A standard way to approximate the distance between any two vertices p and
q on a mesh is to compute, in the associated graph, 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 of
the graph yields an efficient computation of approximate distances between any
two vertices. One standard method for choosing k 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 FFPSâ 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 FFPSâ can be bounded in terms
of the minimal value Fâ of the stretch factor obtained using an
optimal placement of k sources as FFPSââ¤2re2âFâ+2re2â+8reâ+1, where reâ 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 k sources that minimize the stretch factor.Comment: 13 pages, 4 figure