Communication complexity of PRAMs

Abstract

We propose a model for the concurrent read exclusive write PRAM that captures its communication and computational requirements. For this model, we present several results, including the following:

Two n×n matrices can be multiplied in O(n ³/p) computation time and O(n ²/p ^2/3) communication delay using p processors (for p≤n ³ / log^3/2 n). Furthermore, these bounds are optimal for arithmetic on semirings (using +, × only). For sorting and for FFT graphs, it is shown that communication delay of Ω(n log n/(p log(n/p)) is required for p≤n/ log n. This bound is tight for FFT graphs; it is also shown to be tight for sorting provided p≤n ^1−ε for any fixed ε>0.

Given a binary tree, τ, with n leaves and height h, let D _opt (τ) denote the minimum communication delay needed to compute τ. It is shown that Ω(log n)≤D _opt (τ)≤$O(\sqrt n )$, and $\Omega (\sqrt h )$≤D _opt ≤O(h), all bounds being the best possible. We also present a simple polynomial algorithm that generates a schedule for computing τ with at most 2D _opt (τ) delay.

It is shown that the a communication delay-computation time tradeoff given by Papadimitriou and Ullman for a diamond dag can be achieved for essentially two values of the computation time. We also present DAGs that exhibit proper tradeoffs for a substantial range of time.

(Preliminary Version)