Computing plurality points and condorcet points in Euclidean space

Abstract

This work concerns two kinds of spatial equilibria. Given a multiset of n points in Euclidean space equipped with the ℓ2-norm, we call a location a plurality point if it is closer to at least as many given points as any other location. A location is called a Condorcet point if there exists no other location which is closer to an absolute majority of the given points. In d-dimensional Euclidean space ℝd , we show that the plurality points and the Condorcet points are equivalent. When the given points are not collinear, the Condorcet point (which is also the plurality point) is unique in ℝd if such a point exists. To the best of our knowledge, no efficient algorithm has been proposed for finding the point if the dimension is higher than one. In this paper, we present an O(n d-1 logn)-time algorithm for any fixed dimension d ≥ 2. © 2013 Springer-Verlag.