Probabilistic analysis of geometric location problems

We analyze the behaviour of thek center and median problems forn points randomly distributed in an arbitrary regionA ofR ^d . Under a mild assumption on the regionA, we show that fork≦k(n)=o(n/logn), the objective function values of the discrete and continuous versions of these problems are equal to each otheralmost surely. For the two-dimensional case, both these problems can be solved by placing the centers or medians in an especially simple regular hexagonal pattern (the ‘honeycomb heuristic’ of Papadimitriou). This yields the exact asymptotic values for thek center and median problem, namely, α(|A|/k)^1/2 and β(|A|/k)^1/2, where |A| denotes the volume ofA, α and β are known constants, and the objective of the median problem is given in terms of the average, rather than the usual total, distance. For the 3- and 4-dimensional case, similar results can be obtained for the center problem to within an accuracy of roughly one percent. As a by-product, we also get asymptotically optimal algorithms for the 2-dimensionalp-normk median problem and for the twin problems of minimizing the maximum number of vertices served by any center and similarly for maximizing the minimum.