Faster Fr\'echet Distance Approximation through Truncated Smoothing

The Fr\'echet distance is a popular distance measure for curves. Computing the Fr\'echet distance between two polygonal curves of $n$ vertices takes roughly quadratic time, and conditional lower bounds suggest that even approximating to within a factor $3$ cannot be done in strongly-subquadratic time, even in one dimension. The current best approximation algorithms present trade-offs between approximation quality and running time. Recently, van der Horst $\textit{et al.}$ (SODA, 2023) presented an $O((n^2 / \alpha) \log^3 n)$ time $\alpha$-approximate algorithm for curves in arbitrary dimensions, for any $\alpha \in [1, n]$. Our main contribution is an approximation algorithm for curves in one dimension, with a significantly faster running time of $O(n \log^3 n + (n^2 / \alpha^3) \log^2 n \log \log n)$. Additionally, we give an algorithm for curves in arbitrary dimensions that improves upon the state-of-the-art running time by a logarithmic factor, to $O((n^2 / \alpha) \log^2 n)$. Both of our algorithms rely on a linear-time simplification procedure that in one dimension reduces the complexity of the reachable free space to $O(n^2 / \alpha)$ without making sacrifices in the asymptotic approximation factor.Comment: 27 pages, 11 figure

Chromatic k-Nearest Neighbor Queries

Let $P$ be a set of $n$ colored points. We develop efficient data structures that store $P$ and can answer chromatic $k$-nearest neighbor ($k$-NN) queries. Such a query consists of a query point $q$ and a number $k$, and asks for the color that appears most frequently among the $k$ points in $P$ closest to $q$. Answering such queries efficiently is the key to obtain fast $k$-NN classifiers. Our main aim is to obtain query times that are independent of $k$ while using near-linear space. We show that this is possible using a combination of two data structures. The first data structure allow us to compute a region containing exactly the $k$-nearest neighbors of a query point $q$, and the second data structure can then report the most frequent color in such a region. This leads to linear space data structures with query times of $O(n^{1 / 2} \log n)$ for points in $\mathbb{R}^1$, and with query times varying between $O(n^{2/3}\log^{2/3} n)$ and $O(n^{5/6} {\rm polylog} n)$, depending on the distance measure used, for points in $\mathbb{R}^2$. Since these query times are still fairly large we also consider approximations. If we are allowed to report a color that appears at least $(1-\varepsilon)f^*$ times, where $f^*$ is the frequency of the most frequent color, we obtain a query time of $O(\log n + \log\log_{\frac{1}{1-\varepsilon}} n)$ in $\mathbb{R}^1$ and expected query times ranging between $\tilde{O}(n^{1/2}\varepsilon^{-3/2})$ and $\tilde{O}(n^{1/2}\varepsilon^{-5/2})$ in $\mathbb{R}^2$ using near-linear space (ignoring polylogarithmic factors).Comment: 37 pages, 9 figure

Computing Minimum Complexity 1D Curve Simplifications under the Fréchet Distance

We consider the problem of simplifying curves under the Fréchet distance. Let P be a curve and ε ≥ 0 be a distance threshold. An ε-simplification is a curve within Fréchet distance ε of P . We consider ε-simplifications of minimum complexity (i.e. minimum number of vertices). Parameterized by ε, we define a continuous family of minimum complexity ε-simplifications P ε of a curve P inone dimension. We present a data structure that after linear preprocessing time can report the ε-simplification in linear output-sensitive time. Moreover, for k ≥ 1, we show how this data structure can be used to report a simplification P ε with at most k vertices that is closest to P in O(k) time

Simply Realising an Imprecise Polyline is NP-hard

We consider the problem of deciding, given a sequence of regions, if there is a choice of points, one for each region, such that the induced polyline is simple or weakly simple, meaning that it can touch but not cross itself. Specifically, we consider the case where each region is a translate of the same shape. We show that the problem is NP-hard when the shape is a unit-disk or unit-square. We argue that the problem is is NP-complete when the shape is a vertical unit-segment

Faster and Deterministic Subtrajectory Clustering

We study the subtrajectory clustering problem. Given a trajectory $T$, the goal is to identify a set of subtrajectories such that each point on $T$ is included in at least one subtrajectory, and subsequently group these subtrajectories together based on similarity under the Fréchet distance. We wish to minimize the set of groups. This problem was shown to be NP-complete by Akitaya, Brüning, Chambers, and Driemel (2021), and the focus has mainly been on approximation algorithms. We study a restricted variant, where we may only pick subtrajectories that start and end at vertices of $T$, and give an approximation algorithm that significantly improves previous algorithms in both running time and space, whilst being deterministic

Robust Bichromatic Classification using Two Lines

Given two sets $\mathit{R}$ and $\mathit{B}$ of at most $\mathit{n}$ points in the plane, we present efficient algorithms to find a two-line linear classifier that best separates the "red" points in $\mathit{R}$ from the "blue" points in $B$ and is robust to outliers. More precisely, we find a region $\mathit{W}_\mathit{B}$ bounded by two lines, so either a halfplane, strip, wedge, or double wedge, containing (most of) the blue points $\mathit{B}$, and few red points. Our running times vary between optimal $O(n\log n)$ and $O(n^4)$, depending on the type of region $\mathit{W}_\mathit{B}$ and whether we wish to minimize only red outliers, only blue outliers, or both.Comment: 19 pages, 11 figures. Updated to include new result

A Subquadratic nε-approximation for the Continuous Fréchet Distance

The Fréchet distance is a commonly used similarity measure between curves. It is known how to compute the continuous Fréchet distance between two polylines with m and n vertices in R^d in O(mn(log log n)²) time; doing so in strongly subquadratic time is a longstanding open problem. Recent conditional lower bounds suggest that it is unlikely that a strongly subquadratic algorithm exists. Moreover, it is unlikely that we can approximate the Fr ́echet distance to within a factor 3 in strongly subquadratic time, even if d = 1. The best current results establish a tradeoff between approximation quality and running time. Specifically, Colombe and Fox (SoCG, 2021) give an O(α)-approximate algorithm that runs in O((n3/α2) log n) time for any α ∈ [√n, n], assuming m = n. In this paper, we improve this result with an O(α)-approximate algorithm that runs in O((n + mn/α) log³ n) time for any α ∈ [1, n], assuming m ≤ n and constant dimension d

Variables associated with in-hospital and postdischarge outcomes after postcardiotomy extracorporeal membrane oxygenation:Netherlands Heart Registration Cohort

Objectives: Extracorporeal membrane oxygenation (ECMO) for postcardiotomy cardiogenic shock has been increasingly used without concomitant mortality reduction. This study aims to investigate determinants of in-hospital and postdischarge mortality in patients requiring postcardiotomy ECMO in the Netherlands. Methods: The Netherlands Heart Registration collects nationwide prospective data from cardiac surgery units. Adults receiving intraoperative or postoperative ECMO included in the register from January 2013 to December 2019 were studied. Survival status was established through the national Personal Records Database. Multivariable logistic regression analyses were used to investigate determinants of in-hospital (3 models) and 12-month postdischarge mortality (4 models). Each model was developed to target specific time points during a patient's clinical course. Results: Overall, 406 patients (67.2% men, median age, 66.0 years [interquartile range, 55.0-72.0 years]) were included. In-hospital mortality was 51.7%, with death occurring in a median of 5 days (interquartile range, 2-14 days) after surgery. Hospital survivors (n = 196) experienced considerable rates of pulmonary infections, respiratory failure, arrhythmias, and deep sternal wound infections during a hospitalization of median 29 days (interquartile range, 17-51 days). Older age (odds ratio [OR], 1.02; 95% CI, 1.0-1.04) and preoperative higher body mass index (OR, 1.08; 95% CI, 1.02-1.14) were associated with in-hospital death. Within 12 months after discharge, 35.1% of hospital survivors (n = 63) died. Postoperative renal failure (OR, 2.3; 95% CI, 1.6-4.9), respiratory failure (OR, 3.6; 95% CI, 1.3-9.9), and re-thoracotomy (OR, 2.9; 95% CI, 1.3-6.5) were associated with 12-month postdischarge mortality. Conclusions: In-hospital and postdischarge mortality after postcardiotomy ECMO in adults remains high in the Netherlands. ECMO support in patients with higher age and body mass index, which drive associations with higher in-hospital mortality, should be carefully considered. Further observations suggest that prevention of re-thoracotomies, renal failure, and respiratory failure are targets that may improve postdischarge outcomes