Compute the area covered by cards randomly put on a table

Question

This is an interview question, the interview has been done.

Given a deck of rectangular cards, put them randomly on a rectangular table whose size is much larger than t

Answer 1

This could be done easily using the union-intersection formula (size of A union B union C = A + B + C - AB - AC - BC + ABC, etc), but that would result in an O(n!) algorithm. There is another, more complicated way that results in O(n^2 (log n)^2).

---

Store each card as a polygon + its area in a list. Compare each polygon in the list to every other polygon. If they intersect, remove them both from the list, and add their union to the list. Continue until no polygons intersect. Sum their areas to find the total area.

The polygons can be concave and have holes, so computing their intersection is not easy. However, there are algorithms (and libraries) available to compute it in O(k log k), where k is the number of vertices. Since the number of vertices can be on the order of n, this means computing the intersection is O(n log n).

Comparing every polygon to every other polygon is O(n^2). However, we can use an O(n log n) sweeping algorithm to find nearest polygons instead, making the overall algorithm O((n log n)^2) = O(n^2 (log n)^2).