Optimal gossiping in square 2D meshes

Gossiping is the communication problem in which each node has a unique message to be transmitted to every other node. The nodes exchange their message by packets. A solution to the problem is judged by how many rounds of packet sending it requires. In this paper, we consider the version of the problem in which small-size packets each carrying exactly one message are used. The nodes of the target meshes are assumed to be all-port (a node’s incident edges can all be active at the same time); and their edges are either half-duplex or full-duplex, which are also known as the H* model and the F* model, respectively. We study the class of 2D square meshes. Soch and Tvrdik (SIROCCO’97, pp. 253–265; Tech. rep. DC-97-04, Dept. of CS&E, Czech Technical University) have obtained optimal algorithms for the F* model (for square or nonsquare meshes). Lau and Zhang (IEEE Trans. Parallel Distribut. Syst. 13 (4) (2002) 349–358) have obtained fast algorithms for the H* model. We present optimal algorithms for both models, with the interesting property that they route their messages along the same paths and in the same order, i.e. for any edge {u,v}, the i-th message from u to v under either model is the same message.