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 most$c \geq 1$ packets each time step. We use the Adversarial Queuing Theory injection model, where a$(\rho,\ \sigma)$-adversary may inject at most$\sigma+\rho\cdot t$ 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$(\rho,\ \sigma)$-adversary with$\rho\leq c.$ 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$\Theta(\sigma+\rho)$ are necessary and sufficient for centralized protocols. The butter requirement of$O(1)$-local protocols was recently proved to be$\Theta(\rho\log D+\sigma)$. In this paper, for any$d \geq 2$, we describe a d-local algorithm whose butter space requirement is$O \left( \left\lceil \frac { \log D } { d } \right\rceil \rho + \sigma \right)$. 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.