On k-enclosing objects in a coloured point set

We introduce the exact coloured k -enclosing object problem: given a set P of n points in R 2 , each of which has an associated colour in f 1 ;:::;t g , and a vec- tor c = ( c 1 ;:::;c t ), where c i 2 Z + for each 1 i t , nd a region that contains exactly c i points of P of colour i for each i . We examine the problems of nd- ing exact coloured k -enclosing axis-aligned rectangles, squares, discs, and two-sided dominating regions in a t -coloured point setPostprint (published version

Output-Sensitive Tools for Range Searching in Higher Dimensions

Let $P$ be a set of $n$ points in ${\mathbb R}^{d}$. A point $p \in P$ is $k$\emph{-shallow} if it lies in a halfspace which contains at most $k$ points of $P$ (including $p$). We show that if all points of $P$ are $k$-shallow, then $P$ can be partitioned into $\Theta(n/k)$ subsets, so that any hyperplane crosses at most $O((n/k)^{1-1/(d-1)} \log^{2/(d-1)}(n/k))$ subsets. Given such a partition, we can apply the standard construction of a spanning tree with small crossing number within each subset, to obtain a spanning tree for the point set $P$, with crossing number $O(n^{1-1/(d-1)}k^{1/d(d-1)} \log^{2/(d-1)}(n/k))$. This allows us to extend the construction of Har-Peled and Sharir \cite{hs11} to three and higher dimensions, to obtain, for any set of $n$ points in ${\mathbb R}^{d}$ (without the shallowness assumption), a spanning tree $T$ with {\em small relative crossing number}. That is, any hyperplane which contains $w \leq n/2$ points of $P$ on one side, crosses $O(n^{1-1/(d-1)}w^{1/d(d-1)} \log^{2/(d-1)}(n/w))$ edges of $T$. Using a similar mechanism, we also obtain a data structure for halfspace range counting, which uses $O(n \log \log n)$ space (and somewhat higher preprocessing cost), and answers a query in time $O(n^{1-1/(d-1)}k^{1/d(d-1)} (\log (n/k))^{O(1)})$, where $k$ is the output size

Data-Centric Distrust Quantification for Responsible AI: When Data-driven Outcomes Are Not Reliable

At the same time that AI and machine learning are becoming central to human life, their potential harms become more vivid. In the presence of such drawbacks, a critical question one needs to address before using these data-driven technologies to make a decision is whether to trust their outcomes. Aligned with recent efforts on data-centric AI, this paper proposes a novel approach to address the trust question through the lens of data, by associating data sets with distrust quantification that specify their scope of use for predicting future query points. The distrust values raise warning signals when a prediction based on a dataset is questionable and are valuable alongside other techniques for trustworthy AI. We propose novel algorithms for computing the distrust values in the neighborhood of a query point efficiently and effectively. Learning the necessary components of the measures from the data itself, our sub-linear algorithms scale to very large and multi-dimensional settings. Besides demonstrating the efficiency of our algorithms, our extensive experiments reflect a consistent correlation between distrust values and model performance. This underscores the message that when the distrust value of a query point is high, the prediction outcome should be discarded or at least not considered for critical decisions

Complete Classification and Efficient Determination of Arrangements Formed by Two Ellipsoids

International audienceArrangements of geometric objects refer to the spatial partitions formed by the objects and they serve as an underlining structure of motion design, analysis and planning in CAD/CAM, robotics, molecular modeling, manufacturing and computer-assisted radio-surgery. Arrangements are especially useful to collision detection, which is a key task in various applications such as computer animation , virtual reality, computer games, robotics, CAD/CAM and computational physics. Ellipsoids are commonly used as bounding volumes in approximating complex geometric objects in collision detection. In this paper we present an in-depth study on the arrangements formed by two ellipsoids. Specifically, we present a classification of these arrangements and propose an efficient algorithm for determining the arrangement formed by any particular pair of ellipsoids. A stratification diagram is also established to show the connections among all the arrangements formed by two ellipsoids. Our results for the first time elucidate all possible relative positions between two arbitrary ellipsoids and provides an efficient and robust algorithm for determining the relative position of any two given ellipsoids, therefore providing the necessary foundation for developing practical and trustworthy methods for processing ellipsoids for collision analysis or simulation in various applications