Output sensitive algorithm for covering many points

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 ρ as the maximum number of points that one disk can cover. In this paper first we solve the problem for m = 2 in O(nρ + ρ3 log ρ)) time. The previous algorithm for this problem runs in O(n3 logn) time. Our algorithm outperforms the previous algorithm because ρ 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(mnρ + (mρ)2m−1 logmρ) time. The previous algorithm for this problem runs in O(n2m−1 logn) time. Our algorithm runs faster than the previous algorithm because mρ 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