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12
Hardness and approximation of gathering in static radio networks
 Parallel Processing Letters
, 2006
"... 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 act ..."
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Cited by 46 (16 self)
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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 timesteps in each of which a round (set of noninterfering 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 NPhard, 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
Cayley Graphs with Complete Rotations
, 1999
"... 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 under ..."
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Cited by 6 (1 self)
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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 definition 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.
Complete Rotations in Cayley Graphs
"... 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 underl ..."
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Cited by 5 (0 self)
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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.
Gossiping and routing in undirected tripleloop networks
 Networks
, 2010
"... Given integers n ≥ 7 and a, b, c with 1 ≤ a, b, c ≤ n − 1 such that a, n − a, b, n − b, c, n − c are pairwise distinct, the (undirected) tripleloop network TLn(a, b, c) is the degreesix graph with vertices 0, 1, 2,..., n − 1 such that each vertex x is adjacent to x ± a, x ± b, and x ± c, where the ..."
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Cited by 3 (1 self)
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Given integers n ≥ 7 and a, b, c with 1 ≤ a, b, c ≤ n − 1 such that a, n − a, b, n − b, c, n − c are pairwise distinct, the (undirected) tripleloop network TLn(a, b, c) is the degreesix graph with vertices 0, 1, 2,..., n − 1 such that each vertex x is adjacent to x ± a, x ± b, and x ± c, where the operation is modulo n. It is known that the maximum order of a connected tripleloop network of the form TLn(a, b, n − (a + b)) with given diameter d ≥ 2 is nd = 3d 2 + 3d + 1, which is achieved by TLnd = TLnd (1, 3d +1, 3d 2−1). In this article, we study the routing and gossiping problems for such optimal tripleloop networks under the storeandforward, allport, and fullduplex model, and prove that they admit “perfect” gossiping and routing schemes which exhibit many interesting features. Using a grouptheoretic approach we develop for TLnd a method for systematically producing such optimal gossiping and routing schemes. Moreover, we determine the minimum gossip time, the edge and arcforwarding indices, and the minimal edge and arcforwarding indices of TLnd, and prove that our routing schemes are optimal with respect to these four indices simultaneously. As a key step towards these results, we prove that TLnd is a Frobenius graph on a Frobenius group with Frobenius kernel Znd, and that TLnd is arctransitive with respect to this Frobenius group. In addition, we show that TLnd admits complete rotations.
Communication Complexity of Gossiping by Packets
 J. OF PARALLEL AND DISTRIBUTED COMPUTING
, 1996
"... 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 NPhard. We also give an approxim ..."
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Cited by 2 (1 self)
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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 NPhard. 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.
Fast Collective Communication by Packets in the Postal Model
, 1995
"... Collective communication operations play an important role in messagepassing systems and have been extensively investigated. We study two widely used collective communication operations: gossiping and alltoall personalized communication. We assume the multiport postal model of communication that ..."
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Cited by 1 (1 self)
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Collective communication operations play an important role in messagepassing systems and have been extensively investigated. We study two widely used collective communication operations: gossiping and alltoall personalized communication. We assume the multiport 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 almostoptimal algorithm for the alltoall personalized communication operation. Work partially supported by the Italian Ministry of the University and Scientific Research, Project: Algoritmi, Modelli di Calcolo e Stru...
Rotational Cayley Graphs on Transposition Generated Groups
, 2000
"... 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, op ..."
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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.
fc World Scientific Publishing Company HARDNESS AND APPROXIMATION OF GATHERING IN STATIC RADIO NETWORKS
, 2005
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
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, 2010
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Optimal Gossiping in Square 2D Meshes ⋆ Abstract
"... 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 probl ..."
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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 smallsize packets each carrying exactly one message are used. The nodes of the target meshes are assumed to be allport (a node’s incident edges can all be active at the same time); and their edges are either halfduplex or fullduplex, 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. DC9704, 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. on Parallel and Distributed Systems, Vol. 13, No. 4, pp. 349–358, 2002) 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 ith message from u to v under either model is the same message.