# Identification of points using disks

## TL;DR

This paper investigates the minimal number of disks needed to uniquely identify points in the plane, providing bounds, complexity results, and efficient algorithms under certain conditions.

## Contribution

It establishes tight bounds on the number of disks for point identification, proves NP-completeness for fixed-radius disks, and offers a linear-time solution for colinear points.

## Key findings

- Approximately n/3 disks suffice under general position
- NP-completeness of fixed-radius disk identification
- Linear-time algorithm for colinear points

## Abstract

We consider the problem of identifying n points in the plane using disks, i.e., minimizing the number of disks so that each point is contained in a disk and no two points are in exactly the same set of disks. This problem can be seen as an instance of the test covering problem with geometric constraints on the tests. We give tight lower and upper bounds on the number of disks needed to identify any set of n points of the plane. In particular, we prove that if there are no three colinear points nor four cocyclic points, then roughly n/3 disks are enough, improving the known bound of (n+1)/2 when we only require that no three points are colinear. We also consider complexity issues when the radius of the disks is fixed, proving that this problem is NP-complete. In contrast, we give a linear-time algorithm computing the exact number of disks if the points are colinear.