Faster Algorithms for Computing Plurality Points

Let V be a set of n points in Rd, which we call voters. A point p ∈ Rd is preferred over another point p′ ∈ Rd by a voter υ ∈ V if dist(υ, p) < dist(υ, 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(nlogn) 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 υ ∈ V has a preference vector ⟨w1(υ),…,wd(υ)⟩ and the distance from υ to any point p ∈ Rd is given by ∑i=1d wi(υ)· |xi(υ)−xi(p)|. For this case we can compute in O(nd−1) time the set of all plurality points of V. When all preference vectors are equal, the running time improves to O(n).