Virtually all theoretical work on message routing in parallel computers has dwelt on packet routing: messages are conveyed as packets, an entire packet can reside at a node of the network, and a packet is sent from the queue of one node to the queue of another node until its reaches its destination. The current trend in multicomputer architecture, however, is to use wormhole routing. In wormhole routing a message is transmitted as a contiguous stream of bits, physically occupying a sequence of nodes/edges in the network. Thus, a message resembles a worm burrowing through the network. The authors give theoretical analyses of simple wormhole routing algorithms, showing them to be nearly optimal for butterfly and mesh connected networks. The analysis requires initial random delays in injecting messages to the network. They report simulation results suggesting that the idea of random initial delays is not only useful for theoretical analysis but may actually improve the performance of wormhole routing algorithms. >

A theory of wormhole routing in parallel computers

In this paper we present randomized algorithms for kk routing, k-k sorting, and cut through routing on the mesh connected processor array. In these three problems, each processor is assumed to contain k packets at the beginning and k packets are destined for any processor node with k ≥ 1. We give two different algorithms for k-k routing that run in kn 2 +o(kn) and k2 n+o(kn) routing steps respectively. We also show that k-k sorting can be accomplished within k2 n + n + o(kn) steps and that cut through routing can be done in kn 2 + n k + 3 2n+ o(kn) steps. The stated resource bounds hold with high probability and for any k ≥ 8. The best known previous algorithms take almost twice as many routing steps in any case. For k ≤ 8 we derive new bounds which come close to the optimum. kn 2 is a known lower bound for all three problems, the bisection bound. Hence, our algorithms are very nearly optimal. All the above mentioned algorithms have optimal queue length, namely k + o(k). These algorithms also extend to higher dimensional meshes. The achieved improvements are made possible by novel algorithmic and analytical techniques.

Matching the bisection bound for routing and sorting on the mesh

A Theory of Wormhole Routing in Parallel Computers (Extended Abstract)

https://repository.upenn.edu/cgi/viewcontent.cgi?article=1355&context=cis_reports

k-k routing, k-k sorting, and cut-through routing on the mesh

Virtually all theoretical work on message routing in parallel computers has dwelt on packet routing: messages are conveyed as packets, an entire packet can reside at a node of the network, and a packet is sent from the queue of one node to the queue of another node until its reaches its destination. A trend in multicomputer architecture, however, is to use wormhole routing. In wormhole routing a message is transmitted as a contiguous stream of bits, physically occupying a sequence of nodes/edges in the network. Thus, a message resembles a worm burrowing through the network. In this paper we give theoretical analyses of simple wormhole routing algorithms, showing them to be nearly optimal for butterfly and mesh connected networks. Our analysis requires initial random delays in injecting messages to the network. We report simulation results suggesting that the idea of random initial delays may have an impact beyond theoretical analysis.

The bit-serial routing problem wherein each packet consists of a sequence of k flits and is thus called a snake, is considered. On the basis of the properties of the snake during the routing, a formal definition is given for three different packet routing models, namely, the store-and-forward model, the cut-through model, and the wormhole model. The wormhole model, which is most commonly used in practice, is studied. The first algorithms (deterministic and probabilistic) based on the wormhole model for the permutation routing problem on a chain, on a square mesh, and on a square torus are given. A new lower bound is derived for distance-limited permutation routing on a ring of processors, and an algorithm that matches this lower bound if the packets are routed independently is given. >

A. Simvonis

Papers

On bit-serial packet routing for the mesh and the torus