Flip Distance to some Plane Configurations

We study an old geometric optimization problem in the plane. Given a perfect matching $M$ on a set of $n$ points in the plane, we can transform it to a non-crossing perfect matching by a finite sequence of flip operations. The flip operation removes two crossing edges from $M$ and adds two non-crossing edges. Let $f(M)$ and $F(M)$ denote the minimum and maximum lengths of a flip sequence on $M$, respectively. It has been proved by Bonnet and Miltzow (2016) that $f(M)=O(n^2)$ and by van Leeuwen and Schoone (1980) that $F(M)=O(n^3)$. We prove that $f(M)=O(n\Delta)$ where $\Delta$ is the spread of the point set, which is defined as the ratio between the longest and the shortest pairwise distances. This improves the previous bound if the point set has sublinear spread. For a matching $M$ on $n$ points in convex position we prove that $f(M)=n/2-1$ and $F(M)={{n/2} \choose 2}$; these bounds are tight. Any bound on $F(\cdot)$ carries over to the bichromatic setting, while this is not necessarily true for $f(\cdot)$. Let $M'$ be a bichromatic matching. The best known upper bound for $f(M')$ is the same as for $F(M')$, which is essentially $O(n^3)$. We prove that $f(M')\le n-2$ for points in convex position, and $f(M')= O(n^2)$ for semi-collinear points. The flip operation can also be defined on spanning trees. For a spanning tree $T$ on a convex point set we show that $f(T)=O(n\log n)$.