An Efficient Algorithm for the Proximity Connected Two Center Problem

TL;DR

This paper introduces a new deterministic algorithm with quadraticithmic complexity for the proximity connected 2-center problem, a generalized version of the 2-center problem with an additional distance constraint between centers.

Contribution

The paper presents the first nontrivial deterministic algorithm for the proximity connected 2-center problem with a time complexity of O(n^2 log n).

Findings

Provides a deterministic algorithm for the PCTC problem.

Achieves an O(n^2 log n) time complexity.

Solves an open problem from 1992 regarding the PCTC problem.

Abstract

Given a set $P$ of $n$ points in the plane, the $k$-center problem is to find $k$ congruent disks of minimum possible radius such that their union covers all the points in $P$. The $2$-center problem is a special case of the $k$-center problem that has been extensively studied in the recent past \cite{CAHN,HT,SH}. In this paper, we consider a generalized version of the $2$-center problem called \textit{proximity connected} $2$-center (PCTC) problem. In this problem, we are also given a parameter $\delta\geq 0$ and we have the additional constraint that the distance between the centers of the disks should be at most $\delta$. Note that when $\delta=0$, the PCTC problem is reduced to the $1$-center(minimum enclosing disk) problem and when $\delta$ tends to infinity, it is reduced to the $2$-center problem. The PCTC problem first appeared in the context of wireless networks in 1992 \cite{ACN0}, but obtaining a nontrivial deterministic algorithm for the problem remained open. In this paper, we resolve this open problem by providing a deterministic $O(n^2\log n)$ time algorithm for the problem.