9 research outputs found
Analysis of Farthest Point Sampling for Approximating Geodesics in a Graph
A standard way to approximate the distance between any two vertices and
on a mesh is to compute, in the associated graph, a shortest path from
to that goes through one of sources, which are well-chosen vertices.
Precomputing the distance between each of the sources to all vertices of
the graph yields an efficient computation of approximate distances between any
two vertices. One standard method for choosing sources, which has been used
extensively and successfully for isometry-invariant surface processing, 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 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 can be bounded in terms
of the minimal value of the stretch factor obtained using an
optimal placement of sources as , where is the ratio of the lengths of
the longest and the shortest edges of the graph. This provides some evidence
explaining why farthest point sampling has been used successfully for
isometry-invariant shape processing. Furthermore, we show that it is
NP-complete to find sources that minimize the stretch factor.Comment: 13 pages, 4 figure
Analysis of Farthest Point Sampling for Approximating Geodesics in a Graph
International audienceA standard way to approximate the distance between two vertices and in a graph is to compute a shortest path from to that goes through one of sources, which are well-chosen vertices. Precomputing the distance between each of the sources to all vertices yields an efficient computation of approximate distances between any two vertices. One standard method for choosing 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 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 can be bounded in terms of the minimal value of the stretch factor obtained using an optimal placement of sources as , where is the length ratio of longest edge over the shortest edge in the graph. We further show that the factor is not an artefact of the analysis by providing a class of graphs for which
Stochastic Minimum Spanning Trees and Related Problems
We investigate the computational complexity of minimum spanning trees and maximum flows in a simple model of stochastic networks, where each node or edge of an undirected master graph can fail with an independent and arbitrary probability. We show that computing the expected length of the MST or the value of the max-flow is #P-Hard, but that for the MST it can be approximated within O(log n) factor for metric graphs. The hardness proof for the MST applies even to Euclidean graphs in 3 dimensions. We also show that the tail bounds for the MST cannot be approximated in general to any multiplicative factor unless P = NP. This stochastic MST problem was mentioned but left unanswered by Bertsimas, Jaillet and Odoni [Operations Research, 1990] in their work on a priori optimization. More generally, we also consider the complexity of linear programming under probabilistic constraints, and show it to be #P-Hard. If the linear program has a constant number of variables, then it can be solved exactly in polynomial time. For general dimensions, we give a randomized algorithm for approximating the probability Uncertainty is a fact of life whether we are dealing with physical or natural systems: devices fail