Analysis of farthest point sampling for approximating geodesics in a graph

作者：Kamousi Pegah; Lazard Sylvain; Maheshwari Anil; Wuhrer Stefanie^*

来源：Computational Geometry-Theory and Applications, 2016, 57: 1-7.

DOI：10.1016/j.comgeo.2016.05.005

• Farthest Point Sampling
• Approximate geodesics
• Shortest paths
• Planar graphs
• Approximation algorithms

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 F-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 F-FPS can be bounded in terms of the minimal value F* of the stretch factor obtained using an optimal placement of k sources as F-FPS <= 2r(e)(2)F* + 2r(e)(2) + 8r(e) + 1, where re is the length ratio of longest edge over the shortest edge in the graph.

• 出版日期2016-8