In this paper, we address the problem of gathering information in a specific node (or sink) of a radio network, where interference constraints are present. We take into account the fact that, when a node transmits, it produces interference in an area bigger than the area in which its message can actually be received. The network is modeled by a graph; a node is able to transmit one unit of information to the set of vertices at distance at most dT in the graph, but when doing so it generates interference that does not allow nodes at distance up to dI (dI ≥ dT) to listen to other transmissions. Time is synchronous and divided into time-steps in each of which a round (set of non-interfering radio transmissions) is performed. We give general lower bounds on the number of rounds required to gather into a sink of a general graph, and present an algorithm working on any graph, with an approximation factor of 4. We also show that the problem of finding an optimal strategy for gathering is NP-hard, for any values of dI and dT. If dI> dT, we show that the problem remains hard when restricted to the uniform case where each vertex in the network has exactly one piece of information to communicate to the sink. 1

: As it is introduced by Bermond, P#rennes, and Kodate and by Fragopoulou and Akl, some Cayley graphs, including most popular models for interconnection networks, admit a special automorphism, called complete rotation. Such an automorphism is often used to derive algorithms or properties of the underlying graph. For example, some optimal gossiping algorithms can be easily designed by using a complete rotation, and the constructions of the best known edge disjoint spanning trees in the toroidal meshes and the hypercubes are based on such an automorphism. Our purpose is to investigate such Cayley graphs. We relate some symmetries of a graph with potential algebraic symmetries appearing in its de#nition as a Cayley graph on a group. In the case of Cayley graphs de#ned on a group generated by transpositions, we characterize the ones admitting a complete rotation. Key-words: networks, Cayley graphs, rotations, transposition graphs, group This work was partially supported by the french #Op#...

As it is introduced by Bermond, Pérennes, and Kodate and by Fragopoulou and Akl, some Cayley graphs, including most popular models for interconnection networks, admit a special automorphism, called complete rotation. Such an automorphism is often used to derive algorithms or properties of the underlying graph. For example, some optimal gossiping algorithms can be easily designed by using a complete rotation, and the constructions of the best known edge disjoint spanning trees in the toroidal meshes and the hypercubes are based on such an automorphism. Our purpose is to investigate such Cayley graphs. We relate some symmetries of a graph with potential algebraic symmetries appearing in its definition as a Cayley graph on a group.

This paper considers the problem of gossiping with packets of limited size in a network with cost function. We show that the problem of determining the minimum cost necessary to perform gossiping among a given set of participants with packets of limited size is NP-hard. We also give an approximate (with respect to the cost) gossiping algorithm. The ratio between the cost of an optimal algorithm and the approximate one is less than 1+2(k \Gamma 1)=n, were n is the number of nodes participating to the gossiping process and k n \Gamma 1 is the upper bound on the number of individual blocks of information that a packet can carry. We also analyze the communication time and communication complexity, i.e., the product of the communication cost and time, of gossiping algorithms.

Collective communication operations play an important role in message-passing systems and have been extensively investigated. We study two widely used collective communication operations: gossiping and all-to-all personalized communication. We assume the multi-port postal model of communication that seems particularly suited for developing fast and portable algorithms on current technology parallel computers. Unlike most of the previous work on the subject, we assume that processors communicate by sending messages of limited size. Indeed, when the maximum size of a message is fixed, the number of rounds required by a communication algorithm gives a realistic measure of the performance of the algorithm. We provide an optimal algorithm for the gossiping operation and an almost-optimal algorithm for the all-to-all personalized communication operation. Work partially supported by the Italian Ministry of the University and Scientific Research, Project: Algoritmi, Modelli di Calcolo e Stru...

As it is introduced by Bermond, Kodate, and Pérennes, some Cayley graphs, including most popular models for interconnection networks, admit a special automorphism, called complete rotation. Such an automorphism is often used to derive algorithms or properties of the underlying graph. For example, optimal gossiping algorithms can be easily designed, and the constructions of the best known edge disjoint spanning trees in the toroidal meshes and the hypercubes are based on such an automorphism. Our purpose is to characterize among Cayley graphs dened on a group generated by transpositions, which admit a complete rotation.