Universal emulations with sublogarithmic slowdown

The existence of bounded degree networks which can emulate the computation of any bounded degree network of the same size with logarithmic slowdown is well-known. The butterfly is an example of such a universal network. Leiserson was the first to introduce the concept of an area-universal network: a network with VLSI layout area A which can emulate any network of the same size and layout area with logarithmic slowdown. His results imply the existence of an N-node network with layout area O(N log² N) which can emulate any N-node planar network with O(log N) slowdown. The main results of this paper are: There exists an N-node network with layout area O(N log² N) which can emulate any N-node planar network with O(loglogN) slowdown. The N-node butterfly (and hypercube) can emulate any network with VLSI layout area N^2-ε (ε>0) with O(loglogN) slowdown. We also discuss sublogarithmic bounds for the slowdown of emulations of arbitrary bounded degree networks