Cover $n$ points in the plane with $k$ circles of radius $1$

Question

I am studying for an algorithms exam and ran into this example problem.

A circular disk with radius $2r$ can be covered by $7$ circular disks with radius $r$; in the following you can use this statement without proof.

Suppose that we are given a set $S$ of $n$ points in the plane, and we want to cover all points in $S$ by a minimum number of circular disks of radius $1$. Let $k$ be the optimal number of disks needed. Propose a polynomial time algorithm to produce a covering with at most $7k$ circular disks of radius $1$.

I do understand the question but I have no idea where to begin or how to approach this. Should a reduction be used? Or is there a simple and obvious answer? How do you cover $n$ points that you know nothing about? Maybe calculate the distances between them?

I do realize that if the distance between all points is greater than $2$ units, then $k = n$ since every circle of radius $2$ can only cover a single point.

I mainly want to understand the problem so that I can attempt a solution on my own, but a solution to the problem would also be nice, so that I can check my answer later.

Answer 1

Use a greedy algorithm to cover your points with circular disks of radius 2. In other words, repeatedly choose an uncovered point, and cover it with a circular disk of radius 2 centered at the point. Now cover each of your disks with seven disks of radius 1.

Why does this work? Suppose that you covered the points using $k$ circular disks of radius 2 centered at the points $p_1,\ldots,p_k$. It is not hard to check that the distance between any two different points $p_i,p_j$ is larger than 2 (otherwise $p_j$ would have been covered by the disk around $p_i$, assuming $i < j$), and so no circular disk of radius 1 can cover both $p_i$ and $p_j$. This implies that $k$ circular disks of radius 1 are needed to cover the points $p_1,\ldots,p_k$, and so at least $k$ are needed to cover the original set of points. The solution produced by the algorithm uses $7k$ circular disks.