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 packetreaches its destination in O(C + D) steps, where C’ is the congestion and D is the dilation. The proof relies on the Lovbz 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 dehvering packets without resorting to the Lovi$sz Local Lemma. We improve the distributed algorithm for problems with relatively high dilation to O(C) + (log* n)”(iog” ‘JD -t 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 A(e). We require that for some small constant e > 0, for every edge e, J(e) S 1 – c. In this case we use the parameter R = mw ri, 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 traf*Supported in part by the NSF PYI award of $va Tardos. Part of this work was performed while visiting the School of OR&IE at Cornell, and while a postdoctoral fellow at the University of Toronto Computer Science Department. Work at the Technion supported in part by the Ruth and David Moskowitz Academic Lectureship award. t Research supported in part by a Packard Fellowship and ZUI NSF PYI award, by NSF through grant DMS 9505155, and ONR through grant NOO014-961-0050. Permission to make digitel/bard copies of all or pad of this material for personal or classroom use is granted without fee provided that the copies are not made or distributed for profit or conunereial advantage, the copyright notice, the title of the publication and its date appear, and notice is given that copyright is by permission of tie ACM, Inc. To c~y otherwise, to republish, to peat on servers or to redlatribute to lists, requmes specific permission and/or fee. STOC’96, Philadelphia PA, USA @ 1996 ACM 0-89791-785.5/96/05. .$3.50 fic for handling bursty communication. The model is motivated by the new adversarial model suggested by Borodin et al.