On Order Types of Random Point Sets

TL;DR

This paper investigates the distribution of order types of random point sets, presents experimental bias analysis for small sets, and establishes bounds on the information needed to determine order types, highlighting concentration issues.

Contribution

It provides experimental evidence of bias in random point set order types and derives bounds on the information complexity for determining order types.

Findings

Experimental bias observed in order types for n up to 16.

Average bits needed to determine order type is approximately 4n log n.

Lower bound on bits required for any algorithm is close to the average, indicating concentration.

Abstract

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.