Determine whether the two classes are linearly separable (algorithmically in 2D)

Question

There are two classes, let's call them X and O. A number of elements belonging to these classes are spread out in the xy-plane. Here is an example where the two classes are not linearly separable. It is not possible to draw a straight line that perfectly divides the Xs and the Os on each side of the line.

How to determine, in general, whether the two classes are linearly separable?. I am interested in an algorithm where no assumptions are made regarding the number of elements or their distribution. An algorithm of the lowest computational complexity is of course preferred.

Answer 1

If you found the convex hull for both the X points and the O points separately (i.e. you have two separate convex hulls at this stage) you would then just need to check whether any segments of the hulls intersected or whether either hull was enclosed by the other.

If the two hulls were found to be totally disjoint the two data-sets would be geometrically separable.

Since the hulls are convex by definition, any separator would be a straight line.

There are efficient algorithms that can be used both to find the convex hull (the qhull algorithm is based on an O(nlog(n)) quickhull approach I think), and to perform line-line intersection tests for a set of segments (sweepline at O(nlog(n))), so overall it seems that an efficient O(nlog(n)) algorithm should be possible.

This type of approach should also generalise to general k-way separation tests (where you have k groups of objects) by forming the convex hull and performing the intersection tests for each group.

It should also work in higher dimensions, although the intersection tests would start to become more challenging...

Hope this helps.