# Constructing disjoint paths on expander graphs

## Abstract

In a typical parallel or distributed computation model processors are connected by a spars interconnection network. To establish open-line communication between pairs of processors that wish to communicate interactively, a set of disjoint paths has to be constructed on the network. Since communication needs vary in time, paths have to be dynamically constructed and destroyed. We study the complexity of constructing disjoint paths between given pairs of vertices on expander interconnection graphs. These graphs have been shown before to possess desirable properties for other communication tasks. We present a sufficient condition for the existence of K≦nQ edge-disjoint paths connecting any set of K pairs of vertices on an expander graph, where n is the number of vertices and ρ{variant}<1 is some constant. We then show that the computational problem of constructing these paths lies in the classes Deterministic-P and Random-P C. Furthermore, we show that the set of paths can be constructed in probabilistic polylog time in the parallel-distributed model of computation, in which the n participating processors reside in the nodes of the communication graph and all communication is done through edges of the graph. Thus, the disjoint paths are constructed in the very computation model that uses them. Finally, we show how to apply variants of our parallel algorithms to find sets of vertex-disjoint paths when certain conditions are satisfied. © 1989 Akadémiai Kiadó.