Analysis of Farthest Point Sampling for Approximating Geodesics in a Graph - Inria - Institut national de recherche en sciences et technologies du numérique Accéder directement au contenu
Article Dans Une Revue Computational Geometry Année : 2016

Analysis of Farthest Point Sampling for Approximating Geodesics in a Graph

Résumé

A standard way to approximate the distance between two vertices $p$ and $q$ in a graph is to compute a shortest path from $p$ to $q$ that goes through one of $k$ sources, which are well-chosen vertices. Precomputing the distance between each of the $k$ sources to all vertices yields an efficient computation of approximate distances between any two vertices. One standard method for choosing $k$ sources is the so-called Farthest Point Sampling (FPS), which starts with a random vertex as the first source, and iteratively selects the farthest vertex from the already selected sources.
In this paper, we analyze the stretch factor $\mathcal{F}_{\text{FPS}}$ of approximate geodesics computed using FPS, which is the maximum, over all pairs of distinct vertices, of their approximated distance over their geodesic distance in the graph. We show that $\mathcal{F}_{\text{FPS}}$ can be bounded in terms of the minimal value $\mathcal{F}^\ast$ of the stretch factor obtained using an optimal placement of $k$ sources as $\mathcal{F}_{\text{FPS}}\leq 2 r_e^2 \mathcal{F}^\ast+ 2 r_e^2 + 8 r_e + 1$, where $r_e$ is the length ratio of longest edge over the shortest edge in the graph. We further show that the factor $r_e$ is not an artefact of the analysis by providing a class of graphs for which $\mathcal{F}_{\text{FPS}} \geq \frac{1}{2} r_e \mathcal{F}^\ast$.
Fichier principal
Vignette du fichier
ShortestPaths_final.pdf (322.34 Ko) Télécharger le fichier
Vignette du fichier
2016 Farthest Point Sampling.png (43.74 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)
Format Figure, Image
Loading...

Dates et versions

hal-01297624 , version 1 (04-04-2016)
hal-01297624 , version 2 (25-05-2016)

Identifiants

Citer

Pegah Kamousi, Sylvain Lazard, Anil Maheshwari, Stefanie Wuhrer. Analysis of Farthest Point Sampling for Approximating Geodesics in a Graph. Computational Geometry, 2016, 57, pp.1-7. ⟨10.1016/j.comgeo.2016.05.005⟩. ⟨hal-01297624v2⟩
405 Consultations
1277 Téléchargements

Altmetric

Partager

Gmail Mastodon Facebook X LinkedIn More