Results 1  10
of
54
A Tight Bound on Approximating Arbitrary Metrics by Tree Metrics
 In Proceedings of the 35th Annual ACM Symposium on Theory of Computing
, 2003
"... In this paper, we show that any n point metric space can be embedded into a distribution over dominating tree metrics such that the expected stretch of any edge is O(log n). This improves upon the result of Bartal who gave a bound of O(log n log log n). Moreover, our result is existentially tight; t ..."
Abstract

Cited by 260 (7 self)
 Add to MetaCart
In this paper, we show that any n point metric space can be embedded into a distribution over dominating tree metrics such that the expected stretch of any edge is O(log n). This improves upon the result of Bartal who gave a bound of O(log n log log n). Moreover, our result is existentially tight; there exist metric spaces where any tree embedding must have distortion#sto n)distortion. This problem lies at the heart of numerous approximation and online algorithms including ones for group Steiner tree, metric labeling, buyatbulk network design and metrical task system. Our result improves the performance guarantees for all of these problems.
On Approximating Arbitrary Metrics by Tree Metrics
 In Proceedings of the 30th Annual ACM Symposium on Theory of Computing
, 1998
"... This paper is concerned with probabilistic approximation of metric spaces. In previous work we introduced the method of ecient approximation of metrics by more simple families of metrics in a probabilistic fashion. In particular we study probabilistic approximations of arbitrary metric spaces by \hi ..."
Abstract

Cited by 251 (13 self)
 Add to MetaCart
This paper is concerned with probabilistic approximation of metric spaces. In previous work we introduced the method of ecient approximation of metrics by more simple families of metrics in a probabilistic fashion. In particular we study probabilistic approximations of arbitrary metric spaces by \hierarchically wellseparated tree" metric spaces. This has proved as a useful technique for simplifying the solutions to various problems.
Network Topology Generators: DegreeBased vs. Structural
, 2002
"... Following the longheld belief that the Internet is hierarchical, the network topology generators most widely used by the Internet research community, TransitStub and Tiers, create networks with a deliberately hierarchical structure. However, in 1999 a seminal paper by Faloutsos et al. revealed tha ..."
Abstract

Cited by 164 (14 self)
 Add to MetaCart
Following the longheld belief that the Internet is hierarchical, the network topology generators most widely used by the Internet research community, TransitStub and Tiers, create networks with a deliberately hierarchical structure. However, in 1999 a seminal paper by Faloutsos et al. revealed that the Internet's degree distribution is a powerlaw. Because the degree distributions produced by the TransitStub and Tiers generators are not powerlaws, the research community has largely dismissed them as inadequate and proposed new network generators that attempt to generate graphs with powerlaw degree distributions.
Spanning Trees Short Or Small
 SIAM JOURNAL ON DISCRETE MATHEMATICS
"... We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number k of nodes are required to be connected in the solution. A prototypical example is the kMST problem in which we require a tree of minimum weight s ..."
Abstract

Cited by 65 (2 self)
 Add to MetaCart
We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number k of nodes are required to be connected in the solution. A prototypical example is the kMST problem in which we require a tree of minimum weight spanning at least k nodes in an edgeweighted graph. We show that the kMST problem is NPhard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio 2 p k for the general edgeweighted case and O(k 1=4 ) for the case of points in the plane. Polynomialtime exact solutions are also presented for the class of treewidthbounded graphs which includes trees, seriesparallel graphs, and bounded bandwidth graphs, and for points on the boundary of a convex region in the Euclidean plane. We also investigate the problem of finding short trees, and more generally, that of finding networks with minimum diameter. A simple technique is used to prov...
LowerStretch Spanning Trees
, 2005
"... ... as a subgraph a spanning tree into which the edges of G can be embedded with average stretch exp (O ( √ log n log log n)), and that there exists an nvertex graph G such that all its spanning trees have average stretch Ω(log n). Closing the exponential gap between these upper and lower bounds i ..."
Abstract

Cited by 63 (10 self)
 Add to MetaCart
... as a subgraph a spanning tree into which the edges of G can be embedded with average stretch exp (O ( √ log n log log n)), and that there exists an nvertex graph G such that all its spanning trees have average stretch Ω(log n). Closing the exponential gap between these upper and lower bounds is listed as one of the longstanding open questions in the area of lowdistortion embeddings of metrics (Matousek 2002). We significantly reduce this gap by constructing a spanning tree in G of average stretch O((log n log log n) 2). Moreover, we show that this tree can be constructed in time O(m log 2 n) in general, and in time O(m log n) if the input graph is unweighted. The main ingredient in our construction is a novel graph decomposition technique. Our new algorithm can be immediately used to improve the running time of the recent solver for diagonally dominant linear systems of Spielman and Teng from to m2 (O( √ log n log log n)) log(1/ɛ) m log O(1) n log(1/ɛ), and to O(n(log n log log n) 2 log(1/ɛ)) when the system is planar. Applying a recent reduction of Boman, Hendrickson and Vavasis, this provides an O(n(log n log log n) 2 log(1/ɛ)) time algorithm for solving the linear systems that arise when applying the finite element method to solve twodimensional elliptic partial differential equations. Our result can also be used to improve several earlier approximation algorithms that use lowstretch spanning trees.
Representing Trees in Genetic Algorithms
 Proceedings of the First IEEE Conference on Evolutionary Computation
, 1994
"... We consider the problem of representing trees (undirected, cyclefree graphs) in Genetic Algorithms. This problem arises, among other places, in the solution of network design problems. After comparing several commonly used representations based on their usefulness in genetic algorithms, we describe ..."
Abstract

Cited by 45 (1 self)
 Add to MetaCart
We consider the problem of representing trees (undirected, cyclefree graphs) in Genetic Algorithms. This problem arises, among other places, in the solution of network design problems. After comparing several commonly used representations based on their usefulness in genetic algorithms, we describe a new representation and show it to be superior in almost all respects to the others. In particular, we show that our representation covers the entire space of solutions, produces only viable offspring, and possesses locality, all necessary features for the effective use of a genetic algorithm. We also show that the representation will reliably produce very good, if not optimal, solutions even when the problem definition is changed. I. Introduction In this paper, we consider the problem of representing trees in genetic algorithms. A tree is an undirected graph which contains no closed cycles. There are many optimization problems which can be phrased in terms of finding the optimal tree wit...
A Polynomial Time Approximation Scheme for Minimum Routing Cost Spanning Trees
, 1998
"... Given an undirected graph with nonnegative costs on the edges, the routing cost of any of its spanning trees is the sum over all pairs of vertices of the cost of the path between the pair in the tree. Finding a spanning tree of minimum routing cost is NPhard, even when the costs obey the triangle i ..."
Abstract

Cited by 42 (6 self)
 Add to MetaCart
Given an undirected graph with nonnegative costs on the edges, the routing cost of any of its spanning trees is the sum over all pairs of vertices of the cost of the path between the pair in the tree. Finding a spanning tree of minimum routing cost is NPhard, even when the costs obey the triangle inequality. We show that the general case is in fact reducible to the metric case and present a polynomialtime approximation scheme valid for both versions of the problem. In particular, we show how to build a spanning tree of an nvertex weighted graph with routing cost within (1 + ffl) from the minimum in time O(n O( 1 ffl ) ). Besides the obvious connection to network design, trees with small routing cost also find application in the construction of good multiple sequence alignments in computational biology. The communication cost spanning tree problem is a generalization of the minimum routing cost tree problem where the routing costs of different pairs are weighted by different r...
On Characterizing Network Topologies and Analyzing Their Impact on Protocol Design
, 2000
"... Recently there have been several papers examining aspects of the Internet topology. This paper follows in that tradition and addresses two issues related to Internet topology. First, we use three properties  expansion, resilience, and distortion  to characterize real and generated networks. For ..."
Abstract

Cited by 40 (3 self)
 Add to MetaCart
Recently there have been several papers examining aspects of the Internet topology. This paper follows in that tradition and addresses two issues related to Internet topology. First, we use three properties  expansion, resilience, and distortion  to characterize real and generated networks. For these metrics, we find that existing network topology generators differ qualitatively from most real networks. Second, we ask what impact topology has on four different multicast design questions. We find that, for many of these questions, a single topology metric appears to influence the answer.
An Approach To A Problem In Network Design Using Genetic Algorithms
, 1995
"... In the work of communications network design there are several recurring themes: maximizing flows, finding circuits, and finding shortest paths or minimal cost spanning trees, among others. Some of these problems appear to be harder than others. For some, effective algorithms exist for solving them, ..."
Abstract

Cited by 31 (0 self)
 Add to MetaCart
In the work of communications network design there are several recurring themes: maximizing flows, finding circuits, and finding shortest paths or minimal cost spanning trees, among others. Some of these problems appear to be harder than others. For some, effective algorithms exist for solving them, for others, tight bounds are known, and for still others, researchers have few clues towards a good approach. One of these latter, nastier problems arises in the design of communications networks: the Optimal Communication Spanning Tree Problem (OCSTP). First posed by Hu in 1974, this problem has been shown to be in the family of NPcomplete problems. So far, a good, generalpurpose approximation algorithm for it has proven elusive. This thesis describes the design of a genetic algorithm for finding reliably good solutions to the OCSTP. The genetic algorithm approach was thought to be an appropriate choice since they are computationally simple, provide a powerful parallel search capability...