Randomized Routing on Fat Trees.

Fat-trees are a class of routing networks for hardware-efficient parallel computation. This paper presents a randomized algorithm for routing messages on a fat-tree. The quality of the algorithm is measured in terms of the load factor of a set of messages to be routed, which is a lower bound on the time required to deliver the messages. This document shows that if a set of messages has load factor lambda on a fat-tree with n processors, the number of delivery cycles routing attempts that the algorithm requires is Olambda lg n lg lg n with probability 1-O1n. The best previous bound was Olambda lg n for the off-line problem where switch settings can be determined in advance. In a VLSI-like model where hardware cost is equated with physical volume, the routing algorithm demonstrates that fat-trees are universal routing networks in the sense that any routing network can be efficiently simulated by a fat-tree of comparable hardware cost.