Calculating the meeting point of scattered robots on weighted terrain surfaces

  • Authors:

  • Mark A. Lanthier;Doron Nussbaum;Tsuo-Jung Wang

  • Affiliations:

  • Carleton University, Ottawa, ON, Canada;Carleton University, Ottawa, ON, Canada;Carleton University, Ottawa, ON, Canada

  • Venue:
  • CATS '05 Proceedings of the 2005 Australasian symposium on Theory of computing - Volume 41
  • Year:
  • 2005

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Abstract

In this paper we discuss the problem of determining a meeting point of a set of scattered robots R = {r1, r2,..., rs} 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 = {VG, EG} which lies on the surface of P. For a chosen vertex p′ ε VG, we define ||II(ri, p′)|| as the minimum weight cost of traveling from ri to p′ We show that minp′εVG{max1≤i≤s {||II(ri,P′)|| ≤ min p*εP{max1≤i≤s{||II(ri, P*)||}} + 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. This error of W | L| is an upper bound which can be stated more precisely as [EQUATION], where m is an adjustable non-zero parameter and k is the maximum number of segments of II(ri, p′) ∀ 1 ≤ i ≤ s. Our algorithm requires O(snm log(snm) + snm2) time to run, where m = n in the Euclidean metric and m = n2 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 (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*.