Chapter 26 Finite Metric Spaces and Partitions

For example, IR 2 with the regular Euclidean distance is a metric space. It is usually of interest to consider the finite case, where X is an a set of n points. Then, the function d can be specified by n 2 real numbers; that is, the distance between every pair of points of X. Alternatively, one can think about (X, d) is a weighted complete graph, where we specify positive weights on the edges, and the resulting weights on the edges comply with the triangle inequality. In fact, finite metric spaces rise naturally from (sparser) graphs. Indeed, let G = (X, E) be an undirected weighted graph defined over X, and let d G (x, y) be the length of the shortest path between x and y in G. It is easy to verify that (X, d G) is a finite metric space. As such if the graph G is sparse, it provides a compact representation to the finite space (X, d G). Definition 26.1.2 Let (X, d) be an n-point metric space. We denote the open ball of radius r about x ∈ X, by b(x, r) = y ∈ X d(x, y) < r. Underling our discussion of metric spaces are algorithmic applications. The hardness of various computational problems depends heavily on the structure of the finite metric space. Thus, given a finite metric space, and a computational task, it is natural to try to map the given metric space into a new metric where the task at hand becomes easy. Example 26.1.3 Consider the problem of computing the diameter, while it is not trivial in two dimensions, it is easy in one dimension. Thus, if we could map points in two dimensions into x This work is licensed under the Creative Commons Attribution-Noncommercial 3.0 License. To view a copy of this license, visit