Abstract

Let S be a set of n points in the plane, and let each point p of S have a positive weight w(p). We consider the problem of positioning a point x inside a compact region R # R 2 such that min{ w(p) -1 d(x, p) ; p # S } is maximized. Based on the parametric search paradigm, we give the first subquadratic algorithms for this problem, with running time O(n log 4 n). Furthermore, we shall introduce the concept of `exact approximation' as the bit model counterpart to parametric search. Exploiting ideas from exact computation, we show that the considered problem can be solved in time O(L(L)n log n), where L denotes the maximal bit-size of input numbers, and (L) the complexity of multiplying two L-bit integers.

Citations