Chapter 25 Finite Metric Spaces and Partitions

x The problem received the title of 'Buridan's sheep.' The biological code was taken from a young merino sheep, by the Casparo-Karpov method, at a moment when the sheep was between two feeding troughs full of mixed fodder. This code, along with additional data about sheep in general, was fed into CODD. The machine was required: a) to predict which trough the merino would choose, and b) to give the psychophysiological basis for this choice. – The mystery of the hind leg, Arkady and Boris Strugatsky. 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).