Full reversal routing as a linear dynamical system

Link reversal is a versatile algorithm design paradigm, originally proposed by Gafni and Bertsekas in 1981 for routing, and subsequently applied to other problems including mutual exclusion and resource allocation. Although these algorithms are well-known, until now there have been only preliminary results on time complexity, even for the simplest link reversal scheme for routing, called Full Reversal (FR). In this paper we tackle this open question for arbitrary communication graphs. Our central technical insight is to describe the behavior of FR as a dynamical system, and to observe that this system is linear in the min-plus algebra. From this characterization, we derive the first exact formula for the time complexity: Given any node in any (acyclic) graph, we present an exact formula for the time complexity of that node, in terms of some simple properties of the graph. These results for FR are instrumental in analyzing a broader class of link reversal routing algorithms, as we show in a companion paper that such algorithms can be reduced to FR. In the current paper, we further demonstrate the utility of our formulas by using them to show the previously unknown fact that FR is time-efficient when executed on trees.