Gossiping in cayley graphs by packets

Abstract

Gossiping (also called total exchange or all-to-all communication) is the process of information di usion in which each node of a network holds a packet that must be communicated to all other nodes in the network. We consider here gossiping in the store-and-forward, fullduplex and -port (or shouting) model. In such a model, the protocol consists of a sequence of rounds and during each round, each node can send (and receive) messages from all its neighbors. The great majority of the previous works on gossiping problems allows the messages to be freely concatenated and so messages of arbitrary length can be transmitted during a round. Here we restrict the problem to the case where at each round communicating nodes can exchange exactly one packet. We give a lower bound of N ?1 , where is the minimum degree, and show that it is attained in Cayley symmetric digraphs with some additional properties. That implies the existence of an optimal gossiping protocol for classical networks like hypercubes, k-dimensional tori, and star-graphs.

Fast collective communication by packets in the postal model

“…Gossiping in interconnection networks The latency l can be seen as representing the amount of time needed to let the receiver get an atomic message with messages of limited size was recently considered in [8,9,14,15,18] under different communication models. initially held by another processor measured in the units Analogous problems in bus networks were considered in rithm on n processors is bounded below by the integer [17,22].…”

Complete Rotations in Cayley Graphs

“…Bermond, Kodate and Perennes define in [4] the concept of complete rotation in Cayley graphs in order to construct a gossip algorithm from a broadcast protocol applied to each vertex simultaneously. Given particular conditions on the orbits of the vertices under the complete rotation, they provide an optimal gossip algorithm.…”

Frobenius Circulant Graphs of Valency Four

“…Here by gossiping we mean an information dissemination process for which each vertex has a distinct message to be sent to all other vertices. In this paper we consider the store-and-forward, all-port and full-duplex model [3]: a vertex must receive a message wholly before retransmitting it to other vertices; a vertex can exchange messages (which may be different) with all of its neighbours at each time step; messages can traverse an edge in both directions simultaneously; no two messages can transmit over the same arc at the same time; [3] Frobenius circulant graphs of valency four 271…”

“…A gossiping scheme is a procedure fulfilling the gossiping under these constraints, and the minimum gossip time [3] of a graph , t ( ), is the minimum number of time steps required by such a scheme.…”

“…A complete rotation [3,8,9] of a Cayley graph Cay(K , S) is a group automorphism of K which fixes S setwise and induces a cyclic permutation on S. A connected graph is said to be G-arc transitive if G ≤ Aut( ) is transitive on Arc( ), the set of arcs of . Let Z * n = {[m] | 0 < m < n, gcd(m, n) = 1} be the multiplicative group of units of the ring Z n .…”