Recognition Algorithms for Triadic and Multiway Distances, with an Application to Approximation into a Distance, an Ultrametric or a Robinson Dissimilarity - Archive ouverte HAL Accéder directement au contenu
Communication Dans Un Congrès Année : 2020

Recognition Algorithms for Triadic and Multiway Distances, with an Application to Approximation into a Distance, an Ultrametric or a Robinson Dissimilarity

Résumé

Given a triadic distance [6, 7] D on a n-set S, we consider the following two recognition problems: (i) determining if D is a perimeter, i.e. if there exists a (ordinary) dissimilarity d such that ∀x, y, z ∈ S, D(x, y, z) = d(x, y) + d(y, z) + d(x, z) and (ii) determining if D is a diameter, i.e. if there exists a dissimilarity d such that ∀x, y, z ∈ S, D(x, y, z) = max{d(x, y), d(y, z), d(x, z)}. In this note, we give optimal O(n 3) algorithms for these problems. Algorithms for Problem (i) can be extended, with the same complexity, to problems like determining if there exist a dissimilarity d such that (iii) ∀x, y, z ∈ S, D(x, y, z) = p d(x, y) p + d(y, z) p + d(x, z) p or (iv) ∀x, y, z ∈ S, D(x, y, z) = d(x, y) • d(y, z) • d(x, z). In addition, all these algorithms can be extended to k-way dissimilarities, with complexity O(n k). From the algorithm for Problem (ii), we derive algorithms to approximate a dissimilarity into a distance, an ultrametric or a Robinson dissimilarity. The general frame of these algorithms is the following: if the dissimilarity is closed to a distance (resp. an ultrametric, resp. a Robinson dissimilarity), return a solution to the approximation problem, otherwise return None.
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Dates et versions

hal-03513229 , version 1 (05-01-2022)

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  • HAL Id : hal-03513229 , version 1

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Pascal Préa, François Brucker. Recognition Algorithms for Triadic and Multiway Distances, with an Application to Approximation into a Distance, an Ultrametric or a Robinson Dissimilarity. Metric Graph Theory and Related Topics, Dec 2020, Marseille, France. ⟨hal-03513229⟩
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