The goal of approximating metric spaces by more simple metric spaces has led to the notion of graph spanners [PU89, PS89] and to low-distortion embeddings in low-dimensional spaces [LLR94], having many algorithmic applications. This paper provides a novel technique for the analysis of randomized algorithms for optimization problems on metric spaces, by relating the randomized performance ratio for any metric space to the randomized performance ratio for a set of "simple" metric spaces. We define a notion of a set of metric spaces that probabilistically-approximates another metric space. We prove that any metric space can be probabilistically-approximated by hierarchically well-separated trees (HST) with a polylogarithmic distortion. These metric spaces are "simple" as being: (1) tree metrics. (2) natural for applying a divide-and-conquer algorithmic approach. The technique presented is of particular interest in the context of on-line computation. A large number of on-line al...

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.

There is an increasing need to quickly and efficiently learn network distances, in terms of metrics such as latency or bandwidth, between Internet hosts. For example, Internet content providers often place data and server mirrors throughout the Internet to improve access latency for clients, and it is necessary to direct clients to the closest mirrors based on some distance metric in order to realize the benefit of mirrors. We suggest a scalable Internet-wide architecture, called IDMaps, which measures and disseminates distance information on the global Internet. Higher-level services can collect such distance information to build a virtual distance map of the Internet and estimate the distance between any pair of IP addresses. We present our solutions to the measurement server placement and distance map construction problems in IDMaps. We show that IDMaps can indeed provide useful distance estimations to applications such as closest-mirror selection. 1 Keywords: network service, distributed algorithms, scalability, modeling. 1

We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and low-dilation graphs. 1 Introduction This survey covers topics in geometric network design theory. The problem is easy to state: connect a collection of sites by a "good" network. For instance, one may wish to connect components of a VLSI circuit by networks of wires, in a way that uses little surface area on the chip, draws little power, and propagates signals quickly. Similar problems come up in other applications such as telecommunications, road network design, and medical imaging [1]. One network design problem, the Traveling Salesman problem, is sufficiently important to have whole books devoted to it [79]. Problems involving some form of geometric minimum or maximum spanning tree also arise in the solution of other geometric problems such as clustering [12], mesh generation [56], and robot motion planning [93]. One can vary the network design problem in many w...

This paper investigates a zero-sum game played on a weighted connected graph G between two players, the tree player and the edge player. At each play, the tree player chooses a spanning tree T and the edge player chooses an edge e. The payoff to the edge player is cost(T; e), defined as follows: If e lies in the tree T then cost(T; e) = 0; if e does not lie in the tree then cost(T; e) = cycle(T; e)=w(e), where w(e) is the weight of edge e and cycle(T; e) is the weight of the unique cycle formed when edge e is added to the tree T. Our main result is that the value of the game on any n-vertex graph is bounded above by exp(O( p log n log log n)). The game arises in connection with the k-server problem on a road network; i.e., a metric space that can be represented as a multigraph G in which each edge e represents a road of length w(e). We show that, if the value of the game on G is V al(G; w), then there is a randomized strategy that achieves a competitive ratio of k(1 + V al(G; w)) against any oblivious adversary. Thus, on any n-vertex road network, there is a randomized algorithm for the k-server problem that is k exp(O( p log n log log n))-competitive against oblivious adversaries. At the heart of our analysis of the game is an algorithm that, for any n-vertex weighted, connected multigraph, constructs a spanning tree T such

Let G = (V

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 per-graph basis, finding trees that achieve better factors is NP-hard.

A tree t-spanner 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 1-spanner, 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 t-spanner in a weighted graph is proven to be NP-complete. For unweighted graphs, it is shown that constructing a tree 2-spanner takes linear time, whereas determining the existence of a tree t-spanner is NP-complete for any fixed t ≥ 4. A theorem which captures the structure of tree 2-spanners is presented for unweighted graphs. For digraphs, an O((m+n)α(m, n)) algorithm is provided for

(Extended Abstract)

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