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Let T be a tessellation composed of equilateral triangular regions, where each region has an associated positive weight. We present two methods that discretize the space based on the placement of Steiner points in the cells of T. Using such a discretization, we can use Dijkstra's algorithm for computing the shortest path in the geometric graph obtained. This will lead us to two approximation algorithms for solving the Weighted Region Problem. For a given parameter ε∈(0,1], the first discretization scheme provides an approximate path that is (1+0.428ε) times better than the approximation given by Aleksandrov et al. Determining approximate shortest paths on weighted polyhedral surfaces. Journal of the ACM, 52(1):25-53, 2005. The other discretization scheme uses at least (ε+2ε+4ε+4)log2e fewer points per segment of the triangulation with the same approximation factor.
Bose et al. (Tue,) studied this question.