Computing Origami Universal Molecules with Cyclic Tournament Forests

Lang's "universal molecule" algorithm solves a variant of the origami design problem. It takes as input a metric tree and a convex polygonal region (the "paper") having a certain metric relationship with the tree. It computes a crease- pattern which allows for the paper to "fold" to a uniaxial base, which is a 3-dimensional shape projecting onto the given tree. Lang's universal molecule algorithm runs in cubic time and quadratic space. We investigate two implementations which improve the running time to sub-cubic time. The first uses a cyclic tournament forest, a new data structure which extends kinetic tournament trees to allow for cycle splitting operations, and the second uses a priority queue to store events.