Output sensitive algorithm for covering many points

Abstract

A set of points and a positive integer $m$ are given and our goal is to cover the maximum number of these point with $m$ disks. We devise the first output sensitive algorithm for this problem. We introduce a parameter $\rho$ as the maximum number of points that one disk can cover. In this paper first we solve the problem for $m=2$ in $O({n\rho} + {\rho ^3}\log \rho ))$ time. The previous algorithm for this problem runs in $O({n^3}\log n)$ time. Our algorithm outperforms the previous algorithm because $\rho$ is much smaller than $n$ in many cases. Then we extend the algorithm for any value of $m$ and we solve the problem in $O(m{n\rho} + {(m\rho )^{2m - 1}}\log m\rho )$ time. The previous algorithm for this problem runs in $O({n^{2m - 1}}\log n)$ time. Our algorithm runs faster than the previous algorithm because $m\rho$ is smaller than $n$ in many cases. Our technique to obtain an output sensitive algorithm is to use a greedy algorithm to confine the areas that we should search to obtain the result. Our technique in this paper may be applicable in other set covering problems that deploy a greedy algorithm, to obtain faster solutions.