Randomized Routing and Sorting on Fixed-Connection Networks

Abstract

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 + log N ) 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(log N ).

Fast Algorithms for Finding O(Congestion + Dilation) Packet Routing Schedules,

“…For the class of leveled networks, Leighton, Maggs, Ranade, and Rao [8] showed that there is a simple on-line randomized algorithm for routing the packets to their destinations within O(c + L + log N) steps, with high probability, where L is the numb e r o f l e v els in the network, and N is the total number of packets.…”

Efficient bufferless packet switching on trees and leveled networks

“…Leveled networks have also been studied in the context of store-and-forward scheduling (with buffers), by Leighton et al [32], where they present an O(C +L+log N) randomized algorithm with constant size buffers. For store-and-forward scheduling, there have been many results on obtaining optimal O(C + D) algorithms for arbitrary networks [31,33,36,40,43].…”

Store-and-Forward Multicast Routing on the Mesh

“…Finally, it is worth pondering the relationship between multicast routing and a form of unicast routing which allows multiple messages sharing a common destination s to be combined into a single message along their way to s [16,10]. In fact, for every instance M of the multicast routing problem in which each message x ∈ M must be delivered from its source s(x) to its various destinations {t 1 (x), t 2 (x), .…”

“…However, none of the known algorithms for unicast routing (with combining) provides the basis for an efficient multicast algorithm. For example, the work by [10] is largely concerned with scheduling the movement of packets along pre-selected paths; the path selection techniques employed (largely based on greedy path selection and random intermediate destinations)…”