Network routing on regular directed graphs from spanning factorizations

TL;DR

This paper introduces spanning factorizations as a method to extend routing schemes from a single node to all nodes in symmetric networks, enhancing network optimization for parallel processing.

Contribution

It defines spanning factorizations and demonstrates their applicability to Cayley and vertex transitive graphs for efficient routing.

Findings

Spanning factorizations enable conflict-free routing schemes.

All Cayley graphs have spanning factorizations.

Many vertex transitive graphs also possess spanning factorizations.

Abstract

Networks with a high degree of symmetry are useful models for parallel processor networks. In earlier papers, we defined several global communication tasks (universal exchange, universal broadcast, universal summation) that can be critical tasks when complex algorithms are mapped to parallel machines. We showed that utilizing the symmetry can make network optimization a tractable problem. In particular, we showed that Cayley graphs have the desirable property that certain routing schemes starting from a single node can be transferred to all nodes in a way that does not introduce conflicts. In this paper, we define the concept of spanning factorizations and show that this property can also be used to transfer routing schemes from a single node to all other nodes. We show that all Cayley graphs and many (perhaps all) vertex transitive graphs have spanning factorizations.