Randomized Routing and Sorting on Fixed-Connection Networks

This paper presents a general paradigm for the design of packet routing algorithms for fixed-connection networks. Its basis is a randomized on-line algorithm for scheduling any set of N packets whose paths have congestion c on any bounded-degree leveled network with depth L in O(c+L+ logN) steps, using constant-size queues. In this paradigm, the design of a routing algorithm is broken into three parts: (1) showing that the underlying network can emulate a leveled network, (2) designing a path selection strategy for the leveled network, and (3) applying the scheduling algorithm. This strategy yields randomized algorithms for routing and sorting in time proportional to the diameter for meshes, butterflies, shuffle-exchange graphs, multidimensional arrays, and hypercubes. It also leads to the construction of an area-universal network: an N -node network with area Θ(N) that can simulate any other network of area O(N) with slowdown O(logN). This research was supported by the Defense Advanced Research Projects Agency under Contracts N00014–87–K–825 and N00014–89–J–1988, the Office of Naval Research under Contracts N00014–86–K–0593 and N00014–86–K–0564, the Air Force under Contract OSR–89–0271, and the Army under Contract DAAL–03–86–K-0171. Tom Leighton is supported by an NSF Presidential Young Investigator Award with a matching funds provided by IBM. Laboratory for Computer Science, MIT, Cambridge, MA. Department of Mathematics, MIT, Cambridge, MA. Department of Electrical Engineering and Computer Science, University of California, Berkeley, CA. Aiken Computation Laboratory, Harvard University, Cambridge, MA. Second and third authors’ current address: NEC Research Institute, Princeton NJ.