Question

Given a set of n points, can we find three points that describe a triangle with minimum area in O(n^2)? If yes, how, and if not, can we do better than O(n^3)?

I have found some papers that state that this problem is at least as hard as the problem that asks to find three collinear points (a triangle with area 0). These papers describe an O(n^2) solution to this problem by reducing it to an instance of the 3-sum problem. I couldn’t find any solution for what I’m interested in however. See this (look for General Position) for such a paper and more information on 3-sum.

Answer 1

There are O(n²) algorithms for finding the minimum area triangle.

For instance you can find one here: http://www.cs.tufts.edu/comp/163/fall09/CG-lecture9-LA.pdf

If I understood that pdf correctly, the basic idea is as follows:

1. For each pair of points AB you find the point that is closest to it.

2. You construct a dual of the points so that lines <-> points.
   Line y = mx + c is mapped to point (m,c)

3. In the dual, for a given point (which corresponds to a segment in original set of points) the nearest line vertically gives us the required point for 1.

Apparently 2 & 3 can be done in O(n²) time.

Also I doubt the papers showed 3SUM-hardness by reducing to 3SUM. It should be the other way round.