Congestion and Dialation

Congestion and Dialation - C OM BIN A TORIC A Bolyai...

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COMBINATORICA Bolyai Society – Springer-Verlag C OMBINATORICA 19 (3) (1999) 375– 401 FAST ALGORITHMS FOR FINDING O(Congestion + Dilation) PACKET ROUTING SCHEDULES TOM LEIGHTON, BRUCE MAGGS and ANDR ´ EA W. RICHA Received July 8, 1996 In 1988, Leighton, Maggs, and Rao showed that for any network and any set of packets whose paths through the network are fixed and edge-simple, there exists a schedule for routing the packets to their destinations in O ( c + d ) steps using constant-size queues, where c is the congestion of the paths in the network, and d is the length of the longest path. The proof, however, used the Lov´asz Local Lemma and was not constructive. In this paper, we show how to find such a schedule in O ( m ( c + d )(log P ) 4 (loglog P )) time, with probability 1 - 1 / P β , for any positive constant β ,where P is the sum of the lengths of the paths taken by the packets in the network, and m is the number of edges used by some packet in the network. We also show how to parallelize the algorithm so that it runs in NC . The method that we use to construct the schedules is based on the algorithmic form of the Lov´asz Local Lemma discovered by Beck. 1. Introduction I n this paper, we consider the problem of scheduling the movements of packets whose paths through a network have already been determined. The problem is formalized as follows. We are given a network with n nodes (switches) and m edges (communication channels). Each node can serve as the source or destination of an arbitrary number of packets (or cells or flits , as they are sometimes referred to). Let N denote the total number of packets to be routed. The goal is to route the N packets from their origins to their destinations via a series of synchronized time steps, where at each step at most one packet can traverse each edge and each packet can traverse at most one edge. Without loss of generality, we assume that all edges in the network are used by the path of some packet, and thus that m gives the number of such edges (all the other edges are irrelevant to our problem). Mathematics Subject Classification (1991): 68M20, 68M10, 68M07, 60C05 Tom Leighton is supported in part by ARPA Contracts N00014-91-J-1698 and N00014-92- J-1799. Bruce Maggs is supported in part by an NSF National Young Investigator Award under Grant No. CCR–94–57766, with matching funds provided by NEC Research Institute, and by ARPA Contract F33615–93–1–1330. Part of this research was conducted while Andr´ ea Richa was at Carnegie Mellon University, supported by NSF National Young Investigator Award under Grant No. CCR–94–57766, with matching funds provided by NEC Research Institute, and ARPA Contract F33615–93–1–1330. 0209–9683/99/$6.00 c ± 1999 J´anos Bolyai Mathematical Society
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376 TOM LEIGHTON, BRUCE MAGGS, ANDR ´ EA W. RICHA Figure 1 shows a 5-node network in which one packet is to be routed to each node. The shaded nodes in the figure represent switches, and the edges between the nodes represent channels. A packet is depicted as a square box containing the label of its destination.
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Congestion and Dialation - C OM BIN A TORIC A Bolyai...

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