A Network Configuration Algorithm Based on Optimization of Kirchhoff Index

Traditionally, a parallel application is partitioned, mapped and then routed on a network of compute nodes where the topology of the interconnection network is fixed and known beforehand. Such a topology often comes with redundant links to accommodate the communication patterns of a wide range of applications. With recent advances in technology for optical circuit switches, it is now possible to construct a network with much fewer links, and to make the link endpoints configurable to suit the communication pattern of a given application. While this is economical (saving both links and the power to run them), it raises the difficult problem of how to configure the network and how to reconfigure it quickly when the application's communication pattern changes. In this paper, we propose the Kirchhoff index (KI) of a certain weighted graph related to the interconnection network as a proxy for its communication throughput. Our usage of this metric is based on a theoretical analogy between resistances in an electrical network and communication loads in the interconnection network. We show how mathematical techniques for reducing KI can be used to configure a network in a dramatically shorter time as compared to the current state-of-the-art scheme.