By: Emil Toftegaard Gæde, Inge Li Gørtz, Ivor van der Hoog, Christoffer Krogh, Eva Rotenberg

The convex hull of a data set $P$ is the smallest convex set that contains $P$. In this work, we present a new data structure for convex hull, that allows for efficient dynamic updates. In a dynamic convex hull implementation, the following traits are desirable: (1) algorithms for efficiently answering queries as to whether a specified point is inside or outside the hull, (2) adhering to geometric robustness, and (3) algorithmic simplicit... more

The convex hull of a data set $P$ is the smallest convex set that contains $P$. In this work, we present a new data structure for convex hull, that allows for efficient dynamic updates. In a dynamic convex hull implementation, the following traits are desirable: (1) algorithms for efficiently answering queries as to whether a specified point is inside or outside the hull, (2) adhering to geometric robustness, and (3) algorithmic simplicity.Furthermore, a specific but well-motivated type of two-dimensional data is rank-based data. Here, the input is a set of real-valued numbers $Y$ where for any number $y\in Y$ its rank is its index in $Y$'s sorted order. Each value in $Y$ can be mapped to a point $(rank, value)$ to obtain a two-dimensional point set. In this work, we give an efficient, geometrically robust, dynamic convex hull algorithm, that facilitates queries to whether a point is internal. Furthermore, our construction can be used to efficiently update the convex hull of rank-ordered data, when the real-valued point set is subject to insertions and deletions. Our improved solution is based on an algorithmic simplification of the classical convex hull data structure by Overmars and van Leeuwen~[STOC'80], combined with new algorithmic insights. Our theoretical guarantees on the update time match those of Overmars and van Leeuwen, namely $O(\log^2 |P|)$, while we allow a wider range of functionalities (including rank-based data). Our algorithmic simplification includes simplifying an 11-case check down to a 3-case check that can be written in 20 lines of easily readable C-code. We extend our solution to provide a trade-off between theoretical guarantees and the practical performance of our algorithm. We test and compare our solutions extensively on inputs that were generated randomly or adversarially, including benchmarking datasets from the literature. less

# Dynamic Dynamic Time Warping

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By: Karl Bringmann, Nick Fischer, Ivor van der Hoog, Evangelos Kipouridis, Tomasz Kociumaka, Eva Rotenberg

The Dynamic Time Warping (DTW) distance is a popular similarity measure for polygonal curves (i.e., sequences of points). It finds many theoretical and practical applications, especially for temporal data, and is known to be a robust, outlier-insensitive alternative to the Fr\'echet distance. For static curves of at most $n$ points, the DTW distance can be computed in $O(n^2)$ time in constant dimension. This tightly matches a SETH-based lo... more

The Dynamic Time Warping (DTW) distance is a popular similarity measure for polygonal curves (i.e., sequences of points). It finds many theoretical and practical applications, especially for temporal data, and is known to be a robust, outlier-insensitive alternative to the Fr\'echet distance. For static curves of at most $n$ points, the DTW distance can be computed in $O(n^2)$ time in constant dimension. This tightly matches a SETH-based lower bound, even for curves in $\mathbb{R}^1$. We study dynamic algorithms for the DTW distance. We accommodate local changes to one or both curves, such as inserting or deleting vertices and, after each operation, can report the updated DTW distance. We give such a data structure with update and query time $O(n^{1.5} \log n)$, where $n$ is the maximum length of the curves. The natural follow-up question is whether this time bound can be improved; under the aforementioned SETH-based lower bound, we could even hope for linear update time. We refute these hopes and prove that our data structure is conditionally optimal, up to subpolynomial factors. More precisely, we prove that, already for curves in $\mathbb{R}^1$, there is no dynamic algorithm to maintain the DTW distance with update and query time $O(n^{1.5 - \delta})$ for any constant~$\delta > 0$, unless the Negative-$k$-Clique Hypothesis fails. This holds even if one of the curves is fixed at all times, whereas the points of the other curve may only undergo substitutions. In fact, we give matching upper and lower bounds for various further trade-offs between update and query time, even in cases where the lengths of the curves differ. The Negative-$k$-Clique Hypothesis is a recent but well-established hypothesis from fine-grained complexity, that generalizes the famous APSP Hypothesis, and successfully led to several lower bounds. less

By: Zhimeng Gao, Sariel Har-Peled

$ \newcommand{\Re}{\mathbb{R}} \newcommand{\reals}{\mathbb{R}} \newcommand{\SetX}{\mathsf{X}} \newcommand{\rad}{r} \newcommand{\Eps}{\Mh{\mathcal{E}}} \newcommand{\p}{\Mh{p}} \newcommand{\q}{\Mh{q}} \newcommand{\Mh}[1]{#1} \newcommand{\query}{q} \newcommand{\eps}{\varepsilon} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\Polygon}{\mathsf{P}} \newcommand{\IntRange}[1]{[ #1 ]} \newcommand{\Space}{\overline{\mathsf{m}}} \newcommand{... more

$ \newcommand{\Re}{\mathbb{R}} \newcommand{\reals}{\mathbb{R}} \newcommand{\SetX}{\mathsf{X}} \newcommand{\rad}{r} \newcommand{\Eps}{\Mh{\mathcal{E}}} \newcommand{\p}{\Mh{p}} \newcommand{\q}{\Mh{q}} \newcommand{\Mh}[1]{#1} \newcommand{\query}{q} \newcommand{\eps}{\varepsilon} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\Polygon}{\mathsf{P}} \newcommand{\IntRange}[1]{[ #1 ]} \newcommand{\Space}{\overline{\mathsf{m}}} \newcommand{\pth}[2][\!]{#1\left({#2}\right)} \newcommand{\polylog}{\mathrm{polylog}} \newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\pt}{p} \newcommand{\distY}[2]{\left\| {#1} - {#2} \right\|} \newcommand{\ptq}{q} \newcommand{\pts}{s}$ For a parameter $\eps \in (0,1)$, we present a new construction of $\eps$-locality-sensitive orderings (<em>LSO</em>s) in $\Re^d$ of size $M = O(\Eps^{d-1} \log \Eps)$, where $\Eps = 1/\eps$. This improves over previous work by a factor of $\Eps$, and is optimal up to a factor of $\log \Eps$. Such a set of LSOs has the property that for any two points, $\p, \q \in [0,1]^d$, there exist an order in the set such that all the points between $\p$ and $\q$ in the order are $\eps$-close to either $\p$ or $\q$. The existence of such LSOs is a fundamental property of low dimensional Euclidean space, conceptually similar to the existence of well-separated pairs decomposition, so the question of how to compute (near) optimal construction of LSOs is quite natural. As a consequence we get a flotilla of improved dynamic geometric algorithms, such as maintaining bichromatic closest pair, and spanners, among others. In particular, for geometric dynamic spanners the new result matches (up to the aforementioned $\log \Eps$ factor) the lower bound, Thus offering a near-optimal simple dynamic data-structure for maintaining spanners under insertions and deletions. less

By: Elizabeth Munch

The Euler characteristic transform (ECT) is a simple to define yet powerful representation of shape. The idea is to encode an embedded shape using sub-level sets of a a function defined based on a given direction, and then returning the Euler characteristics of these sublevel sets. Because the ECT has been shown to be injective on the space of embedded simplicial complexes, it has been used for applications spanning a range of disciplines, ... more

The Euler characteristic transform (ECT) is a simple to define yet powerful representation of shape. The idea is to encode an embedded shape using sub-level sets of a a function defined based on a given direction, and then returning the Euler characteristics of these sublevel sets. Because the ECT has been shown to be injective on the space of embedded simplicial complexes, it has been used for applications spanning a range of disciplines, including plant morphology and protein structural analysis. In this survey article, we present a comprehensive overview of the Euler characteristic transform, highlighting the main idea on a simple leaf example, and surveying its its key concepts, theoretical foundations, and available applications. less

By: Sergio Salinas-Fernández, Nancy Hitschfeld-Kahler

Polylla is a polygonal mesh algorithm that generates meshes with arbitrarily shaped polygons using the concept of terminal-edge regions. Until now, Polylla has been limited to 2D meshes, but in this work, we extend Polylla to 3D volumetric meshes. We present two versions of Polylla 3D. The first version generates terminal-edge regions, converts them into polyhedra, and repairs polyhedra that are joined by only an edge. This version differs ... more

Polylla is a polygonal mesh algorithm that generates meshes with arbitrarily shaped polygons using the concept of terminal-edge regions. Until now, Polylla has been limited to 2D meshes, but in this work, we extend Polylla to 3D volumetric meshes. We present two versions of Polylla 3D. The first version generates terminal-edge regions, converts them into polyhedra, and repairs polyhedra that are joined by only an edge. This version differs from the original Polylla algorithm in that it does not have the same phases as the 2D version. In the second version, we define two new concepts: longest-face propagation path and terminal-face regions. We use these concepts to create an almost direct extension of the 2D Polylla mesh with the same three phases: label phase, traversal phase, and repair phase. less

By: Sergio Cabello, Siu-Wing Cheng, Otfried Cheong, Christian Knauer

Let $P$ be a set of at most $n$ points and let $R$ be a set of at most $n$ geometric ranges, such as for example disks or rectangles, where each $p \in P$ has an associated supply $s_{p} > 0$, and each $r \in R$ has an associated demand $d_{r} > 0$. An assignment is a set $\mathcal{A}$ of ordered triples $(p,r,a_{pr}) \in P \times R \times \mathbb{R}_{>0}$ such that $p \in r$. We show how to compute a maximum assignment that satisfies the c... more

Let $P$ be a set of at most $n$ points and let $R$ be a set of at most $n$ geometric ranges, such as for example disks or rectangles, where each $p \in P$ has an associated supply $s_{p} > 0$, and each $r \in R$ has an associated demand $d_{r} > 0$. An assignment is a set $\mathcal{A}$ of ordered triples $(p,r,a_{pr}) \in P \times R \times \mathbb{R}_{>0}$ such that $p \in r$. We show how to compute a maximum assignment that satisfies the constraints given by the supplies and demands. Using our techniques, we can also solve minimum bottleneck problems, such as computing a perfect matching between a set of $n$ red points~$P$ and $n$ blue points $Q$ that minimizes the length of the longest edge. For the $L_\infty$-metric, we can do this in time $O(n^{1+\varepsilon})$ in any fixed dimension, for the $L_2$-metric in the plane in time $O(n^{4/3 + \varepsilon})$, for any $\varepsilon > 0$. less

By: Bernd Gärtner, Vishwas Kalani, Meghana M. Reddy, Wouter Meulemans, Bettina Speckmann, Miloš Stojaković

In information visualization, the position of symbols often encodes associated data values. When visualizing data elements with both a numerical and a categorical dimension, positioning in the categorical axis admits some flexibility. This flexibility can be exploited to reduce symbol overlap, and thereby increase legibility. In this paper we initialize the algorithmic study of optimizing symbol legibility via a limited displacement of the ... more

In information visualization, the position of symbols often encodes associated data values. When visualizing data elements with both a numerical and a categorical dimension, positioning in the categorical axis admits some flexibility. This flexibility can be exploited to reduce symbol overlap, and thereby increase legibility. In this paper we initialize the algorithmic study of optimizing symbol legibility via a limited displacement of the symbols. Specifically, we consider unit square symbols that need to be placed at specified $y$-coordinates. We optimize the drawing order of the symbols as well as their $x$-displacement, constrained within a rectangular container, to maximize the minimum visible perimeter over all squares. If the container has width and height at most $2$, there is a point that stabs all squares. In this case, we prove that a staircase layout is arbitrarily close to optimality and can be computed in $O(n\log n)$ time. If the width is at most $2$, there is a vertical line that stabs all squares, and in this case, we give a 2-approximation algorithm that runs in $O(n\log n)$ time. As a minimum visible perimeter of $2$ is always trivially achievable, we measure this approximation with respect to the visible perimeter exceeding $2$. We show that, despite its simplicity, the algorithm gives asymptotically optimal results for certain instances. less