Linear transformation distance for bichromatic matchings

Let $P=B\cup R$ be a set of $2n$ points in general position, where $B$ is a set of $n$ blue points and $R$ a set of $n$ red points. A \emph{$BR$-matching} is a plane geometric perfect matching on $P$ such that each edge has one red endpoint and one blue endpoint. Two $BR$-matchings are compatible if their union is also plane. The \emph{transformation graph of $BR$-matchings} contains one node for each $BR$-matching and an edge joining two such nodes if and only if the corresponding two $BR$-matchings are compatible. In SoCG 2013 it has been shown by Aloupis, Barba, Langerman, and Souvaine that this transformation graph is always connected, but its diameter remained an open question. In this paper we provide an alternative proof for the connectivity of the transformation graph and prove an upper bound of $2n$ for its diameter, which is asymptotically tight

Geodesic-Preserving Polygon Simplification

Polygons are a paramount data structure in computational geometry. While the complexity of many algorithms on simple polygons or polygons with holes depends on the size of the input polygon, the intrinsic complexity of the problems these algorithms solve is often related to the reflex vertices of the polygon. In this paper, we give an easy-to-describe linear-time method to replace an input polygon $\mathcal{P}$ by a polygon $\mathcal{P}'$ such that (1) $\mathcal{P}'$ contains $\mathcal{P}$, (2) $\mathcal{P}'$ has its reflex vertices at the same positions as $\mathcal{P}$, and (3) the number of vertices of $\mathcal{P}'$ is linear in the number of reflex vertices. Since the solutions of numerous problems on polygons (including shortest paths, geodesic hulls, separating point sets, and Voronoi diagrams) are equivalent for both $\mathcal{P}$ and $\mathcal{P}'$, our algorithm can be used as a preprocessing step for several algorithms and makes their running time dependent on the number of reflex vertices rather than on the size of $\mathcal{P}$

Empty Monochromatic Simplices

Let $S$ be a $k$-colored (finite) set of $n$ points in $\mathbb{R}^d$, $d\geq 3$, in general position, that is, no {$(d + 1)$} points of $S$ lie in a common $(d - 1)$}-dimensional hyperplane. We count the number of empty monochromatic $d$-simplices determined by $S$, that is, simplices which have only points from one color class of $S$ as vertices and no points of $S$ in their interior. For $3 \leq k \leq d$ we provide a lower bound of $\Omega(n^{d-k+1+2^{-d}})$ and strengthen this to $\Omega(n^{d-2/3})$ for $k=2$. On the way we provide various results on triangulations of point sets in $\mathbb{R}^d$. In particular, for any constant dimension $d\geq3$, we prove that every set of $n$ points ($n$ sufficiently large), in general position in $\mathbb{R}^d$, admits a triangulation with at least $dn+\Omega(\log n)$ simplices

Empty monochromatic simplices

Let S be a k-colored (finite) set of n points in , da parts per thousand yen3, in general position, that is, no (d+1) points of S lie in a common (d-1)-dimensional hyperplane. We count the number of empty monochromatic d-simplices determined by S, that is, simplices which have only points from one color class of S as vertices and no points of S in their interior. For 3a parts per thousand currency signka parts per thousand currency signd we provide a lower bound of and strengthen this to Omega(n (d-2/3)) for k=2.; On the way we provide various results on triangulations of point sets in . In particular, for any constant dimension da parts per thousand yen3, we prove that every set of n points (n sufficiently large), in general position in , admits a triangulation with at least dn+Omega(logn) simplices.Postprint (author’s final draft

Asymptotic tracking by funnel control with internal models

Funnel control achieves output tracking with guaranteed tracking performance for unknown systems and arbitrary reference signals. In particular, the tracking error is guaranteed to satisfy time-varying error bounds for all times (it evolves in the funnel). However, convergence to zero cannot be guaranteed, but the error often stays close to the funnel boundary, inducing a comparatively large feedback gain. This has several disadvantages (e.g. poor tracking performance and sensitivity to noise due to the underlying high-gain feedback principle). In this paper, therefore, the usually known reference signal is taken into account during funnel controller design, i.e. we propose to combine the well-known internal model principle with funnel control. We focus on linear systems with linear reference internal models and show that under mild adjustments of funnel control, we can achieve asymptotic tracking for a whole class of linear systems (i.e. without relying on the knowledge of system parameters)