On The Limits To Speed Up Parallel Machines By Large Hardware And Unbounded Communication

Lower bounds for sequential and parallel random access machines (RAM's, WRAM's) and distributed systems of RAM's (DRAM's) are proved. We show that, when p processors instead of one are available, the computation of certain functions cannot be speeded up by a factor p but only by a factor 0 (log(p)). For DRAM's with communication graph of degree c a maximal speedup 0 (log(c)) can be achieved for these problems. We apply these results to testing the solvability of linear diophantine equations. This generalizes a lower bond of Yao for parallel computation trees. Improving results of Dobkin/Lipton and Klein/Meyer auf der Heide, we establish large lower bounds for the above problem on RAM's. Finnaly we prove that at least log (n) + 1 steps are necessary for computing the sum of n integers by a WRAM regardless of the number of processors and the solution of write conflicts.