Computing the Rectilinear Center of Uncertain Points in the Plane

TL;DR

This paper introduces an optimal algorithm to find a point minimizing the maximum expected rectilinear distance to uncertain points in the plane, effectively solving a complex probabilistic geometric problem.

Contribution

The paper presents the first optimal $O(mn)$ time algorithm for computing the rectilinear center of uncertain points in the plane.

Findings

Algorithm runs in $O(mn)$ time, matching input size.

Provides an exact solution for the probabilistic rectilinear center problem.

Establishes optimality of the algorithm based on input size.

Abstract

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 $q^*$ in the plane which minimizes the maximum expected rectilinear distance from $q^*$ to all uncertain points of $P$, and $q^*$ 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 $\Theta(mn)$, our algorithm is optimal.