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113
Tree spanners
 SIAM J. Discrete Math
, 1995
"... A tree tspanner T of a graph G is a spanning tree in which the distance between every pair of vertices is at most t times their distance in G. This notion is motivated by applications in communication networks, distributed systems, and network design. This paper studies graph theoretic, algorithmic ..."
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Cited by 58 (1 self)
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A tree tspanner T of a graph G is a spanning tree in which the distance between every pair of vertices is at most t times their distance in G. This notion is motivated by applications in communication networks, distributed systems, and network design. This paper studies graph theoretic, algorithmic and complexity issues about tree spanners. It is shown that a tree 1spanner, if it exists, in a weighted graph with m edges and n vertices is a minimum spanning tree and can be found in O(m log β(m, n)) time, where β(m, n) = min{i  log (i) n ≤ m/n}. On the other hand, for any fixed t> 1, the problem of determining the existence of a tree tspanner in a weighted graph is proven to be NPcomplete. For unweighted graphs, it is shown that constructing a tree 2spanner takes linear time, whereas determining the existence of a tree tspanner is NPcomplete for any fixed t ≥ 4. A theorem which captures the structure of tree 2spanners is presented for unweighted graphs. For digraphs, an O((m+n)α(m, n)) algorithm is provided for
On the Hardness of Approximating Spanners
 Algorithmica
, 1999
"... A k\Gammaspanner of a connected graph G = (V; E) is a subgraph G 0 consisting of all the vertices of V and a subset of the edges, with the additional property that the distance between any two vertices in G 0 is larger than the distance in G by no more than a factor of k. This paper concerns ..."
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Cited by 55 (16 self)
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A k\Gammaspanner of a connected graph G = (V; E) is a subgraph G 0 consisting of all the vertices of V and a subset of the edges, with the additional property that the distance between any two vertices in G 0 is larger than the distance in G by no more than a factor of k. This paper concerns the hardness of finding spanners with a number of edges close to the optimum. It is proved that for every fixed k, approximating the spanner problem is at least as hard as approximating the set cover problem We also consider a weighted version of the spanner problem, and prove an essential difference between the approximability of the case k = 2, and the case k 5. Department of Computer Science, The Open University, 16 Klauzner st., Ramat Aviv, Israel, guyk@shaked.openu.ac.il. 1 Introduction The concept of graph spanners has been studied in several recent papers in the context of communication networks, distributed computing, robotics and computational geometry [ADDJ90, C94, CK94,...
Routing in Distributed Networks: Overview and Open Problems
 ACM SIGACT News  Distributed Computing Column
, 2001
"... This article focuses on routing messages in distributed networks with efficient data structures. After an overview of the various results of the literature, we point some interestingly open problems. ..."
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Cited by 49 (12 self)
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This article focuses on routing messages in distributed networks with efficient data structures. After an overview of the various results of the literature, we point some interestingly open problems.
Memory Requirement for Routing in Distributed Networks
 IN 15 TH ANNUAL ACM SYMPOSIUM ON PRINCIPLES OF DISTRIBUTED COMPUTING (PODC
, 1995
"... In this paper, we deal with the compact routing problem on distributed networks, that is implementing routing schemes that use a minimum memory size on each node. We prove that for every shortest path routing scheme, for any constant ", 0 ! " ! 1, and for every integer d such that 3 d "n, there ..."
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Cited by 36 (7 self)
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In this paper, we deal with the compact routing problem on distributed networks, that is implementing routing schemes that use a minimum memory size on each node. We prove that for every shortest path routing scheme, for any constant ", 0 ! " ! 1, and for every integer d such that 3 d "n, there exists a nnode network of maximum degree d that locally requires at least \Theta(n log d) bits of memory on \Theta(n) nodes. This optimal lower bound means that whatever the routing scheme (interval routing, boolean routing, prefix routing, : : : ), there exists a network on which one can not do better than routing tables. Moreover, we prove that, for the wellknown interval routing scheme, there exists a nnode network of bounded degree d, for every d 3, that requires \Theta(n) intervals on \Theta(n) links to code any shortest path routing function. This tight lower bound shows that, for networks of bounded degree, the interval routing scheme can be worst than the routing tables ...
The Complexity of Interval Routing on Random Graphs
 THE COMPUTER JOURNAL
, 1995
"... Several methods exist for routing messages in a network without using complete routing tables (compact routing). In kinterval routing schemes (kIR.S), links carry up to k intervals each. A message is routed over certain link if its destination belongs to one of the intervals of the link. We giv ..."
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Cited by 32 (4 self)
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Several methods exist for routing messages in a network without using complete routing tables (compact routing). In kinterval routing schemes (kIR.S), links carry up to k intervals each. A message is routed over certain link if its destination belongs to one of the intervals of the link. We give some results for the necessary value of k in order to achieve shortest path routing. Even though for very structured networks low values of suce, we show that for 'general graphs' interval routing cannot significantly reduce the spacerequirements for shortest path routing. In particular we show that for suitably large n, there are suitable values of p such that for randomly chosen graphs G 6 ,p the following holds, with high probability: if G admits an optimal kIIS, then k = The result is obtained by means of a novel matrix representation for the shortest paths in a network.
Topology Aggregation for Directed Graph
 IEEE/ACM Trans. Networking
, 1997
"... This paper addresses the problem of aggregating the topology of a subnetwork in a compact way with minimum distortion. The problem arises from networks that have a hierarchical structure, where each subnetwork must advertise the cost of routing between each pair of its border nodes. The straightf ..."
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Cited by 29 (4 self)
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This paper addresses the problem of aggregating the topology of a subnetwork in a compact way with minimum distortion. The problem arises from networks that have a hierarchical structure, where each subnetwork must advertise the cost of routing between each pair of its border nodes. The straightforward solution of advertising the exact cost for each pair has a quadratic cost which is not practical. We look at the realistic scenario of networks where all links are bidirectional, but their cost (or distance) in the opposite directions might differ significantly. The paper present a solution with distortion that is bounded by the logarithm of the number of border nodes and the squareroot of the asymmetry in the cost of a link. This is the first time that a theoretical bound is given to an undirected graph. We show how to apply our solution to PNNI.
Memory Requirement for Universal Routing Schemes
 In 14 th Annual ACM Symposium on Principles of Distributed Computing (PODC
, 1995
"... In this paper, we deal with the compact routing problem, that is implementing routing schemes that use a minimum memory size on each router. In [20], Peleg and Upfal showed that there is no hope to do that with less than a total \Omega\Gamma n 1+1=(2s+4) ) memory bits for any stretch factor s 1. ..."
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Cited by 28 (8 self)
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In this paper, we deal with the compact routing problem, that is implementing routing schemes that use a minimum memory size on each router. In [20], Peleg and Upfal showed that there is no hope to do that with less than a total \Omega\Gamma n 1+1=(2s+4) ) memory bits for any stretch factor s 1. We improve this bound for stretch factors s ! 2 by proving that any nearshortest path routing scheme uses a total of \Omega\Gamma n 2 ) memory bits. 1 Introduction The general routing problem in a network (as opposed to the permutation routing problem [18] or the broadcasting problem [8]) consists of finding a routing protocol or routing function such that, for any sourcedestination pair, any message from the source can be routed to the destination. XY routing [2] or ecube routing [3] are such protocols. The efficiency of a protocol is measured in terms of latency (related to the length of the paths) and/or throughput (related to link or node congestion). Finding shortest paths in a n...
A General Theory for Deadlock Avoidance in WormholeRouted Networks
 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS
, 1998
"... Most machines of the last generation of distributed memory parallel computers possess specific routers which are used to exchange messages between nonneighboring nodes in the network. Among the several technologies, wormhole routing is usually prefered because it allows low channelsetup time, and ..."
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Cited by 23 (2 self)
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Most machines of the last generation of distributed memory parallel computers possess specific routers which are used to exchange messages between nonneighboring nodes in the network. Among the several technologies, wormhole routing is usually prefered because it allows low channelsetup time, and reduces the dependency between latency and internode distance. However, wormhole routing is very susceptible to deadlock because messages are allowed to hold many resources while requesting others. Therefore, designing deadlockfree routing algorithms using few hardware facilities is a major problem for wormholerouted networks. In this paper, we describe a general theoretical framework for the study of deadlockfree routing functions. We give a general definition of what can be a routing function. This definition captures many specific definitions of the literature (e.g., vertexdependent, inputdependent, sourcedependent, pathdependent, etc.). Using our definition, we give a necessary an...
Efficient algorithms for constructing (1 + ɛ, β)spanners in the distributed and streaming models
 Distributed Computing
, 2004
"... For an unweighted undirected graph G = (V, E), and a pair of positive integers α ≥ 1, β ≥ 0, a subgraph G ′ = (V, H), H ⊆ E, is called an (α, β)spanner of G if for every pair of vertices u, v ∈ V, distG ′(u, v) ≤ α · distG(u, v) + β. It was shown in [20] that for any ɛ> 0, κ = 1, 2,..., there exi ..."
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Cited by 21 (6 self)
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For an unweighted undirected graph G = (V, E), and a pair of positive integers α ≥ 1, β ≥ 0, a subgraph G ′ = (V, H), H ⊆ E, is called an (α, β)spanner of G if for every pair of vertices u, v ∈ V, distG ′(u, v) ≤ α · distG(u, v) + β. It was shown in [20] that for any ɛ> 0, κ = 1, 2,..., there exists an integer β = β(ɛ, κ) such that for every nvertex graph G there exists a (1 + ɛ, β)spanner G ′ with O(n 1+1/κ) edges. An efficient distributed protocol for constructing (1+ ɛ, β)spanners was devised in [18]. The running time and the communication complexity of that protocol are O(n 1+ρ) and O(En ρ), respectively, where ρ is an additional control parameter of the protocol that affects only the additive term β. In this paper we devise a protocol with a drastically improved running time (O(n ρ) as opposed to O(n 1+ρ)) for constructing (1 + ɛ, β)spanners. Our protocol has the same communication complexity as the protocol of [18], and it constructs spanners with essentially the same properties as the spanners that are constructed by the protocol of [18]. We also show that our protocol for constructing (1+ɛ, β)spanners can be adapted to the streaming model, and devise a streaming algorithm that uses a constant number of passes and O(n 1+1/κ · log n) bits of space for computing allpairsalmostshortestpaths of length at most by a multiplicative factor (1 + ɛ) and an additive term of β greater than the shortest paths. Our algorithm processes each edge in time O(n ρ), for an arbitrarily small ρ> 0. The only
Affiliation Networks
"... In the last decade, structural properties of several naturally arising networks (the Internet, social networks, the web graph, etc.) have been studied intensively with a view to understanding their evolution. In recent empirical work, Leskovec, Kleinberg, and Faloutsos identify two new and surprisin ..."
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Cited by 21 (3 self)
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In the last decade, structural properties of several naturally arising networks (the Internet, social networks, the web graph, etc.) have been studied intensively with a view to understanding their evolution. In recent empirical work, Leskovec, Kleinberg, and Faloutsos identify two new and surprising properties of the evolution of many realworld networks: densification (the ratio of edges to vertices grows over time), and shrinking diameter (the diameter reduces over time to a constant). These properties run counter to conventional wisdom, and are certainly inconsistent with graph models prior to their work. In this paper, we present the first model that provides a simple, realistic, and mathematically tractable generative model that intrinsically explains all the wellknown properties of the social networks, as well as densification and shrinking diameter. Our model is based on ideas studied empirically in the social sciences, primarily on the groundbreaking work of Breiger (1973) on bipartite models of social networks that capture the affiliation of agents to societies. We also present algorithms that harness the structural consequences of our model. Specifically, we show how to overcome the bottleneck of densification in computing shortest paths between vertices by producing sparse subgraphs that preserve or approximate shortest distances to all or a distinguished subset of vertices. This is a rare example of an algorithmic benefit derived from a realistic graph model. Finally, our work also presents a modular approach to connecting random graph paradigms (preferential attachment, edgecopying, etc.) to structural consequences (heavytailed degree distributions, shrinking diameter, etc.).