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(log4n) 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.