An Improved Algorithm and Analysis for Finding the Simpson Point of a Convex Polygon In Competitive Location

We consider the problem of finding the Simpson Point, also known as the (1|1) Centroid, of a market that is uniformly distributed over a convex polygon. Diaz and O’Rourke (1994) developed an iterative search algorithm for the case when the demand distribution is uniform; they also observed that it did not appear to converge pointwise, and modified it to do so. We present an enhancement of their algorithm that improves its time complexity from O(log 1/ε + n log 1/ε) to O(n log 1/ε). We also prove that both our algorithm and the original unmodified Diaz-O’Rourke algorithm, appropriately interpreted, do converge pointwise and we derive explicit bounds on convergence rates. Finally, we explore how our algorithm might be modified to find the Simpson Point when the demand distribution is non-uniform.