Smoothed Analysis of Order Types
Ivor van der Hoog, Tillmann Miltzow, van Schaik, Martijn · arXiv (Cornell University) · 2019
Consider an ordered point set $P = (p_1,\ldots,p_n)$, its order type (denoted by $χ_P$) is a map which assigns to every triple of points a value in $\{+,-,0\}$ based on whether the points are collinear(0), oriented clockwise(-) or counter-clockwise(+). An abstract order type is a map $χ: \left[\substack{n\\3}\right] \rightarrow \{+,-,0\}$ (where $\left[\substack{n\\3}\right]$ is the collection of all triples of a set of $n$ elements) that satisfies the following condition: for every set of five elements $S\subset [n]$ its induced order type $χ_{|S}$ is realizable by a point set. To be precise, a point set $P$ realizes an order type $χ$,if $χ_P(p_i,p_j,p_k) = χ(i,j,k)$, for all $i