). We then present a pipelined form of optimal schedules with d2N=(p + 1)e + d(p 0 1)=2e 0 1 time, which takes a constant overhead of d(p 0 1)=2e time more than the optimal schedules. Parallel prefix is an important common operation in many algorithms including the evaluation of polynomials, general Hornor expressions, carry look-ahead circuits and ranking and packing problems. A most important application of parallel prefix is loop parallelizing transformation. 1 Introduction Given x 1 ; . . . ; xN , parallel prefix computes x 1 ffi x 2 ffi . . . ffi x...