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15
Optimal strategies in the average consensus problem
- in Proceedings of the IEEE Conference on Decision and Control
, 2007
"... Abstract — We prove that for a set of communicating agents to compute the average of their initial positions (average consensus problem), the optimal topology of communication is given by a de Bruijn’s graph. Consensus is then reached in a finitely many steps. A more general family of strategies, co ..."
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Abstract — We prove that for a set of communicating agents to compute the average of their initial positions (average consensus problem), the optimal topology of communication is given by a de Bruijn’s graph. Consensus is then reached in a finitely many steps. A more general family of strategies, constructed by block Kronecker products, is investigated and compared to Cayley strategies. I.
The Inframetric Model for the Internet
, 2007
"... Abstract—A large amount of algorithms has recently been designed for the Internet under the assumption that the distance defined by the round-trip delay (RTT) is a metric. Moreover, many of these algorithms (e.g., overlay network construction, routing scheme design, sparse spanner construction) rely ..."
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Cited by 12 (4 self)
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Abstract—A large amount of algorithms has recently been designed for the Internet under the assumption that the distance defined by the round-trip delay (RTT) is a metric. Moreover, many of these algorithms (e.g., overlay network construction, routing scheme design, sparse spanner construction) rely on the assumption that the metric has bounded ball growth or bounded doubling dimension. This paper analyzes the validity of these assumptions and proposes a tractable model matching experimental observations. On the one hand, based on Skitter data collected by CAIDA and King matrices of Meridian and P2PSim projects, we verify that the ball growth of the Internet, as well as its doubling dimension, can actually be quite large. Nevertheless, we observed that the doubling dimension is much smaller when restricting the measures to balls of large enough radius. Moreover, by computing the number of balls of radius r required to cover balls of radius R> r, we observed that this number grows with R much slower than what is predicted by a large doubling dimension. On the other hand, based on data collected on the PlanetLab platform by the All-Sites-Pings project, we confirm that the triangle inequality does not hold for a significant fraction of the nodes. Nevertheless, we demonstrate that RTT measures satisfy a weak version of the triangle inequality: there exists a small constant ρ such that for any triple u, v, w, we have RTT(u,v) ≤ ρ ·max{RTT(u,w), RTT(w,v)}. (Smaller bounds on ρ can even be obtained when the triple u, v, w is skewed). We call inframetric a distance function satisfying this latter inequality. Inframetrics subsume standard metrics and ultrametrics. Based on inframetrics and on our observations concerning the doubling dimension, we propose an analytical model for Internet RTT latencies. This model is tuned by a small set of parameters concerning the violation of the triangle inequality and the geometrical dimension of the network. We demonstrate the tractability of our model by designing a simple and efficient compact routing scheme with low stretch. Precisely, the scheme has constant multiplicative stretch and logarithmic additive stretch. I.
Bake: A balanced kautz tree structure for peer-to-peer networks
- In Proc. 27th IEEE INFOCOM
, 2008
"... Abstract—In order to improve scalability and reduce maintenance overhead for structured Peer-to-Peer systems, researchers design optimal architectures with constant degree and logarithmical diameter. The expected topologies, however, require the number of peers to be some given values determined by ..."
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Cited by 7 (3 self)
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Abstract—In order to improve scalability and reduce maintenance overhead for structured Peer-to-Peer systems, researchers design optimal architectures with constant degree and logarithmical diameter. The expected topologies, however, require the number of peers to be some given values determined by the average degree and the diameter. Hence, existing designs fail to address the issue due to the fact that 1) we cannot guarantee how many peers to join a P2P system at a given time, and 2) a P2P system is typically dynamic with peers frequently coming and leaving. In this work, we propose BAKE scheme based on balanced Kautz tree structure with logdn diameter and constant degree even the number of peers is an arbitrary value. Resources that are similar in single or multi-dimensional attributes space are stored on a same peer or neighboring peers. Through formal analysis and comprehensive simulations, we show that BAKE achieves optimal diameter and good connectivity as the Kautz digraph does. Indeed, the concepts of balanced Kautz tree introduced in this work can also be extended and applied to other interconnection networks after minimal modifications, for example, de Bruijn digraph. I.
Labeling Schemes with Queries
, 2006
"... We study the question of “how robust are the known lower bounds of labeling schemes when one increases the number of consulted labels”. Let f be a function on pairs of vertices. An f-labeling scheme for a family of graphs F labels the vertices of all graphs in F such that for every graph G ∈ F and e ..."
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We study the question of “how robust are the known lower bounds of labeling schemes when one increases the number of consulted labels”. Let f be a function on pairs of vertices. An f-labeling scheme for a family of graphs F labels the vertices of all graphs in F such that for every graph G ∈ F and every two vertices u, v ∈ G, the value f(u, v) can be inferred by merely inspecting the labels of u and v. This paper introduces a natural generalization: the notion of f-labeling schemes with queries, in which the value f(u, v) can be inferred by inspecting not only the labels of u and v but possibly the labels of some additional vertices. We show that inspecting the label of a single additional vertex (one query) enables us to reduce the label size of many labeling schemes significantly. In particular, we show that to support the distance function on n-node trees as well as the flow function on n-node general graphs, O(log n + log W)-bit labels are sufficient and necessary, where W is the maximum (integral) capacity of an edge. We note that it was shown that any labeling scheme (without queries) supporting either the flow function on general graphs or the distance function on trees, must have label size Ω(log 2 n + log n log W). Using a single query, we also show a routing
Degree-Optimal Routing for P2P Systems
- THEORY COMPUT SYST
, 2007
"... We define a family of Distributed Hash Table systems whose aim is to combine the routing efficiency of randomized networks—e.g. optimal average path length O(log 2 n/δ log δ) with δ degree—with the programmability and startup efficiency of a uniform overlay—that is, a deterministic system in which ..."
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Cited by 1 (1 self)
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We define a family of Distributed Hash Table systems whose aim is to combine the routing efficiency of randomized networks—e.g. optimal average path length O(log 2 n/δ log δ) with δ degree—with the programmability and startup efficiency of a uniform overlay—that is, a deterministic system in which the overlay network is transitive and greedy routing is optimal. It is known that �(log n) is a lower bound on the average path length for uniform overlays with O(log n) degree (Xu et al., IEEE J. Sel. Areas Commun. 22(1), 151–163, 2004). Our work is inspired by neighbor-of-neighbor (NoN) routing, a recently introduced variation of greedy routing that allows us to achieve optimal average path length in randomized networks. The advantage of our proposal is that of allowing the NoN technique to be implemented without adding any overhead to the corresponding deterministic network. We propose a family of networks parameterized with a positive integer c which measures the amount of randomness that is used. By varying the value c, the system goes from the deterministic case (c = 1) to an “almost uniform ” system. Increasing c to relatively low values allows for routing with asymptotically optimal average path length while retaining most of the advantages of a uniform system, such as easy programmability and quick bootstrap of the nodes entering the system.
CoMMEDIA: Separating Scaramouche from Harlequin to Accurately Estimate Items Frequency in Distributed Data Streams
, 2013
"... Abstract. In this paper, we investigate the problem of estimating the number of times data items that recur in very large distributed data streams. We present an alternative approach to the well-known Count-Min Sketch in order to reduce the impact of collisions on the accuracy of the estimation. We ..."
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Abstract. In this paper, we investigate the problem of estimating the number of times data items that recur in very large distributed data streams. We present an alternative approach to the well-known Count-Min Sketch in order to reduce the impact of collisions on the accuracy of the estimation. We propose to decrease, for each concerned item, the over-estimation that results from these collisions. Our sketch, called CoMMEDIETTA, keeps track of the most frequent items of the stream, and removes their weight from the one of the items with which these frequent items collide. By doing so, we significantly improve upon the Count-Min Sketch by achieving a randomized (ε, δ)-approximation algorithm. We then propose to judiciously distribute this local sketch to estimate the global frequency of any item that may recur in multiple streams. This distributed sketch, called CoMMEDIA (for Count-Min Sketch-based Estimation of Data Items Arrival frequency), organizes nodes of the system in a distributed hash table (DHT) such that each node implements a tiny local sketch on a reduced number of items. By doing so we guarantee a significantly more accurate estimation of item frequencies. Simulations both on synthetic and real traces confirm the accuracy of CoMMEDIA.
CDACAN: A Scalable Structured P2P Network Based on Continuous Discrete Approach and CAN
"... Abstract. CAN is a famous structured peer-to-peer network based on ddimensional torus topology with constant degree and logarithmical diameter, but suffers from poor scalability when N>>2 d, N is the number of peers. To address this issue, we proposes a novel scalable structured peer-to-peer o ..."
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Abstract. CAN is a famous structured peer-to-peer network based on ddimensional torus topology with constant degree and logarithmical diameter, but suffers from poor scalability when N>>2 d, N is the number of peers. To address this issue, we proposes a novel scalable structured peer-to-peer overlay network, CDACAN that embeds the one-dimensional discrete distance halving graph into each dimension of CAN. The out-degree and average routing path length of CDACAN are O(d) and O(log (N)), respectively, and are better than that of CAN. On the other hand, we analyze the optimal value of dimensions and the smooth division method of d-dimensional Cartesian coordinate space when handling the dynamic operations of peers. The smooth division of multidimensional space can improve the routing performance, and also is helpful to keep load balance among peers. Those properties and protocols are carefully evaluated by formal proofs or simulations. Furthermore, we present a layered improving scheme to decrease the out-degree of each peer in the future work. The expected topology will keep 8 out-degree and O(log 2(N)+d) routing path length. 1
Recently constant-degree Distributed Hash Tables
"... Most proposed DHTs have their unique maintenance mechanisms specific to the static graphs on which they are based. In this paper we propose distributed line graphs (DLG), a universal framework for building DHTs based on arbitrary constant-degree graphs. We prove that in a DLG-enabled, N-node DHT, th ..."
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Most proposed DHTs have their unique maintenance mechanisms specific to the static graphs on which they are based. In this paper we propose distributed line graphs (DLG), a universal framework for building DHTs based on arbitrary constant-degree graphs. We prove that in a DLG-enabled, N-node DHT, the outdegree is d, the in-degree is between 1 and 2d, and the diameter is less than 2(log dN−log dN 0+D 0+1), where d, D0 and N0 represent the degree, diameter and number of nodes of the initial graph, respectively. The maintenance cost of DLG-enabled DHTs is O(log dN). We show the power of DLG technique by applying it to Kautz graphs to propose a new DHT scheme.
