An Efficient Algorithm for 2D Euclidean 2-Center with Outliers

TL;DR

This paper introduces efficient algorithms for the 2-center with outliers problem in 2D, extending to (p,k)-center variants under different metrics, with randomized and polynomial-time solutions.

Contribution

It presents a novel randomized algorithm for the (2,k)-center problem with outliers and provides solutions for (p,k)-center problems under the _-metric for p=4 and p=5.

Findings

Expected running time for (2,k)-center: O(n k^7 log^3 n)

Solutions for (p,k)-center with p=4 and p=5 in polynomial time

Effective algorithms for covering points with minimal radius disks

Abstract

For a set P of n points in R^2, the Euclidean 2-center problem computes a pair of congruent disks of the minimal radius that cover P. We extend this to the (2,k)-center problem where we compute the minimal radius pair of congruent disks to cover n-k points of P. We present a randomized algorithm with O(n k^7 log^3 n) expected running time for the (2,k)-center problem. We also study the (p,k)-center problem in R}^2 under the \ell_\infty-metric. We give solutions for p=4 in O(k^{O(1)} n log n) time and for p=5 in O(k^{O(1)} n log^5 n) time.