Diameter in linear time for constant-dimension median graphs

Bergé, Pierre; Habib, Michel

doi:10.1016/j.procs.2021.11.015

Cited by 3 publications

(35 citation statements)

References 13 publications

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“…A more recent contribution introduces a quasilinear time algorithm -running in O(n log O(1) n) -for the diameter on cube-free median graphs [24], using distance and routing labeling schemes proposed in [25]. Eventually, a linear-time algorithm [26] for the diameter on constant-dimension median graphs was proposed, i.e. for median graphs satisfying d = O (1).…”

Section: State Of the Artmentioning

confidence: 99%

“…Furthermore, we define a new discrete structure on median graphs: the maximal outgoing POFs (MOPs), generalizing the POFs used throughout the paper and defined in [26]. We propose an alternative procedure to compute all eccentricities, based on the enumeration of MOPs.…”

Section: Contributionsmentioning

confidence: 99%

“…Lemma 7 (Orthogonal⇔Incompatible [26]) Given two Θ-classes E i and E j of a median graph G, the following statements are equivalent:…”

Section: Orthogonal θ-Classes and Hypercubesmentioning

confidence: 99%

“…Lemma 8 (Squares [5,26]) Let xu ∈ E i and uy ∈ E j . If E i and E j are orthogonal, then there is a vertex v such that uyvx is a square.…”

Section: Orthogonal θ-Classes and Hypercubesmentioning

confidence: 99%

“…Some parts of this subsection are redundant with [26], however we keep this subsection self-contained. The new outcomes presented are Theorems 7 and 9.…”

Section: Lemma 16 ([26]mentioning

confidence: 99%

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Subquadratic-time algorithm for the diameter and all eccentricities on median graphs

Bergé

Ducoffe

Habib

2022

Preprint

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On sparse graphs, Roditty and Williams [2013] proved that no O(n2−ε)- time algorithm achieves an approximation factor smaller than 3/2 for the diameter problem unless SETH fails. In this article, we solve an open question formulated in the literature: can we use the structural properties of median graphs to break this global quadratic barrier? We propose the first combinatiorial algorithm computing exactly all eccentricities of a median graph in truly subquadratic time. Median graphs constitute the family of graphs which is the most studied in metric graph theory because their structure represents many other discrete and geometric concepts, such as CAT(0) cube complexes. Our result generalizes a recent one, stating that there is a linear-time algorithm for all eccentricities in median graphs with bounded dimension d, i.e. the dimension of the largest induced hypercube. This prerequisite on d is not necessarily anymore to determine all eccentricities in subquadratic time. The execution time of our algorithm is O(n1.6456 logO(1) n). We provide also some satellite outcomes related to this general result. In particular, restricted to simplex graphs, this algorithm enumerates all eccentricities with a quasilinear running time. Moreover, an algorithm is proposed to compute exactly all reach centralities in time O(23dn logO(1) n).

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Section: State Of the Artmentioning

confidence: 99%

Section: Contributionsmentioning

confidence: 99%

“…Lemma 7 (Orthogonal⇔Incompatible [26]) Given two Θ-classes E i and E j of a median graph G, the following statements are equivalent:…”

Section: Orthogonal θ-Classes and Hypercubesmentioning

confidence: 99%

“…Lemma 8 (Squares [5,26]) Let xu ∈ E i and uy ∈ E j . If E i and E j are orthogonal, then there is a vertex v such that uyvx is a square.…”

Section: Orthogonal θ-Classes and Hypercubesmentioning

confidence: 99%

“…Some parts of this subsection are redundant with [26], however we keep this subsection self-contained. The new outcomes presented are Theorems 7 and 9.…”

Section: Lemma 16 ([26]mentioning

confidence: 99%

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Subquadratic-time algorithm for the diameter and all eccentricities on median graphs

Bergé

Ducoffe

Habib

2022

Preprint

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Subquadratic-time Algorithm for the Diameter and all Eccentricities on Median Graphs

Bergé,

Ducoffe,

Habib

2023

Theory Comput Syst

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On sparse graphs, Roditty and Williams [2013] proved that no O(n 2−ε )-time algorithm achieves an approximation factor smaller than 3 2 for the diameter problem unless SETH fails. We answer here an open question formulated in the literature: can we use the structural properties of median graphs to break this global quadratic barrier?We propose the first combinatorial algorithm computing exactly all eccentricities of a median graph in truly subquadratic time. Median graphs constitute the family of graphs which is the most studied in metric graph theory because their structure represents many other discrete and geometric concepts, such as CAT(0) cube complexes. Our result generalizes a recent one, stating that there is a linear-time algorithm for computing all eccentricities in median graphs with bounded dimension d, i.e. the dimension of the largest induced hypercube (note that 1-dimensional median graphs are exactly the forests). This prerequisite on d is not necessarily anymore to determine all eccentricities in subquadratic time. The execution time of our algorithm is O(n 1.6456 log O(1) n).

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