An Optimal Algorithm for Strict Circular Seriation

Armstrong, Santiago; Guzmán, Cristóbal; Sing, Long, Carlos A.

doi:10.1137/21m139356x

Cited by 7 publications

(5 citation statements)

References 23 publications

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“…Due to this, even if optimal, this algorithm is not simple. PQ-trees are also used in the general case of the Atkins et al [3] recognition algorithm based on the spectral approach and in the recent algorithm of [2] for strict circular seriation. Our optimal recognition algorithm is simple and was relatively easy to implement in OCaml, as it should also be in any mainstream programming language.…”

Section: Discussionmentioning

confidence: 99%

“…More recently, Aracena and Thraves Caro [1] presented a parametrized algorithm for the NP-complete problem of recognition of Robinson incomplete matrices. Armstrong et al [2] presented an optimal O(n 2 ) time algorithm for the recognition of strict circular Robinson dissimilarities (Hubert et al [21] defined circular seriation first and it was studied also in the papers [31] and [23]).…”

Section: Introductionmentioning

confidence: 99%

“…2 ) in the total complexity; indeed, applyingLemma 7.3, the first call takes α⋅|X| 2 −∑ k i=1 |C i | 2 (for some constant α), while the cost of recursiveRefine in the recursive calls are at most α ∑ k i=1 |C i | 2 by induction, summing to α|C| 2 . • refine contributes O(|C| 2 ) in the total complexity, because it follows the recurrence relation described in Lemma 7.6.…”

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confidence: 99%

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Modules in Robinson Spaces

Carmona¹,

Chepoi²,

Naves³

et al. 2022

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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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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Modules in Robinson Spaces

Carmona¹,

Chepoi²,

Naves³

et al. 2022

Preprint

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“…We call such dissimilarities quasicircular Robinson. Recently, Armstrong, Guzmán, and Sing Long [1] presented an optimal O(n 2 ) time algorithm for the recognition of strict quasi-circular Robinson dissimilarities, i.e., for which the ball hypergraph is a hypercycle and x ⋖ y ⋖ z ⋖ t implies that d(x, z) > min{d(x, y), d(x, t)}. Among other tools, their algorithm uses PQ-trees.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we give a very simple O(n log n) time algorithm which builds a compatible circular order for all versions of strict circular Robinson dissimilarities, introduced and investigated in the papers [1,4,12]. Then the adjunction of a verification step gives an optimal O(n 2 ) time algorithm to recognize these dissimilarities.…”

Section: Introductionmentioning

confidence: 99%

A simple and optimal algorithm for strict circular seriation

Carmona¹,

Chepoi²,

Naves³

et al. 2022

Preprint

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Recently, Armstrong, Guzmán, and Sing Long (2021), presented an optimal O(n 2 ) time algorithm for strict circular seriation (called also the recognition of strict quasi-circular Robinson spaces). In this paper, we give a very simple O(n log n) time algorithm for computing a compatible circular order for strict circular seriation. When the input space is not known to be strict quasi-circular Robinson, our algorithm is complemented by an O(n 2 ) time verification of compatibility of the returned order. This algorithm also works for recognition of other types of strict circular Robinson spaces known in the literature. We also prove that the circular Robinson dissimilarities (which are defined by the existence of compatible orders on one of the two arcs between each pair of points) are exactly the pre-circular Robinson dissimilarities (which are defined by a four-point condition).

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