Medians in Median Graphs and Their Cube Complexes in Linear Time

Bénéteau, Laurine; Chalopin, Jérémie; Chepoi, Victor; Vaxès, Yann

doi:10.4230/lipics.icalp.2020.10

Cited by 3 publications

(4 citation statements)

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“…We believe that classes of graphs with connected or G 2 -connected medians are good candidates in which the median problem can be solved faster than in O(nm) time. Our belief is based on the fact that all known such algorithms (for median graphs [20], for planar bridged triangulations [38], for Helly graphs [46], and for basis graphs of matroids [9]) use the unimodality of median functions. However, designing such minimization algorithms is not a so easy problem because they cannot use the entire distance matrix of the graph.…”

Section: Discussionmentioning

confidence: 99%

“…First, median graphs are exactly the retracts of hypercubes [4] and second, finite median graphs are exactly the graphs obtained from hypercubes (Cartesian products of edges) by successive gated amalgams. Median graphs are graphs with connected medians [5,69] and this property together with the majority rule were used in our linear-time algorithm [20] for computing medians in median graphs. For properties and characterizations of median graphs, see the survey [10].…”

Section: 3mentioning

confidence: 99%

“…The fact that the median function of median graphs is weakly convex was established in [69] (it was previously shown in [5] that the median function on median graphs is unimodal). We used these results to compute medians in median graphs in linear time [20]. Using the unimodality of the median function on Helly graphs [9], Ducoffe [46] presented an algorithm with complexity Õ(m √ n) for computing medians of Helly graphs with n vertices and m edges.…”

Section: Introductionmentioning

confidence: 99%

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Graphs with $G^p$-connected medians

Bénéteau¹,

Chalopin²,

Chepoi³

et al. 2022

Preprint

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The median of a graph G with weighted vertices is the set of all vertices x minimizing the sum of weighted distances from x to the vertices of G. For any integer p ≥ 2, we characterize the graphs in which, with respect to any non-negative weights, median sets always induce connected subgraphs in the pth power G p of G. This extends some characterizations of graphs with connected medians (case p = 1) provided by Bandelt and Chepoi (2002). The characteristic conditions can be tested in polynomial time for any p. We also show that several important classes of graphs in metric graph theory, including bridged graphs (and thus chordal graphs), graphs with convex balls, bucolic graphs, and bipartite absolute retracts, have G 2 -connected medians. Extending the result of Bandelt and Chepoi that basis graphs of matroids are graphs with connected medians, we characterize the isometric subgraphs of Johnson graphs and of halved-cubes with connected medians.

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Section: Discussionmentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Graphs with $G^p$-connected medians

Bénéteau¹,

Chalopin²,

Chepoi³

et al. 2022

Preprint

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“…In the full version of [11], we managed to encode the cube-free median graphs in O(n log n) time instead of O(n 2 log n). This improvement uses a recent result of [5] allowing to compute a median vertex of a median graph in linear time. We also compute in cube-free median graphs the partition into fibers, the gates (equivalent of the entrances) and the imprints (equivalent to the exits) in fibers in linear time, with a BFS-like algorithm.…”

Section: Discussionmentioning

confidence: 99%

Distance labeling schemes for $K_4$-free bridged graphs

Chepoi¹,

Labourel²,

Ratel³

2020

Preprint

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k-Approximate distance labeling schemes are schemes that label the vertices of a graph with short labels in such a way that the k-approximation of the distance between any two vertices u and v can be determined efficiently by merely inspecting the labels of u and v, without using any other information. One of the important problems is finding natural classes of graphs admitting exact or approximate distance labeling schemes with labels of polylogarithmic size. In this paper, we describe an approximate distance labeling scheme for the class of K4-free bridged graphs. This scheme uses labels of poly-logarithmic length O(log n 3 ) allowing a constant decoding time. Given the labels of two vertices u and v, the decoding function returns a value between the exact distance dG(u, v) and its quadruple 4dG(u, v).

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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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Medians in Median Graphs and Their Cube Complexes in Linear Time

Cited by 3 publications

References 0 publications

Graphs with $G^p$-connected medians

Graphs with $G^p$-connected medians

Distance labeling schemes for $K_4$-free bridged graphs

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

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