Assigning Weights to Minimize the Covering Radius in the Plane

TL;DR

This paper presents two algorithms for assigning weights to points in a plane to minimize the maximum weighted distance from a center, improving efficiency especially when the number of weights is small.

Contribution

It introduces two algorithms with different complexities for the weighted covering radius minimization problem, including a near-linear time solution for constant weight counts.

Findings

First algorithm runs in O(k^2 n^2 log^3 n) time.

Second algorithm runs in O(k^5 n log^3 k + n log^3 n) time.

For constant k, the second algorithm achieves near-linear time complexity.

Abstract

Given a set $P$ of $n$ points in the plane and a multiset $W$ of $k$ weights with $k\leq n$, we assign each weight in $W$ to a distinct point in $P$ to minimize the maximum weighted distance from the weighted center of $P$ to any point in $P$. In this paper, we give two algorithms which take $O(k^2n^2\log^3 n)$ time and $O(k^5n\log^3k+kn\log^3 n)$ time, respectively. For a constant $k$, the second algorithm takes only $O(n\log^3n)$ time, which is near linear.