Trees, Tight-Spans and Point Configuration

Tight-spans of metrics were first introduced by Isbell in 1964 and rediscovered and studied by others, most notably by Dress, who gave them this name. Subsequently, it was found that tight-spans could be defined for more general maps, such as directed metrics and distances, and more recently for diversities. In this paper, we show that all of these tight-spans as well as some related constructions can be defined in terms of point configurations. This provides a useful way in which to study these objects in a unified and systematic way. We also show that by using point configurations we can recover results concerning one-dimensional tight-spans for all of the maps we consider, as well as extend these and other results to more general maps such as symmetric and unsymmetric maps.Comment: 21 pages, 2 figure

Optimally fast incremental Manhattan plane embedding and planar tight span construction

We describe a data structure, a rectangular complex, that can be used to represent hyperconvex metric spaces that have the same topology (although not necessarily the same distance function) as subsets of the plane. We show how to use this data structure to construct the tight span of a metric space given as an n x n distance matrix, when the tight span is homeomorphic to a subset of the plane, in time O(n^2), and to add a single point to a planar tight span in time O(n). As an application of this construction, we show how to test whether a given finite metric space embeds isometrically into the Manhattan plane in time O(n^2), and add a single point to the space and re-test whether it has such an embedding in time O(n).Comment: 39 pages, 15 figure

On-line algorithms for the K-server problem and its variants.

by Chi-ming Wat.Thesis (M.Phil.)--Chinese University of Hong Kong, 1995.Includes bibliographical references (leaves 77-82).Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Performance analysis of on-line algorithms --- p.2Chapter 1.2 --- Randomized algorithms --- p.4Chapter 1.3 --- Types of adversaries --- p.5Chapter 1.4 --- Overview of the results --- p.6Chapter 2 --- The k-server problem --- p.8Chapter 2.1 --- Introduction --- p.8Chapter 2.2 --- Related Work --- p.9Chapter 2.3 --- The Evolution of Work Function Algorithm --- p.12Chapter 2.4 --- Definitions --- p.16Chapter 2.5 --- The Work Function Algorithm --- p.18Chapter 2.6 --- The Competitive Analysis --- p.20Chapter 3 --- The weighted k-server problem --- p.27Chapter 3.1 --- Introduction --- p.27Chapter 3.2 --- Related Work --- p.29Chapter 3.3 --- Fiat and Ricklin's Algorithm --- p.29Chapter 3.4 --- The Work Function Algorithm --- p.32Chapter 3.5 --- The Competitive Analysis --- p.35Chapter 4 --- The Influence of Lookahead --- p.41Chapter 4.1 --- Introduction --- p.41Chapter 4.2 --- Related Work --- p.42Chapter 4.3 --- The Role of l-lookahead --- p.43Chapter 4.4 --- The LRU Algorithm with l-lookahead --- p.45Chapter 4.5 --- The Competitive Analysis --- p.45Chapter 5 --- Space Complexity --- p.57Chapter 5.1 --- Introduction --- p.57Chapter 5.2 --- Related Work --- p.59Chapter 5.3 --- Preliminaries --- p.59Chapter 5.4 --- The TWO Algorithm --- p.60Chapter 5.5 --- Competitive Analysis --- p.61Chapter 5.6 --- Remarks --- p.69Chapter 6 --- Conclusions --- p.70Chapter 6.1 --- Summary of Our Results --- p.70Chapter 6.2 --- Recent Results --- p.71Chapter 6.2.1 --- The Adversary Models --- p.71Chapter 6.2.2 --- On-line Performance-Improvement Algorithms --- p.73Chapter A --- Proof of Lemma1 --- p.75Bibliography --- p.7

On-line algorithms for robot navigation and server problems

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1994.Includes bibliographical references (p. 83-88).by Jon Michael Kleinberg.M.S

A New Approach to the Server Problem

A new method for dealing with the server problem is proposed. The technique consists of embedding the given metric space M into a bigger metric space cl (M) called the closure of M , and allowing our servers to move in cl (M ). We show how this technique can be applied to give a new optimal algorithm for two servers. 1 Introduction The k server problem can be formulated as follows: Let M be a metric space, in which we have k mobile servers that can occupy points of M . Initially all servers are on some k specified points of M (called the initial configuration). At each time step we are given a request, specified by a location r 2 M , and we have to choose which server to move to r to "serve" the request. Our measure of cost is the distance traveled by our servers, and the task is to design algorithms that minimize that cost. The problem is that the requests have to be served on-line, that is, the choice of the server at the current step cannot depend on the future requests. It is kno..