Time-Optimal Simulations of Networks by Universal Parallel Computers

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 (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 innuence 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+" log n) processors are needed (" > 0 arbitrary) whereas O(npolylog(n)) processors suuce for networks with polynomial broadcast-capability (e.g. k-dimensional grids). (iii) The UPC is potentially innnite and has multiuser capabilities, i.e. it can be arbitrarily partitioned into nite UPCs each with the above eeciency. This construction generalizes a UPC described in MadH2], where, given a xed degree c, for each n a UPC M 0 is constructed which needs O(n 1+" log n) processors to achieve constant time-loss for simulating networks with n processors and degree c.