Space-Optimal Packet Routing on Trees

We consider packet forwarding on a tree with all packets destined for the root, assuming each link may forward at mostc ≥ 1 packets each time step. We use the Adversarial Queuing Theory injection model, where a (ρ, σ)-adversary may inject at most σ+ρ· t packets into the network at arbitrary locations during any time interval of length t. The goal is to find a forwarding protocol that minimizes the maximal butter space required to avoid overflows against a (ρ,σ)-adversary with ρ≤c. We consider protocols from the locality viewpoint. A protocol is called d-local if the actions of a node depend only on the current state of nodes at distance at most d. A D-local protocol, where D is the network diameter, is called centralized. It is known that buffers of size Θ(σ+ρ) are necessary and sufficient for centralized protocols. The butter requirement ofO(1) -local protocols was recently proved to be Θ(ρ\log D+σ). In this paper, for anyd ≥ 2, we describe a d-local algorithm whose butter space requirement isO left( left lceil \log D } { d } \right\rceil ρ + σ right). This result is tight, up to constant factors. In particular, it implies that O(\log D) locality is sufficient to achieve the best worst-case performance possible even for centralized algorithms. We also give evidence suggesting that the butter requirement of a local algorithm designed for trees is good also when the routes do not constitute a single-destination tree.