Lower Bounds on the Broadcasting and Gossiping Time of Restricted Protocols

In this paper we extend the technique provided in 6] to allow the determination of lower bounds on the broadcasting and gossiping time required by the so-called restrictedd protocols. Informally, a protocol is (i; o)-restricted at a given processor if every outgoing activation of an arc depends on at most i previous incoming activations and any incoming activation innuences at most o successive outgoing activations. Examples of restricted protocols are those running on bounded degree networks or systolic. We thus derive improved lower bounds on the broadcasting time in several well-known networks, such as Butterry, de Bruijn and Kautz graphs. Moreover, we derive the rst general lower bound on the gossiping time of d-bounded degree networks in the directed and half-duplex cases. Improved lower bounds on gossiping are also obtained for Butterry, de Bruijn and Kautz graphs. Finally, as a corollary we obtain the same lower bounds on s-systolic protocols obtained in 6]. All the results are also extended to other communication models, such as c-port and/or postal and optical. Bornes inffrieures du temps de diiusion de protocoles contraints RRsumm : Nous ggnnralisons les techniques introduites dans 6] aan de minorer le temps de diiusion et d''change total de protocoles satisfaisant certaines contraintes. Intuitivement un protocole est (i; o) contraint si un message mis ddpend uniquement des i derniers messages reeus et si un message reeu innuence au plus les o envois suivants. Cette contrainte est utile car elle apparaat la fois dans le cas des protocoles systoliques, et des protocoles fonctionnant dans des rrseaux de degrr bornn. Les rrsultats sont appliquus aan de ddterminer de nouvelles bornes inffrieures pour la diiusion et llechange total dans des rrseaux classiquess. Nous retrouvons aussi les rrsultats de 6] et prouvons la premiire borne inffrieure non triviale sur le temps de lllchange total dans les rrseaux de degrr bornn. Ennn nous montrons comment la technique introduite peut s'appliquer des moddles de communication divers (moddles postal ou optique)