Easy computation of eccentricity approximating trees

Ducoffe, Guillaume

doi:10.1016/j.dam.2019.01.006

Cited by 5 publications

(5 citation statements)

References 18 publications

(27 reference statements)

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“…In what follows, we present two right-sided additive eccentricity approximation schemes for all vertices, using a notion of eccentricity approximating spanning tree introduced in [48] and investigated in [20,25,29,31,34]. We get for m-edge α i -metric graphs a O(m) time right-sided additive (9i + 5)-approximation and a O(im) time right-sided additive (4i + 2)-approximation.…”

Section: Approximating All Eccentricities In α I -Metric Graphsmentioning

confidence: 99%

“…An eccentricity 2-approximating spanning tree of a chordal graph can be computed in linear time [25]. An eccentricity k-approximating spanning tree with minimum k can be found in O(nm) time for any n-vertex, m-edge graph G [34]. It is also known [20,29] that if G is a δ-hyperbolic graph, then G admits an eccentricity (4δ + 1)-approximating spanning tree constructible in O(δm) time and an eccentricity (6δ)-approximating spanning tree constructible in O(m) time.…”

Section: Approximating All Eccentricities In α I -Metric Graphsmentioning

confidence: 99%

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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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Section: Approximating All Eccentricities In α I -Metric Graphsmentioning

confidence: 99%

Section: Approximating All Eccentricities In α I -Metric Graphsmentioning

confidence: 99%

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

Dragan

Ducoffe

Guarnera

2021

Lecture Notes in Computer Science

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“…In what follows, we illustrate two right-sided additive eccentricity approximations for all vertices using a notion of eccentricity approximating spanning tree introduced in [25] and investigated in [11,14,17,18]. We get a O(|E|) time right-sided additive (6δ)-approximations and a O(δ|E|) time rightsided additive (4δ + 1)-approximations.…”

Section: Right-sided Additive Approximations Of All Vertex Eccentrici...mentioning

confidence: 99%

“…An eccentricity 2-approximating spanning tree of a chordal graph can be computed in linear time [14]. An eccentricity k-approximating spanning tree with minimum k can be found in O(|V ||E|) time for any graph G [18]. It is also known [11] that if G is a τ -thin graph, then G admits an eccentricity (2τ )-approximating spanning tree constructible in O(τ |E|) time and an eccentricity (6τ + 1)-approximating spanning tree constructible in O(|E|) time.…”

Section: Right-sided Additive Approximations Of All Vertex Eccentrici...mentioning

confidence: 99%

Eccentricity terrain of $δ$-hyperbolic graphs

Dragan,

Guarnera

2020

Preprint

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A graph G = (V, E) is δ-hyperbolic if for any four vertices u, v, w, x, the two larger of the three distance sums d(u, v) + d(w, x), d(u, w) + d(v, x), and d(u, x) + d(v, w) differ by at most 2δ ≥ 0. 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 δ. This paper describes the eccentricity terrain of a δ-hyperbolic graph. The eccentricity function eThe paper studies the eccentricity layers of vertices along shortest paths, identifying such terrain features as hills, plains, valleys, terraces, and plateaus. It introduces the notion of β-pseudoconvexity, which implies Gromov's -quasiconvexity, and illustrates the abundance of pseudoconvex sets in δ-hyperbolic graphs. In particular, it shows that all sets C ≤k (G) = {v ∈ V : e G (v) ≤ rad(G) + k}, k ∈ N, are (2δ − 1)-pseudoconvex. Additionally, several bounds on the eccentricity of a vertex are obtained which yield a few approaches to efficiently approximating all eccentricities. An O(δ|E|) time eccentricity approximation ê(v), for all v ∈ V , is presented that uses distances to two mutually distant vertices and satisfies e G (v) − 2δ ≤ ê(v) ≤ e G (v). It also shows existence of two eccentricity approximating spanning trees T , one constructible in O(δ|E|) time and the other in O(|E|) time, which satisfy e G (v) ≤ e T (v) ≤ e G (v) + 4δ + 1 and e G (v) ≤ e T (v) ≤ e G (v) + 6δ, respectively. Thus, the eccentricity terrain of a tree gives a good approximation (up-to an additive error O(δ)) of the eccentricity terrain of a δ-hyperbolic graph.

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“…Finally, we compute, for 1 ≤ i ≤ k, all eccentricities in T i . Again, this can be done in total O(N ) time (e.g., see [29,50]). This concludes the pre-processing phase.…”

Section: Algorithmsmentioning

confidence: 99%

Isometric embeddings in trees and their use in the diameter problem

Ducoffe¹

2020

Preprint

Self Cite

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A well-studied topic in Metric Graph Theory, as well as for more general metric spaces, is the existence of isometric embeddings in some product, distribution or system of trees. We initiate the study of a query-answering problem on a finite system of trees whose definition has been tailored for the fast resolution of distance and facility location problems on such "tree-like" spaces. Doing so, we prove that given a discrete space with n points which is either embedded in a system of k trees, or the Cartesian product of k trees, we can compute all eccentricities in 1) ) time, where N is the cumulative total order over all these k trees. This is near optimal under the Strong Exponential-Time Hypothesis, even in the very special case of an n-vertex graph embedded in a system of ω(log n) spanning trees. However, given such an embedding in the strong product of k trees, there is a much faster O(N + kn)-time algorithm for this problem. All our positive results can be turned into approximation algorithms for the graphs and finite spaces with a quasi isometric embedding in trees, if such embedding is given as input, where the approximation factor (resp., the approximation constant) depends on the distortion of the embedding (resp., of its stretch).The existence of embeddings in the Cartesian product of finitely many trees has been thoroughly investigated for cube-free median graphs. We give the first-known quasi linear-time algorithm for computing the diameter within this graph class. It does not require an embedding in a product of trees to be given as part of the input. On our way, being given an n-node tree T , we propose a data structure with O(n log n) pre-processing time in order to compute in O(k log 2 n) time the eccentricity of any subset of k nodes. We combine the latter technical contribution, of independent interest, with a recent distance-labeling scheme that was designed for cube-free median graphs.

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Easy computation of eccentricity approximating trees

Cited by 5 publications

References 18 publications

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

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

Eccentricity terrain of $δ$-hyperbolic graphs

Isometric embeddings in trees and their use in the diameter problem

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