Distance and Routing Labeling Schemes for Cube-Free Median Graphs

Chepoi, Victor; Labourel, Arnaud; Ratel, Sébastien

doi:10.1007/s00453-020-00756-w

Cited by 12 publications

(42 citation statements)

References 44 publications

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“…If we choose the star centered at a median vertex of G, then the following lemma shows that the number of vertices in every fiber is bounded by |V |/2 (the proof is similar to the proof of [9,Lemma 8]).…”

Section: Lemmamentioning

confidence: 95%

“…The global structure of our distance labeling scheme for K 4 -free bridged graphs is similar to the one described in [9] for cube-free median graphs. Namely, the scheme is based on a recursive partitioning of the graph into a star and its fibers (which are classified as panels and cones).…”

Section: Main Ideas Of the Schemementioning

confidence: 99%

“…More precisely, it looks for the part i of the labels corresponding to the step in which m became a median. As noticed in [9], it is possible to find this median vertex m in constant time by adding particular O(log 2 n) bits information to the head of each label. Once the right part of label is found, the decoding function determines that two vertices are 1pc-neighboring if and only if the identifier (i.e., the star-label in St(m) of the fiber of one of the two vertices u, v is included in the identifier of the other).…”

Section: Distance Queriesmentioning

confidence: 99%

“…Several graph classes containing trees also admit DLS with labels of length O(log 2 n): bounded tree-width graphs [16], distance-hereditary graph [14], bounded cliquewidth graphs [10], or planar graphs of non-positive curvature [8]. More recently, in [9] we designed DLS with labels of length O(log 3 n) for cube-free median graphs. Other families of graphs have been considered such as interval graphs, permutation graphs, and their generalizations [4,15] for which an optimal bound of Θ(log n) bits was given, and planar graphs for which there is a lower bound of Ω(n 1 3 ) bits [16] and an upper bound of O( √ n) bits [17].…”

Section: Introductionmentioning

confidence: 99%

“…Finding natural classes of graphs admitting exact or approximate distance labeling schemes with labels of polylogarithmic size is an important and challenging problem. In this note we continue the line of research we started in [9] to investigate classes of graphs with rich metric properties, and we design approximate distance labeling schemes of polylogarithmic size for K 4 -free bridged graphs. Together with hyperbolic, median, and Helly graphs, bridged graphs constitute the most important classes of graphs in metric graph theory [3,5].…”

Section: Introductionmentioning

confidence: 99%

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Distance Labeling Schemes for $$K_4$$-Free Bridged Graphs

Chepoi¹,

Labourel²,

Ratel³

2020

Structural Information and Communication Complexity

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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 kapproximation 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 show that the class of K4-free bridged graphs on n nodes enjoys 4-approximate distance labeling scheme with labels of O(log 3 n) bits.

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Section: Lemmamentioning

confidence: 95%

Section: Main Ideas Of the Schemementioning

confidence: 99%

Section: Distance Queriesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Distance Labeling Schemes for $$K_4$$-Free Bridged Graphs

Chepoi¹,

Labourel²,

Ratel³

2020

Structural Information and Communication Complexity

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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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