## Section: New Results

### Miscellaneous

#### On Order Types of Random Point Sets

Participant : Marc Glisse.

In collaboration with Olivier Devillers and Xavier Goaoc (Inria team Gamble) and Philippe Duchon (LaBRI, Université de Bordeaux).

Let $P$ be a set of $n$ random points chosen uniformly in the unit square. In this paper [41], we examine the typical resolution of the order type of $P$. First, we show that with high probability, $P$ can be rounded to the grid of step $\frac{1}{{n}^{3+\u03f5}}$ without changing its order type. Second, we study algorithms for determining the order type of a point set in terms of the number of coordinate bits they require to know. We give an algorithm that requires on average $4n{log}_{2}n+O\left(n\right)$ bits to determine the order type of $P$, and show that any algorithm requires at least $4n{log}_{2}n-O(nloglogn)$ bits. Both results extend to more general models of random point sets.