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31
Simultaneous Optimization for Concave Costs: Single Sink Aggregation or Single Source BuyatBulk
 In Proc. of the 14 th Symposium on Discrete Algorithms (SODA
, 2003
"... We consider the problem of finding efficient trees to send information from k sources to a single sink in a network where information can be aggregated at intermediate nodes in the tree. Specifically, we assume that if information from j sources is traveling over a link, the total information tha ..."
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Cited by 103 (3 self)
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We consider the problem of finding efficient trees to send information from k sources to a single sink in a network where information can be aggregated at intermediate nodes in the tree. Specifically, we assume that if information from j sources is traveling over a link, the total information that needs to be transmitted is f(j). One natural and important (though not necessarily comprehensive) class of functions is those which are concave, nondecreasing, and satisfy f(0) = 0. Our goal is to find a tree which is a good approximation simultaneously to the optimum trees for all such functions. This problem is motivated by aggregation in sensor networks, as well as by buyatbulk network design.
Balancing Minimum Spanning and Shortest Path Trees
, 1993
"... Efficient algorithms are known for computing a minimum spann.ing tree, or a shortest path. tree (with a fixed vertex as the root). The weight of a shortest path tree can be much more than the weight of a minimum spa,nning tree. Conversely, the distance bet,ween the root, and any vertex in a minimum ..."
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Cited by 64 (1 self)
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Efficient algorithms are known for computing a minimum spann.ing tree, or a shortest path. tree (with a fixed vertex as the root). The weight of a shortest path tree can be much more than the weight of a minimum spa,nning tree. Conversely, the distance bet,ween the root, and any vertex in a minimum spanning tree may be much more than the distance bet#ween the two vertices in the graph. Consider the problem of balancing between the two kinds of trees: Does every graph contain a tree that is “light ” (at most a constant times heavier than the minimum spanning t,ree), such that the distance from the root to any vertex in t,he tree is no more than a constant times the true distance? This paper answers the question in the affirmative. It is shown that there is a continuous tradeoff between the two parameters. For every y> 0, there is a tree in the graph whose total weight is at most 1 + $? times the weight of a minimum spanning tree, such that the di&nce in the tree between the root, and any vertex is at, most 1 + &y times the true distance. Efficient sequential and parallel algorithms achieving these factors are provided. The algorithms are shown to be optimal in two ways. First, it is shown that no algorithm can achieve better factors in all graphs, because there a.re graphs that do not have better trees. Second, it is shown that even on a pergraph basis, finding trees that achieve better factors is NPhard.
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 59 (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 51 (12 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,...
Nearlinear time construction of sparse neighborhood covers
 SIAM Journal on Computing
, 1998
"... Abstract. This paper introduces a nearlinear time sequential algorithm for constructing a sparse neighborhood cover. This implies analogous improvements (from quadratic to nearlinear time) for any problem whose solution relies on network decompositions, including small edge cuts in planar graphs, ..."
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Cited by 40 (3 self)
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Abstract. This paper introduces a nearlinear time sequential algorithm for constructing a sparse neighborhood cover. This implies analogous improvements (from quadratic to nearlinear time) for any problem whose solution relies on network decompositions, including small edge cuts in planar graphs, approximate shortest paths, and weight and distancepreserving graph spanners. In particular, an O(log n) approximation to the kshortest paths problem on an nvertex, Eedge graph is obtained that runs in Õ (n + E + k) time.
Balancing Minimum Spanning Trees and ShortestPath Trees
, 2002
"... We give a simple algorithm to find a spanning tree that simultaneously approximates a shortestpath tree and a minimum spanning tree. The algorithm provides a continuous tradeoff: given the two trees and a fl? 0, the algorithm returns a spanning tree in which the distance between any vertex and the ..."
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Cited by 38 (1 self)
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We give a simple algorithm to find a spanning tree that simultaneously approximates a shortestpath tree and a minimum spanning tree. The algorithm provides a continuous tradeoff: given the two trees and a fl? 0, the algorithm returns a spanning tree in which the distance between any vertex and the root of the shortestpath tree is at most 1 + p 2fl times the shortestpath distance, and yet the total weight of the tree is at most 1 + p 2=fl times the weight of a minimum spanning tree. Our algorithm runs in linear time and obtains the bestpossible tradeoff. It can be implemented on a CREW PRAM to run in logarithmic time using one processor per vertex.
PrimDijkstra Tradeoffs for Improved PerformanceDriven Routing Tree Design
, 1995
"... Analysis of Elmore delay in distributed RC tree structures shows the influence of both tree cost and tree radius on signal delay in VLSI interconnects. We give new and efficient interconnection tree constructions that smoothly combine the minimum cost and the minimum radius objectives, by combining ..."
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Cited by 32 (4 self)
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Analysis of Elmore delay in distributed RC tree structures shows the influence of both tree cost and tree radius on signal delay in VLSI interconnects. We give new and efficient interconnection tree constructions that smoothly combine the minimum cost and the minimum radius objectives, by combining respectively optimal algorithms due to Prim and Dijkstra. Previous "shallowlight" techniques [2, 3, 8, 13] are both less direct and less effective: in practice, our methods achieve uniformly superior costradius tradeoffs. Detailed timing simulations for a range of IC and MCM interconnect technologies show that our wirelength savings yield reduced signal delays when compared to shallowlight or standard minimum spanning tree and Steiner tree routing.
Approximating the Weight of Shallow Steiner Trees
 DAMATH: Discrete Applied Mathematics and Combinatorial Operations Research and Computer Science
, 1998
"... This paper deals with the problem of constructing Steiner trees of minimum weight with diameter bounded by d, spanning a given set of k vertices in a graph. Exact solutions or logarithmic ratio approximation algorithms were known before for the cases of d <= 5. Here we give a polynomial time appr ..."
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Cited by 29 (3 self)
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This paper deals with the problem of constructing Steiner trees of minimum weight with diameter bounded by d, spanning a given set of k vertices in a graph. Exact solutions or logarithmic ratio approximation algorithms were known before for the cases of d <= 5. Here we give a polynomial time approximation algorithm of ratio O(log k) for constant d, which is asymptotically optimal unless P = NP , and an algorithm of ratio O( k^{\epsilon})), for any fixed 0 < \epsilon < 1, for general d. Keywords: NPhard problems, approximation algorithms, Steiner trees 1 Introduction 1.1 The problem This paper considers the problem of finding low diameter Steiner trees of minimum weight. Given an nvertex graph G(V
Fast Distributed Network Decompositions and Covers
 Journal of Parallel and Distributed Computing
, 1996
"... This paper presents deterministic sublineartime distributed algorithms for network decomposition and for constructing a sparse neighborhood cover of a network. The latter construction leads to improved distributed preprocessing time for a number of distributed algorithms, including allpairs shorte ..."
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Cited by 24 (4 self)
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This paper presents deterministic sublineartime distributed algorithms for network decomposition and for constructing a sparse neighborhood cover of a network. The latter construction leads to improved distributed preprocessing time for a number of distributed algorithms, including allpairs shortest paths computation, load balancing, broadcast, and bandwidth management. A preliminary version of this paper appeared in the Proceedings of the Eleventh Annual ACM Symposium on the Principles of Distributed Computing. y Lab. for Computer Science, MIT, Cambridge, MA 02139. Supported by Air Force Contract AFOSR F4962092J0125, NSF contract 9114440CCR, DARPA contracts N0001491J1698 and N00014J921799, and a special grant from IBM. z Dept. of Mathematics and Lab. for Computer Science, MIT. Supported in part by an NSF Postdoctoral Research Fellowship and an ONR grant provided to the Radcliffe Bunting Institute. x Dept. of Math Sciences, Johns Hopkins University, Baltimore, MD 21...