More general parallel tree contraction: Register allocation and broadcasting in a tree

Abstract

We extend the classic parallel tree-contraction technique of Miller and Reif to handle the evaluation of a class of expression trees that does not fit their original framework. We discuss applications to the following problems: (1) Register allocation, i.e., computing the number of registers needed to evaluate a given expression if all intermediate results must be kept in registers; and (2) Broadcasting in a tree, i.e., computing the number of steps needed to transmit a message from the root to all other nodes in a given tree if each node is a processor that can communicate with a single neighbor in each step. We show that on inputs of size n, both problems can be solved with optimal speedup in 0((log n)2) time on an EREW PRAM, in O(log n log log n) time on a CREW PRAM, and in O(log n) time on a CRCW PRAM.