Recognizing Weakly Simple Polygons

We present an O(n log n)-time algorithm that determines whether a given planar n-gon is weakly simple. This improves upon an O(n^2 log n)-time algorithm by [Chang, Erickson, and Xu, SODA, 2015]. Weakly simple polygons are required as input for several geometric algorithms. As such, how to recognize simple or weakly simple polygons is a fundamental question

Conflict-Free Coloring Made Stronger

In FOCS 2002, Even et al. showed that any set of $n$ discs in the plane can be Conflict-Free colored with a total of at most $O(\log n)$ colors. That is, it can be colored with $O(\log n)$ colors such that for any (covered) point $p$ there is some disc whose color is distinct from all other colors of discs containing $p$. They also showed that this bound is asymptotically tight. In this paper we prove the following stronger results: \begin{enumerate} \item [(i)] Any set of $n$ discs in the plane can be colored with a total of at most $O(k \log n)$ colors such that (a) for any point $p$ that is covered by at least $k$ discs, there are at least $k$ distinct discs each of which is colored by a color distinct from all other discs containing $p$ and (b) for any point $p$ covered by at most $k$ discs, all discs covering $p$ are colored distinctively. We call such a coloring a {\em $k$-Strong Conflict-Free} coloring. We extend this result to pseudo-discs and arbitrary regions with linear union-complexity. \item [(ii)] More generally, for families of $n$ simple closed Jordan regions with union-complexity bounded by $O(n^{1+\alpha})$, we prove that there exists a $k$-Strong Conflict-Free coloring with at most $O(k n^\alpha)$ colors. \item [(iii)] We prove that any set of $n$ axis-parallel rectangles can be $k$-Strong Conflict-Free colored with at most $O(k \log^2 n)$ colors. \item [(iv)] We provide a general framework for $k$-Strong Conflict-Free coloring arbitrary hypergraphs. This framework relates the notion of $k$-Strong Conflict-Free coloring and the recently studied notion of $k$-colorful coloring. \end{enumerate} All of our proofs are constructive. That is, there exist polynomial time algorithms for computing such colorings

Optimal Time-Convex Hull under the Lp Metrics

We consider the problem of computing the time-convex hull of a point set under the general $L_p$ metric in the presence of a straight-line highway in the plane. The traveling speed along the highway is assumed to be faster than that off the highway, and the shortest time-path between a distant pair may involve traveling along the highway. The time-convex hull ${TCH}(P)$ of a point set $P$ is the smallest set containing both $P$ and \emph{all} shortest time-paths between any two points in ${TCH}(P)$. In this paper we give an algorithm that computes the time-convex hull under the $L_p$ metric in optimal $O(n\log n)$ time for a given set of $n$ points and a real number $p$ with $1\le p \le \infty$

Algorithms for Colourful Simplicial Depth and Medians in the Plane

The colourful simplicial depth of a point x in the plane relative to a configuration of n points in k colour classes is exactly the number of closed simplices (triangles) with vertices from 3 different colour classes that contain x in their convex hull. We consider the problems of efficiently computing the colourful simplicial depth of a point x, and of finding a point, called a median, that maximizes colourful simplicial depth. For computing the colourful simplicial depth of x, our algorithm runs in time O(n log(n) + k n) in general, and O(kn) if the points are sorted around x. For finding the colourful median, we get a time of O(n^4). For comparison, the running times of the best known algorithm for the monochrome version of these problems are O(n log(n)) in general, improving to O(n) if the points are sorted around x for monochrome depth, and O(n^4) for finding a monochrome median.Comment: 17 pages, 8 figure

Melody recognition with learned edit distances

In a music recognition task, the classification of a new melody is often achieved by looking for the closest piece in a set of already known prototypes. The definition of a relevant similarity measure becomes then a crucial point. So far, the edit distance approach with a-priori fixed operation costs has been one of the most used to accomplish the task. In this paper, the application of a probabilistic learning model to both string and tree edit distances is proposed and is compared to a genetic algorithm cost fitting approach. The results show that both learning models outperform fixed-costs systems, and that the probabilistic approach is able to describe consistently the underlying melodic similarity model.This work was funded by the French ANR Marmota project, the Spanish PROSEMUS project (TIN2006-14932-C02), the research programme Consolider Ingenio 2010 (MIPRCV, CSD2007-00018), and the Pascal Network of Excellence

Unsplittable coverings in the plane

A system of sets forms an {\em $m$-fold covering} of a set $X$ if every point of $X$ belongs to at least $m$ of its members. A $1$-fold covering is called a {\em covering}. The problem of splitting multiple coverings into several coverings was motivated by classical density estimates for {\em sphere packings} as well as by the {\em planar sensor cover problem}. It has been the prevailing conjecture for 35 years (settled in many special cases) that for every plane convex body $C$, there exists a constant $m=m(C)$ such that every $m$-fold covering of the plane with translates of $C$ splits into $2$ coverings. In the present paper, it is proved that this conjecture is false for the unit disk. The proof can be generalized to construct, for every $m$, an unsplittable $m$-fold covering of the plane with translates of any open convex body $C$ which has a smooth boundary with everywhere {\em positive curvature}. Somewhat surprisingly, {\em unbounded} open convex sets $C$ do not misbehave, they satisfy the conjecture: every $3$-fold covering of any region of the plane by translates of such a set $C$ splits into two coverings. To establish this result, we prove a general coloring theorem for hypergraphs of a special type: {\em shift-chains}. We also show that there is a constant $c>0$ such that, for any positive integer $m$, every $m$-fold covering of a region with unit disks splits into two coverings, provided that every point is covered by {\em at most} $c2^{m/2}$ sets

Pushing Lines Helps: Efficient Universal Centralised Transformations for Programmable Matter

In this paper, we study a discrete system of entities residing on a two-dimensional square grid. Each entity is modelled as a node occupying a distinct cell of the grid. The set of all $n$ nodes forms initially a connected shape $A$. Entities are equipped with a linear-strength pushing mechanism that can push a whole line of entities, from 1 to $n$, in parallel in a single time-step. A target connected shape $B$ is also provided and the goal is to \emph{transform} $A$ into $B$ via a sequence of line movements. Existing models based on local movement of individual nodes, such as rotating or sliding a single node, can be shown to be special cases of the present model, therefore their (inefficient, $\Theta(n^2)$) \emph{universal transformations} carry over. Our main goal is to investigate whether the parallelism inherent in this new type of movement can be exploited for efficient, i.e., sub-quadratic worst-case, transformations. As a first step towards this, we restrict attention solely to centralised transformations and leave the distributed case as a direction for future research. Our results are positive. By focusing on the apparently hard instance of transforming a diagonal $A$ into a straight line $B$, we first obtain transformations of time $O(n\sqrt{n})$ without and with preserving the connectivity of the shape throughout the transformation. Then, we further improve by providing two $O(n\log n)$-time transformations for this problem. By building upon these ideas, we first manage to develop an $O(n\sqrt{n})$-time universal transformation. Our main result is then an $O(n \log n)$-time universal transformation. We leave as an interesting open problem a suspected $\Omega(n\log n)$-time lower bound.Comment: 40 pages, 27 figure

Acute intestinal failure: international multicenter point-of-prevalence study

Background & aims: Intestinal failure (IF) is defined from a requirement or intravenous supplementation due to failing capacity to absorb nutrients and fluids. Acute IF is an acute, potentially reversible form of IF. We aimed to identify the prevalence, underlying causes and outcomes of acute IF. Methods: This point-of-prevalence study included all adult patients hospitalized in acute care hospitals and receiving parenteral nutrition (PN) on a study day. The reason for PN and the mechanism of IF (if present) were documented by local investigators and reviewed by an expert panel. Results: Twenty-three hospitals (19 university, 4 regional) with a total capacity of 16,356 acute care beds and 1237 intensive care unit (ICU) beds participated in this study. On the study day, 338 patients received PN (21 patients/1000 acute care beds) and 206 (13/1000) were categorized as acute IF. The categorization of reason for PN was revised in 64 cases (18.9% of total) in consensus between the expert panel and investigators. Hospital mortality of all study patients was 21.5%; the median hospital stay was 36 days. Patients with acute IF had a hospital mortality of 20.5% and median hospital stay of 38 days (P > 0.05 for both outcomes). Disordered gut motility (e.g. ileus) was the most common mechanism of acute IF, and 71.5% of patients with acute IF had undergone abdominal surgery. Duration of PN of ≥42 days was identified as being the best cut-off predicting hospital mortality within 90 days. PN ≥ 42 days, age, sepsis and ICU admission were independently associated with 90-day hospital mortality. Conclusions: Around 2% of adult patients in acute care hospitals received PN, 60% of them due to acute IF. High 90-day hospital mortality and long hospital stay were observed in patients receiving PN, whereas presence of acute IF did not additionally influence these outcomes. Duration of PN was associated with increased 90-day hospital mortality

Coloring geometric range spaces

SCOPUS: cp.kinfo:eu-repo/semantics/publishe