Analysis of Farthest Point Sampling for Approximating Geodesics in a Graph

Pegah Kamousi 1 Sylvain Lazard 2 Anil Maheshwari 3 Stefanie Wuhrer 4
2 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
4 MORPHEO - Capture and Analysis of Shapes in Motion
Inria Grenoble - Rhône-Alpes, LJK - Laboratoire Jean Kuntzmann, INPG - Institut National Polytechnique de Grenoble
Abstract : 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 $\mathcal{F}_{\text{FPS}} \geq \frac{1}{2} r_e \mathcal{F}^\ast$.
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Submitted on : Wednesday, May 25, 2016 - 4:00:54 PM
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Pegah Kamousi, Sylvain Lazard, Anil Maheshwari, Stefanie Wuhrer. Analysis of Farthest Point Sampling for Approximating Geodesics in a Graph. Computational Geometry, Elsevier, 2016, 57, pp.1-7. ⟨10.1016/j.comgeo.2016.05.005⟩. ⟨hal-01297624v2⟩

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