Faster Algorithms for Computing Plurality Points

Let V be a set of n points in R, which we call voters, where d is a fixed constant. A point p ∈ R is preferred over another point p′ ∈ R by a voter v ∈ V if dist(v, p) < dist(v, p′). A point p is called a plurality point if it is preferred by at least as many voters as any other point p′. We present an algorithm that decides in O(n logn) time whether V admits a plurality point in the L2 norm and, if so, finds the (unique) plurality point. We also give efficient algorithms to compute a minimum-cost subset W ⊂ V such that V \W admits a plurality point, and to compute a so-called minimum-radius plurality ball. Finally, we consider the problem in the personalized L1 norm, where each point v ∈ V has a preference vector 〈w1(v), . . . , wd(v)〉 and the distance from v to any point p ∈ R is given by ∑d i=1 wi(v) · |xi(v)−xi(p)|. For this case we can compute in O(n d−1) time the set of all plurality points of V . When all preference vectors are equal, the running time improves to O(n). 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems