An Optimal Algorithm for Reconstructing Point Set Order Types from Radial Orderings

Abstract. Given a set P of n labeled points in the plane, the radial system of P describes, for each p ∈ P , the radial ordering of the other points around p. This notion is related to the order type of P , which describes the orientation (clockwise or counterclockwise) of every ordered triple of P . Given only the order type of P , it is easy to reconstruct the radial system of P , but the converse is not true. Aichholzer et al. (Reconstructing Point Set Order Types from Radial Orderings, in Proc. ISAAC 2014) defined T (R) to be the set of order types with radial system R and showed that sometimes |T (R)| = n − 1. They give polynomial-time algorithms to compute T (R) when only given R. We describe an optimal O(n 2 ) time algorithm for computing T (R). The algorithm constructs the convex hulls of all possible point sets with the given radial system, after which sidedness queries on point triples can be answered in constant time. This set of convex hulls can be found in O(n) time. Our results generalize to abstract order types

The Complexity of Order Type Isomorphism

The order type of a point set in $R^d$ maps each $(d{+}1)$-tuple of points to its orientation (e.g., clockwise or counterclockwise in $R^2$). Two point sets $X$ and $Y$ have the same order type if there exists a mapping $f$ from $X$ to $Y$ for which every $(d{+}1)$-tuple $(a_1,a_2,\ldots,a_{d+1})$ of $X$ and the corresponding tuple $(f(a_1),f(a_2),\ldots,f(a_{d+1}))$ in $Y$ have the same orientation. In this paper we investigate the complexity of determining whether two point sets have the same order type. We provide an $O(n^d)$ algorithm for this task, thereby improving upon the $O(n^{\lfloor{3d/2}\rfloor})$ algorithm of Goodman and Pollack (1983). The algorithm uses only order type queries and also works for abstract order types (or acyclic oriented matroids). Our algorithm is optimal, both in the abstract setting and for realizable points sets if the algorithm only uses order type queries.Comment: Preliminary version of paper to appear at ACM-SIAM Symposium on Discrete Algorithms (SODA14

Induced Ramsey-type results and binary predicates for point sets

Let $k$ and $p$ be positive integers and let $Q$ be a finite point set in general position in the plane. We say that $Q$ is $(k,p)$-Ramsey if there is a finite point set $P$ such that for every $k$-coloring $c$ of $\binom{P}{p}$ there is a subset $Q'$ of $P$ such that $Q'$ and $Q$ have the same order type and $\binom{Q'}{p}$ is monochromatic in $c$. Ne\v{s}et\v{r}il and Valtr proved that for every $k \in \mathbb{N}$, all point sets are $(k,1)$-Ramsey. They also proved that for every $k \ge 2$ and $p \ge 2$, there are point sets that are not $(k,p)$-Ramsey. As our main result, we introduce a new family of $(k,2)$-Ramsey point sets, extending a result of Ne\v{s}et\v{r}il and Valtr. We then use this new result to show that for every $k$ there is a point set $P$ such that no function $\Gamma$ that maps ordered pairs of distinct points from $P$ to a set of size $k$ can satisfy the following "local consistency" property: if $\Gamma$ attains the same values on two ordered triples of points from $P$, then these triples have the same orientation. Intuitively, this implies that there cannot be such a function that is defined locally and determines the orientation of point triples.Comment: 22 pages, 3 figures, final version, minor correction

Reordering for Improved Constrained Reconstruction from Undersampled k-Space Data

Recently, there has been a significant interest in applying reconstruction techniques, like constrained reconstruction or compressed sampling methods, to undersampled k-space data in MRI. Here, we propose a novel reordering technique to improve these types of reconstruction methods. In this technique, the intensities of the signal estimate are reordered according to a preprocessing step when applying the constraints on the estimated solution within the iterative reconstruction. The ordering of the intensities is such that it makes the original artifact-free signal monotonic and thus minimizes the finite differences norm if the correct image is estimated; this ordering can be estimated based on the undersampled measured data. Theory and example applications of the method for accelerating myocardial perfusion imaging with respiratory motion and brain diffusion tensor imaging are presented

Modave lectures on bulk reconstruction in AdS/CFT

These lecture notes are based on a series of lectures given at the XIII Modave summer school in mathematical physics. We review the construction due to Hamilton, Kabat, Lifschytz and Lowe for reconstructing local bulk operators from CFT operators in the context of AdS/CFT and show how to recover bulk correlation functions from this definition. Building on the work of these authors, it has been noted that the bulk displays quantum error correcting properties. We will discuss tensor network toy models to exemplify these remarkable features. We will discuss the role of gauge invariance and of diffeomorphism symmetry in the reconstruction of bulk operators. Lastly, we provide another method of bulk reconstruction specified to AdS$_3$/CFT$_2$ in which bulk operators create cross-cap states in the CFT.Comment: 35 pages, 8 figures, lecture notes, v4: a few minor improvements upon the published proceedings version (version 3 of these lecture notes in arXiv) have been implemente

Following the light:Novel event reconstruction techniques for neutrino oscillation analyses in KM3NeT/ORCA

Neutrinos are tiny, subatomic particles which currently present some outstanding questions in the field of particle physics. Though neutrino oscillations are now an understood phenomenon, efforts are still underway to measure the neutrino oscillation parameters even more precisely. Furthermore, the ordering of the three neutrino mass states relative to one another - the neutrino mass ordering - is still unknown. The KM3NeT/ORCA detector is currently being built in the Mediterranean Sea to address such questions. This infrastructure surrounds huge volumes of seawater with photodetectors, bypassing the tiny interaction cross section of these particles, and detecting the Cherenkov radiation of products of neutrino interactions in the water. In this thesis, the software used to simulate atmospheric muons in the detector using parametric formulae is tuned to KM3NeT/ORCA data, resulting in an improved simulation of the atmospheric muons, which form the main background for neutrino analyses. A novel neutrino event reconstruction algorithm is developed and explored in this thesis, aiming to reconstruct neutrino events with both a track-like and particle shower-like component. The estimate of the reconstructed neutrino energy is improved upon with this technique, as well as directly reconstructing the fractional energy transfer to the hadronic shower component of the interaction. This reconstruction technique also shows the potential for identifying different neutrino interaction channels. The improved energy estimate and the potential to identify the interaction channel pave the way for future analyses, leading to an improved measurement of the neutrino oscillation parameters and determination of the yet-unknown neutrino mass ordering