I am given as input n pairs of integers which describe points in the 2D plane that are known ahead of time to be vertices of some convex polygon. I'd like to efficiently sort these points in a clockwise or counter-clockwise fashion.
At first I thought of doing something like the initial step of Graham Scan, but I can't see a simple way to break ties for vertices that make the same angle with the anchor point. Notice that, as you walk along the sides of the polygon, sometimes these vertices may be getting closer to the anchor point, and sometimes they may be getting farther.
Something that does seem to work is producing a point in the interior of the polygon (for instance, the average of the n points) and using it is an anchor point for radial sorting of the input. Indeed, because the anchor point lies in the interior, any ray emanating from it contains at most one input point, so there will be no ties.
The overall complexity is not affected: computing the midpoint is an O(n) task, and the bottleneck is the actual sorting.
This does involve a few more operations than the hopeful version of Graham Scan (where we assume there are no ties to be broken), but the bigger loss is leaving integer arithmethic behind by introducing division into the mix.
This in turn can be remedied with scaling everything by a factor of n, but at this point it seems like grasping at straws.
Am I missing something? Is there a simpler, efficient way to solve this sorting problem, preferrably one that can avoid floating point calculations?
