Handouts:
4/30/08: Arrangements and Duality

Recall: Applications of Duality:
(1) Stabbing a set of line segments (dual of a segment, dual of a triangle): In the dual, we look at the arrangement of double wedges corresponding to each segment, and we label each face with the number of double wedges present there. Then, the problem of finding a line that stabs the most segments is equivalent to finding a point in a face that has the highest label.
(2) angular sorting (which applies to visibility graphs): How to sort angularly points around each of n points. Naively this can be done in time O(n^2 log n), but the n sorts are not completely independent of each other. Using duality, and constructing the arrangement of the duals to the n points, I showed how to do the n sorts in total time O(n^2).
(3) Ham Sandwich Theorem: mentioned it only briefly with an example.
(4) Degeneracy testing: Determine if there are 3 points that are collinear among an input set of n points.