Distributed packet switching in arbitrary networks

In a seminal paper Leighton, Maggs, and Rao consider the packet scheduling problem when a single packet has to traverse each path. They show that there exists a schedule where each packet reaches its destination in 0(C + D) steps, where C is the congestion and D is the dilation. The proof relies on the Lovász Local Lemma, and hence is not algorithmic. In a followup paper Leighton and Maggs use an algorithmic version of the Local Lemma due to Beck to give centralized algorithms for the problem. Leighton, Maggs, and Rao also give a distributed randomized algorithm where all packets reach their destinations with high probability in O(C + D log n) steps. In this paper we develop techniques to guarantee the high probability of delivering packets without resorting to the Lovász Local Lemma. We improve the distributed algorithm for problems with relatively high dilation to O(C) + (log^∗ n)^O(log^{∗ n})D + poly(log n). We extend the techniques to handle the case of infinite streams of regularly scheduled packets along every path. Here we measure the congestion on an edge e by the sum of the rates of the packet streams that use the edge, denoted by λ(e). We require that for some small constant ∈ > 0, for every edge e, A(e) ≤ 1 - ∈. In this case we use the parameter R = max_i r_i, the maximum distance between packets of the same stream, instead of the congestion C above. We notice that max{R, D} is a worst case lower bound on the maximum delay of a packet. We also extend the results to a model of packet traffic for handling bursty communication. The model is motivated by the new adversarial model suggested by Borodin et al.