Triangle partitioning

Question

This was a problem in the 2010 Pacific ACM-ICPC contest. The gist of it is trying to find a way to partition a set of points inside a triangle into three subtriangles such that

Answer 1

Here's an O(n log n) algorithm. Let's assume no degeneracy.

The high-level idea is, given a triangle PQR,

   P
  C \
 /  S\
R-----Q

we initially place the center point C at P. Slide C toward R until there are n points inside the triangle CPQ and one (S) on the segment CQ. Slide C toward Q until either triangle CRP is no longer deficient (perturb C and we're done) or CP hits a point. In the latter case, slide C away from P until either triangle CRP is no longer deficient (we're done) or CQ hits a point, in which case we begin sliding C toward Q again.

Clearly the implementation cannot “slide” points, so for each triangle involving C, for each vertex S of that triangle other than C, store the points inside the triangle in a binary search tree sorted by angle with S. These structures suffice to implement this kinetic algorithm.

I assert without proof that this algorithm is correct.

As for the running time, each event is a point-line intersection and can be handled in time O(log n). The angles PC and QC and RC are all monotonic, so each of O(1) lines hits each point at most once.