Analysis of farthest point sampling for approximating geodesics in a graph
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 FFPS 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 FFPS can be bounded in terms of the minimal value F∗ of the stretch factor obtained using an optimal placement of k sources as FFPS≤2re2F∗ +2re2+8re+1, where re is the length ratio of longest edge over the shortest edge in the graph. We further show that the factor re is not an artefact of the analysis by providing a class of graphs for which FFPS≥12reF∗.
|Keywords||Approximate geodesics, Approximation algorithms, Farthest Point Sampling, Planar graphs, Shortest paths|
Kamousi, P. (Pegah), Lazard, S. (Sylvain), Maheshwari, A, & Wuhrer, S. (Stefanie). (2016). Analysis of farthest point sampling for approximating geodesics in a graph. Computational Geometry, 57, 1–7. doi:10.1016/j.comgeo.2016.05.005