Lower time bounds for solving linear diophantine equations on several parallel computational models

Abstract

We consider parallel random access machines (PRAM's) with p processors and distributed systems of random access machines (DRAM's) with p processors being partially joint by wires according to a communication graph. For these computational models we prove lower bounds for testing the solvability of linear Diophantine equations and related problems including the knapsack problem. These bounds are achieved by generalizing and simplifying a lower bound for parallel computation trees due to Yao, introducing a new type of computation trees which models computations of DRAM's, and by generalizing a technique used by Paul and Simon and Klein and Meyer auf der Heide to carry over lower bounds from computation trees to RAM's. Thereby we prove that for many problems, p processors cannot speed up a computation by a factor O(p) but only by a factor O(log(p + 1)) and in the case of DRAM's whose communication network has degree c by a factor O(log(c + 1)) only.