, algorithm solves the 2D-ordering problem in O((n + k) 2 )

, D(y, z)}. Ultrametrics have many applications in classification because they are in one-to-one correspondence with hierarchies (see Barthélemy and Guénoche 1991 and Critchley and Fichet, S, D(x, y) ? max{D(x, z), 1994.

, If D is Robinsonian, verify that its PQ-tree has only P-nodes

, We define the dissimilarity D k by: D k (x, y) = D(x, y) if D(x, y) is k-known (i.e. if x ? ? k (y) or y ? ? k (x))

, D k (x, y) = ? otherwise

, D is k-Robinsonian if its graph G k is an interval graph. This generalizes Robinsonian dissimilarities, since (n ? 1)-Robinsonian is equivalent to Robinsonian. In addition, every dissimilarity is 0-Robinsonian and if a dissimilarity D is k-Robinsonian, then D is (k ? 1)-Robinsonian. So, with a dichotomic search

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