Two simple but efficient algorithms to recognize Robinson dissimilarities
Résumé
A dissimilarity d on a set S of size n is said to be Robinson if its matrix can be symmetrically permuted so that its elements do not decrease when moving away from the main diagonal along any row or column. Equivalently, S admits a total order < such that i < j < k implies that d(i, j) ≤ d(i, k) and d(j, k) ≤ d(i, k). Intuitively, d is Robinson if S can be represented by points on a line. Recognizing Robinson dissimilarities has numerous applications in seriation and classification. Robinson dissimilarities also play an important role in the recognition of tractable cases for TSP. In this paper, we present two simple algorithms (inspired by Quicksort) to recognize Robinson dissimilarities. One of these algorithms runs in O(n 2 log n), the other one runs in O(n 3) in worst case and in O(n 2) on average.
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