Link Reversal Routing with Binary Link Labels: Work Complexity

Abstract

Full Reversal and Partial Reversal are two well-known routing algorithms that were introduced by Gafni and Bertsekas [IEEE Trans. Commun., 29 (1981), pp. 11-18]. By reversing the directions of some links of the graph, these algorithms transform a connected input DAG (directed acyclic graph) into an output DAG in which each node has at least one path to a distinguished destination node. We present a generalization of these algorithms, called the link reversal (LR) algorithm, based on a novel formalization that assigns binary labels to the links of the input DAG. We characterize the legal link labelings for which LR is guaranteed to establish routes. Moreover, we give an exact expression for the number of steps-called work complexity-taken by each node in every execution of LR from any legal input graph. Exact expressions for the per-node work complexity of Full Reversal and Partial Reversal follow from our general formula; this is the first exact expression known for Partial Reversal. Our binary link labels formalism facilitates comparison of the work complexity of certain link labelings-including those corresponding to Full Reversal and Partial Reversal-using game theory. We consider labelings in which all incoming links of a given node i are labeled with the same binary value μ i. Finding initial labelings that induce good work complexity can be considered as a game in which to each node i a player is associated who has strategy μ i. In this game, one tries to minimize the cost, i.e., the number of steps. Modeling the initial labelings as this game allows us to compare the work complexity of Full Reversal and Partial Reversal in a way that provides a rigorous basis for the intuition that Partial Reversal is better than Full Reversal with respect to work complexity.

Time Complexity of Link Reversal Routing

“…We now recall the main properties of the LR algorithm established in Charron-Bost et al [2013], and we briefly sketch out their proofs. For the proof, one first observes that from the rules R1 and R2 of the algorithm, it follows that after at most two steps by a node, each incident link must have been reversed at least once; that is, each neighbor must have taken a step.…”

“…This allows us to give a simple expression of the work vector W(t) and to prove that the sequence W(t) t≥0 is stationary. We thus obtain an expression of the limit vector in terms of the sole link-labeled input graph, providing a direct and more conceptual proof of the work complexity result in Charron-Bost et al [2013]. For the time complexity, we consider the dual vector to W(t), namely, the time vector T (n), whose ith component is equal to the time at which node i makes its nth reversal.…”

“…Recently, Charron-Bost et al [2013] have taken a different approach for link reversal, which does not use node heights. Instead, they consider an initial directed graph, assign binary labels to the links of the graph, and use the labels on the incident links to decide which links to reverse and which labels to change.…”

“…In particular, the exact work complexities for each node in Full Reversal and Partial Reversal were obtained. An analysis of the work-complexity tradeoffs between Full Reversal and Partial Reversal based on game theory also appeared in Charron-Bost et al [2013].…”

“…Previous work [Charron-Bost et al 2013] exploited the formulas for work complexity to show that Partial Reversal is better than Full Reversal regarding work complexity. In contrast, using the time complexity formulas developed in this article, we show that Full Reversal is better than Partial Reversal in some aspects of time complexity: in particular, Full Reversal has linear time complexity on trees, while Partial Reversal on trees can take quadratic time.…”