Time-optimal simulations of networks by universal parallel computers

Abstract

For technological reasons, in a realistic parallel computer the processors have to communicate via a communication network with bounded degree. Thus the question for a “good” communication network comes up. In this paper we present such a network, a universal parallel computer (UPC) with the following properties:

• (i) It has optimal time-loss, namely O(log c) for simulating networks of degree c. (We also prove the lower bound Ω(log c) for the time-loss.)

• (ii) We introduce the broadcast-capability (how many processors can be reached by one processor in i steps?) and demonstrate its influence on the number of processors needed for a simulation of a network with n processors. E.g. for broadcast-capability O(c ⁱ) (e.g. networks with degree c), O(n ^1+ε log n) processors are needed (ε>0 arbitrary) whereas O(n · polylog(n)) processors suffice for networks with polynomial broadcast-capability (e.g. k-dimensional grids).

• (iii) The UPC is potentially infinite and has multi-user capabilities, i.e., it can be arbitrarily partitioned into finite UPC's each with the above efficiency.

This construction generalizes a UPC described in [MadH2], where, given a fixed degree c, for each n a UPC M ₀ is constructed which needs O(n ^1+ε log n) processors to achieve constant time-loss for simulating networks with n processors and degree c.

Supported in part by DFG-grants Me 872/1–2 and We 1066/2-1