Modules in Robinson Spaces
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A Robinson space is a dissimilarity space (X,d) (i.e., a set X of size n and a dissimilarity d on X) for which there exists a total order < on X such that x<y<z implies that d(x,z)≥max{ d(x,y), d(y,z)}. Recognizing if a dissimilarity space is Robinson has numerous applications in seriation and classification. An mmodule of (X,d) (generalizing the notion of a module in graph theory) is a subset M of X which is not distinguishable from the outside of M, i.e., the distance from any point of X∖ M to all points of M is the same. If p is any point of X, then { p} and the maximal by inclusion mmodules of (X,d) not containing p define a partition of X, called the copoint partition. In this paper, we investigate the structure of mmodules in Robinson spaces and use it and the copoint partition to design a simple and practical divide-and-conquer algorithm for recognition of Robinson spaces in optimal O(n^2) time.
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