Computing an approximation of the 1-center problem on weighted terrain surfaces

Abstract

In this article, we discuss the problem of determining a meeting point of a set of scattered robots R = { r 1 , r 2 ,…, r s } in a weighted terrain P, which has n > s triangular faces. Our algorithmic approach is to produce a discretization of P by producing a graph G = { V G , E G }, which lies on the surface of P. For a chosen vertex p′ ∈ V G , we define ‖Π( r i , p′ )‖ as the minimum weight cost of traveling from r i to p′ . We show that min p′ ∈ V G {max 1≤ i ≤ s {‖Π( r i , p′ )‖}} ≤ min p *∈P {max 1≤ i ≤ s {‖Π( r i , p *)‖}} + 2 W | L |, where L is the longest edge of P, W is the maximum cost weight of a face of P, and p * is the optimal solution. Our algorithm requires O ( snm log( snm ) + snm 2 ) time to run, where m = n in the Euclidean metric and m = n 2 in the weighted metric. However, we show, through experimentation, that only a constant value of m is required (e.g., m = 8) in order to produce very accurate solutions (< 1% error). Hence, for typical terrain data, the expected running time of our algorithm is O ( sn log( sn )). Also, as part of our experiments, we show that by using geometrical subsets (i.e., 2D/3D convex hulls, 2D/3D bounding boxes, and random selection) of the robots we can improve the running time for finding p′ , with minimal or no additional accuracy error when comparing p′ to p *.