Balanced Generalized Hypercubes: Complexity and Cost/Performance Analysis

The BGHC is a generalized hypercube that has exactly w nodes along each of the d dimensions for a total of wd nodes. A BGHC is said to be maximal if the w nodes along each dimension form a complete directed graph. A BGHC is said to be minimal if the w nodes along each dimension form a unidirectional ring. Lower bound complexities are derived for three intensive communication patterns assuming the balanced generalized hypercube (BGHC) topology. A maximal N node BGHC with a node degree equal to αlog2N, where α≥2, can process certain intensive communication patterns α(α-1) times faster than an N node binary hypercube (which has a node degree equal to log2N). On the other hand, a minimal N node BGHC with a node degree equal to , where β≥2, is 2β times slower at processing certain intensive communication patterns than an N node binary hypercube. For certain communication patterns, increasing one unit cost gains a normalized speedup to the binary hypercube by wlog2w times for the maximal BGHC. For the minimal BGHC, reducing one unit cost gains times speedup normalized to the binary hypercube.