Spanders: Distributed spanning expanders

We consider self-stabilizing and self-organizing distributed construction of a spanner that forms an expander. We use folklore results to randomly define an expander graph. Given the randomized nature of our algorithms, a monitoring technique is presented for ensuring the desired results. The monitoring is based on the fact that expanders have a rapid mixing time and the possibility of examining the rapid mixing time by O(nlogn) short (O(log⁴ n) length) random walks even for non regular expanders. We then employ our results to construct a hierarchical sequence of spanders, each of them an expander spanning the previous one. Such a sequence of spanders may be used to achieve different quality of service assurances in different applications. Several snap-stabilizing algorithms that are used to utilize the monitoring are presented, including reset and token tracing algorithms for message passing systems.