Computing the Rectilinear Center of Uncertain Points in the Plane

In this paper, we consider the rectilinear one-center problem on uncertain points in the plane. In this problem, we are given a set $P$ of $n$ (weighted) uncertain points in the plane and each uncertain point has $m$ possible locations each associated with a probability for the point appearing at that location. The goal is to find a point $q^{\ast }$ in the plane which minimizes the maximum expected rectilinear distance from $q^{\ast }$ to all uncertain points of $P$, and $q^{\ast }$ is called a rectilinear center. We present an algorithm that solves the problem in $O(mn)$ time. Since the input size of the problem is $\mathrm{\Theta }(mn)$, our algorithm is optimal.