Smoothed Analysis of Order Types

Abstract

Consider an ordered point set $P=(p_1,\dots ,p_n)$, its order type (denoted by ${\chi }_P$) 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 $\chi :\left[\begin{array}{c}n\\ 3\end{array}\right]\to \{+,-,0\}$ (where $\left[\begin{array}{c}n\\ 3\end{array}\right]$ 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 ${\chi }_{|S}$ is realizable by a point set. To be precise, a point set $P$ realizes an order type $\chi$,if ${\chi }_P(p_i,p_j,p_k)=\chi (i,j,k)$, for all $i<j<k$. Planar point sets are among the most basic and natural geometric objects of study in Discrete and Computational Geometry. Properties of point sets are relevant in theory and practice alike. It is known, that deciding if an abstract order type is realizable is complete for the existential theory of the reals. Our results show that order type realizability is much easier for realistic instances than in the worst case. In particular, we can recognize instances in "expected \NP-time". This is one of the first $\exists \mathbb{R}$-complete problems analyzed under the lens of Smoothed Analysis.

15 pages, 6 figures, long introduction and short proofs