Question

This is a question that I was asked on a job interview some time ago. And I still can’t figure out sensible answer.

Question is:

you are given set of points (x,y). Find 2 most distant points. Distant from each other.

For example, for points: (0,0), (1,1), (-8, 5) – the most distant are: (1,1) and (-8,5) because the distance between them is larger from both (0,0)-(1,1) and (0,0)-(-8,5).

The obvious approach is to calculate all distances between all points, and find maximum. The problem is that it is O(n^2), which makes it prohibitively expensive for large datasets.

There is approach with first tracking points that are on the boundary, and then calculating distances for them, on the premise that there will be less points on boundary than “inside”, but it’s still expensive, and will fail in worst case scenario.

Tried to search the web, but didn’t find any sensible answer – although this might be simply my lack of search skills.

Answer 1

For this specific problem, with just a list of Euclidean points, one way is to find the convex hull of the set of points. The two distant points can then be found by traversing the hull once with the rotating calipers method.

Here is an O(N log N) implementation:

• http://mukeshiiitm.wordpress.com/2008/05/27/find-the-farthest-pair-of-points/

If the list of points is already sorted, you can remove the sort to get the optimal O(N) complexity.

---

For a more general problem of finding most distant points in a graph:
Algorithm to find two points furthest away from each other

The accepted answer works in O(N^2).