Finite Metric Spaces and Partitions

For example, IR with the regular Euclidean distance is a metric space. It is usually of interest to consider the finite case, where X is an n-point set. Then, the function d can be specified by ( n 2 ) real numbers. 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 dG(x, y) be the length of the shortest path between x and y in G. It is easy to verify that (X,dG) is a finite metric space. As such if the graph G is sparse, it provides a compact representation to the finite space (X,dG).