Length and Number of Buses for Gossiping in Networks Length and Number of Buses for Gossiping in Networks Length and Number of Buses for Gossiping in Networks

Pierre Fraigniaud, Christian Laforest, Pierre Fraigniaud, Christian Laforest, Pierre Fraigniaud, Christian Laforest
1994 unpublished
Gossiping is an information dissemination problem in which e a c h n o d e o f a c o m m unication network has a unique piece of information that must be transmitted to all the other nodes. A bus network is a network of processing elements that communicate by sending messages along buses in a sequence of calls. We assume that (i) each node can participate to at most one call at a time, (ii) a node can either read or write on a bus, (iii) no more than one node can write on a given bus at a given
more » ... iven bus at a given time, and (iv) communicating a message on a bus takes a unit of time. This model extends the telegraph model in allowing the number of nodes connected to each bus to be as large as needed, instead on being bounded by 2. In this paper, we are interested in minimizing the \hardware" of a bus network in keeping optimal the communication performances for solving the gossiping problem. More precisely, w e compute the minimum number of buses required for a gossiping to be optimal. Similarly, w e g i v e upper bounds on the minimum length of buses required for a gossiping to be optimal. Finally, w e combine the two approaches in trying to minimize both parameters: length and number of buses. R esum e L' echange total consiste en des echanges de donn ees entre les processeurs d'un r eseau d'interconnexion oo u c haque processeur poss ede une information qu'il doit diiuser a l'ensemble des autres processeurs. Un r eseau par bus est un r eseau d'interconnexion dont l e s el ements echangent leurs informations au moyen de bus. Dans cet article, nous supposons que (i) chaque processeur ne peut participer qu" a une communication a u n i n s t a n t donn e, (ii) chaque processeur peut soit lire ou ecrire sur un bus, mais pas les deux a la fois, (iii) au plus un processeur peut ecrire sur un bus donn e e a un instant donn e, et (iv) la communication d'une information sur un bus co^ ute une unit e de temps. Ce mod ele peut ^ etre vu comme une extension du mod ele t el egraphe en supposant que le nombre de processeurs connect es a u n m ^ eme bus peut ^ etre aussi grand que souhait e a u l i e u d ' ^ etre limit e e a deux. Dans cet article, nous nous sommes int eress es a minimiser le \mat eriel" d'un r eseau par bus tout en permettant d e r esoudre de faa con optimale le probl eme de l' echange total. Plus pr ecis ement, nous calculons le nombre minimum de bus que requiert un r eseau par bus aan de r ealiser l' echange total de faa con optimale. De la m^ eme mani ere, nous donnons des bornes sup erieures sur la longueur des bus que requiert un r eseau par bus aan de r ealiser l' echange total de faa con optimale. Finalement, nous combinons les deux approches en essayant de minimiser ces deux param etres : longueur et nombre de bus. Abstract Gossiping is an information dissemination problem in which each n o d e o f a c o m m unication network has a unique piece of information that must be transmitted to all the other nodes. A bus network is a network of processing elements that communicate by sending messages along buses in a sequence of calls. We assume that (i) each node can participate to at most one call at a time, (ii) a node can either read or write on a bus, (iii) no more than one node can write on a given bus at a given time, and (iv) communicating a message on a bus takes a unit of time. This model extends the telegraph model in allowing the number of nodes connected to each b u s to be as large as needed, instead on being bounded by 2. In this paper, we are interested in minimizing the \hardware" of a bus network in keeping optimal the communication performances for solving the gossiping problem. More precisely, w e compute the minimum number of buses required for a gossiping to be optimal. Similarly, w e give upper bounds on the minimum length of buses required for a gossiping to be optimal. Finally, we c o m bine the two approaches in trying to minimize both parameters: length and number of buses.
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