On Order Types of Random Point Sets

A simple method to produce a random order type is to take the order type of a random point set. We conjecture that many probability distributions on order types defined in this way are heavily concentrated and therefore sample inefficiently the space of order types. We present two results on this question. First, we study experimentally the bias in the order types of $n$ random points chosen uniformly and independently in a square, for $n$ up to $16$. 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 $4n \log\_2 n+O(n)$ bits to determine the order type of $P$, and show that any algorithm requires at least $4n \log\_2 n - O(n \log\log n)$ bits. This implies that the concentration conjecture cannot be proven by an "efficient encoding" argument

On Order Types of Random Point Sets

A simple method to produce a random order type is to take the order type of a random point set. We conjecture that many probabilitydistributions on order types defined in this way are heavily concentrated and therefore sample inefficiently the space of order types. We present two results on this question. First, we study experimentally the bias in the order types of $n$ random points chosen uniformly and independently in a square, for $n$ up to $16$. 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 $4n \log_2 n+O(n)$ bits to determine the order type of $P$, and show that any algorithm requires at least $4n \log_2 n - O(n \log\log n)$ bits. This implies that the concentration conjecture cannot be proven by an "efficient encoding'' argument

On Order Types of Random Point Sets

Let $P$ be a set of $n$ random points chosen uniformly in the unit square. In this paper, 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 $\frac{1}{n^{3+\epsilon}}$ without changing its order type. Second, we study algorithms for determining the order type of a point set in terms of the the number of coordinate bits they require to know. We give an algorithm that requires on average $4n \log_2 n + O(n)$ bits to determine the order type of $P$, and show that any algorithm requires at least $4n \log_2 n − O(n \log \log n)$ bits. Both results extend to more general models of random point sets