Circuit-Switched Broadcast in Multi-Port 2D Tori

This paper studies the one-to-all broadcast in a circuit-switched 2D torus of any size with-port capability. This is a generalization of the one-port and all-port models. Existing results, as compared to ours, can only solve very restricted sizes of tori, and use more numbers of steps. 1 Introduction One primary communication in an interconnection network is the one-to-all broadcast, where a source node needs to send a message to all other nodes in the network. This paper studies the scheduling of one-to-all broadcast in a 2D circuit-switched torus. The network is assumed to use the-port communication model, in which a node can send up to messages and simultaneously receive up to messages at a time, where 1 4. This is a generalization of the one-port model (= 1) and the all-port model (= 2k). Following the formulation in many works 2, 3, 4, 5], this is achieved by a sequence of steps, where a step consists of a set of congestion-free communication paths each indicating a message delivery. The goal is to minimize the total number of steps used. One-to-all broadcast has been studied for meshes and torus based on diierent port models and switching models 1, 2, 3, 4, 5, 6]. Some of these results are compared and summarized in Table 1. Our result improves over existing results 1, 2, 4] in both the number of broadcast steps used and its applicability in network sizes. A 2D torus of size n 1 n 2 is denoted as T n1n2. Each node is denoted as p x1;x2 , 0 x i < n i. The torus is assumed to have ports, in that a node can send up to messages along any of its 4 ports. Lemma 1. In a 2D-port torus T n1n2 , a lower bound on the number of steps to perform one-to-all broadcast is dlog +1 n 1 n 2 e. Deenition 1. In Z 2 , given a node x, an m-tuple of vectors B = (b 1 ; b 2