This paper addresses point-to-point packet routing in undirected networks, which is the most important communication primitive in most networks. The main result proves the existence of routing tables that guarantee a polylog-competitive completion-time $\textbf{deterministically}$: in any undirected network, it is possible to give each node simple stateless deterministic local forwarding rules, such that, any adversarially chosen set of packets are delivered as fast as possible, up to polylog factors. All previous routing strategies crucially required randomization for both route selection and packet scheduling. The core technical contribution of this paper is a new local packet scheduling result of independent interest. This scheduling strategy integrates well with recent sparse semi-oblivious path selection strategies. Such strategies deterministically select not one but several candidate paths for each packet and require a global coordinator to select a single good path from those candidates for each packet. Another challenge is that, even if a single path is selected for each packet, no strategy for scheduling packets along low-congestion paths that is both local and deterministic is known. Our novel scheduling strategy utilizes the fact that every semi-oblivious routing strategy uses only a small (polynomial) subset of candidate routes. It overcomes the issue of global coordination by furthermore being provably robust to adversarial noise. This avoids the issue of having to choose a single path per packet because congestion caused by ineffective candidate paths can be treated as noise. Our results imply the first deterministic universally-optimal algorithms in the distributed supported-CONGEST model for many important global distributed tasks, including computing minimum spanning trees, approximate shortest paths, and part-wise aggregates.

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