Eccentricity approximating trees

Dragan, Feodor F.; Khler, Ekkehard; Alrasheed, Hend

doi:10.1016/j.dam.2017.07.017

Cited by 10 publications

(19 citation statements)

References 20 publications

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“…Theorem 1 generalizes recent results from [51,81] that chordal graphs and some of their generalizations admit eccentricity 2-approximating spanning trees.…”

Section: ⊓ ⊔supporting

confidence: 76%

“…A spanning tree T of a graph G is called an eccentricity k-approximating spanning tree if for every vertex v of G ecc T (v) ≤ ecc G (v) + k holds [51,81]. Thus, by Proposition 5, we get.…”

Section: ⊓ ⊔mentioning

confidence: 94%

“…It is interesting to note that the equality ecc G (x) = d G (x, C(G)) + rad(G) holds for every vertex of a graph G if and only if the eccentricity function ecc G (·) on G is unimodal (that is, every local minimum is a global minimum) [48]. A slightly weaker condition holds for all chordal graphs [51]: for every vertex x of a chordal graph G, ecc G…”

Section: ⊓ ⊔mentioning

confidence: 99%

“…Similar results are known also for all eccentricities: a 5/3-approximation to the eccentricities of all vertices can be computed inÕ(m 3/2 ) time [31] and, under plausible assumptions, no O(n 2−ǫ ) time algorithm can exist that (5/3 − ǫ ′ )-approximates (for ǫ, ǫ ′ > 0) the eccentricities of all vertices in sparse graphs [8]. Better approximation algorithms are known for some special classes of graphs [27,34,35,42,43,50,51,54,94]. A number of heuristics for approximating diameters, radii and eccentricities in real-world graphs were proposed and investigated in [10,21,22,23,69,24,52].…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, although for every k there is a chordal graph without an additive tree k-spanner [70,80], yet as Prisner demonstrated in [81], every chordal graph has an eccentricity 2-approximating spanning tree. Later this result was extended in [51] to a larger family of graphs which includes all chordal graphs and all plane triangulations with inner vertices of degree at least 7. Both those classes belong to the class of 1-hyperbolic graphs.…”

Section: Introductionmentioning

confidence: 99%

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Fast Approximation of Centrality and Distances in Hyperbolic Graphs

Chepoi¹,

Dragan²,

Habib³

et al. 2018

Combinatorial Optimization and Applications

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We show that the eccentricities (and thus the centrality indices) of all vertices of a δhyperbolic graph G = (V, E) can be computed in linear time with an additive one-sided error of at most cδ, i.e., after a linear time preprocessing, for every vertex v of G one can compute in O(1) time an estimateê(v) of its eccentricity eccG(v) such that eccG(v) ≤ê(v) ≤ eccG(v) + cδ for a small constant c. We prove that every δ-hyperbolic graph G has a shortest path tree, constructible in linear time, such that for every vertexThese results are based on an interesting monotonicity property of the eccentricity function of hyperbolic graphs: the closer a vertex is to the center of G, the smaller its eccentricity is. We also show that the distance matrix of G with an additive one-sided error of at most c ′ δ can be computed in O(|V | 2 log 2 |V |) time, where c ′ < c is a small constant. Recent empirical studies show that many real-world graphs (including Internet application networks, web networks, collaboration networks, social networks, biological networks, and others) have small hyperbolicity. So, we analyze the performance of our algorithms for approximating centrality and distance matrix on a number of real-world networks. Our experimental results show that the obtained estimates are even better than the theoretical bounds.

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“…Theorem 1 generalizes recent results from [51,81] that chordal graphs and some of their generalizations admit eccentricity 2-approximating spanning trees.…”

Section: ⊓ ⊔supporting

confidence: 76%

Section: ⊓ ⊔mentioning

confidence: 94%

Section: ⊓ ⊔mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Fast Approximation of Centrality and Distances in Hyperbolic Graphs

Chepoi¹,

Dragan²,

Habib³

et al. 2018

Combinatorial Optimization and Applications

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Fast Deterministic Algorithms for Computing All Eccentricities in (Hyperbolic) Helly Graphs

Dragan

Ducoffe

Guarnera

2021

Lecture Notes in Computer Science

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We extend known results on chordal graphs and distance-hereditary graphs to much larger graph classes by using only a common metric property of these graphs. Specifically, a graph is called αi-metric (i ∈ N ) if it satisfies the following αi-metric property for every vertices u, w, v and x: if a shortest path between u and w and a shortest path between x and v share a terminal edge vw,Roughly, gluing together any two shortest paths along a common terminal edge may not necessarily result in a shortest path but yields a "near-shortest" path with defect at most i. It is known that α0-metric graphs are exactly ptolemaic graphs, and that chordal graphs and distance-hereditary graphs are αi-metric for i = 1 and i = 2, respectively. We show that an additive O(i)-approximation of the radius, of the diameter, and in fact of all vertex eccentricities of an αi-metric graph can be computed in total linear time. Our strongest results are obtained for α1-metric graphs, for which we prove that a central vertex can be computed in subquadratic time, and even better in linear time for so-called (α1, ∆)-metric graphs (a superclass of chordal graphs and of plane triangulations with inner vertices of degree at least 7). The latter answers a question raised in (Dragan, IPL, 2020). Our algorithms follow from new results on centers and metric intervals of αi-metric graphs. In particular, we prove that the diameter of the center is at most 3i + 2 (at most 3, if i = 1). The latter partly answers a question raised in (Yushmanov & Chepoi, Mathematical Problems in Cybernetics, 1991).

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$$\alpha _i$$-Metric Graphs: Radius, Diameter and all Eccentricities

Dragan,

Ducoffe

2023

Lecture Notes in Computer Science

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Eccentricity approximating trees

Cited by 10 publications

References 20 publications

Fast Approximation of Centrality and Distances in Hyperbolic Graphs

Fast Approximation of Centrality and Distances in Hyperbolic Graphs

Fast Deterministic Algorithms for Computing All Eccentricities in (Hyperbolic) Helly Graphs

$$\alpha _i$$-Metric Graphs: Radius, Diameter and all Eccentricities

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