A Simple Sweep Line Algorithm for Counting Triangulations and Pseudo-triangulations

Let P ⊂ R 2 be a set of n points. In [1] and [2] an algorithm for counting triangulations and pseudo-triangulations of P, respectively, is shown. Both algorithms are based on the divide-and-conquer paradigm, and both work by finding sub-structures on triangulations and pseudo-triangulations that allow the problems to be split. These sub-structures are called triangulation paths for triangulations, or T-paths for short, and zig-zag paths for pseudo-triangulations, or PT-paths for short. Those two algorithms have turned out to be very difficult to analyze, to the point that no good analysis of their running time has been presented so far. The interesting thing about those algorithms, besides their simplicity, is that they experimentally indicate that counting can be done significantly faster than enumeration. In this paper we show two new algorithms, one to compute the number of triangulations of P, and one to compute the number of pseudo-triangulations of P. They are also based on T-paths and PT-paths respectively, but use the sweep line paradigm and not divide-and-conquer. The important thing about our algorithms is that they admit a good analysis of their running times. We will show that our algorithms run in time O � (t(P)) and O � (pt(P)) respectively, where t(P) and pt(P) is the largest number of T-paths and PT-paths, respectively, that the algorithms encounter during their execution. Moreover, we show that t(P) = O � (9 n ), which is the first non-trivial bound on t(P) to be known.