Optimum simulation of meshes by small hypercubes

Abstract

We consider optimum simulations of large mesh networks by hypercubes. For any arbitrary mesh M, let "M's optimum hypercube" be the smallest hypercube that has at least as many processors as M and, for any k>0, let Q(M/2^k) be "M's 1/2^k-size hypercube", which has 1/2^k as many processors as M's optimum hypercube. The ratio MI/Q(M/2^k) is called M's 1/2^k- density. We show that (a) for every 2-D mesh M, if M's 1/2-density≤1.828, then M can be embedded into its 1/2-size hypercube with dilation 1 and load factor 2, (b) for every 2-D mesh M, if M's 1/4-density≤3.809, then M can be embedded into its 1/4-size hypercube with dilation 1 and load factor 4, and if M's 1/4-density≤2.8125, then M can be embedded into its 1/4-size hypercube with dilation 1 and load factor 3, (c) If every 2-D mesh M with 1/2^k ₁-density≤a can be embedded into its 1/2^k ₁-size hypercube with dilation 1 and load factor l₁, and every 2-D mesh M with 1/2^k ₂-density ≤b can be embedded into its 1/2^k ₂-size hypercube with dilation 1 and load factor l₂, then we can obtain the densities for load factor l₁+l₂ and load factor l₁×l₂ based on a, b.