We study the following range searching problem in high-dimensional Euclidean spaces: given a finite set P ⊂ ℝ^d, where each p ∈ P is assigned a weight w_p, and radius r > 0, we need to preprocess P into a data structure such that when a new query point q ∈ ℝ^d arrives, the data structure reports the cumulative weight of points of P within Euclidean distance r from q. Solving the problem exactly seems to require space usage that is exponential to the dimension, a phenomenon known as the curse of dimensionality. Thus, we focus on approximate solutions where points up to (1+ε)r away from q may be taken into account, where ε > 0 is an input parameter known during preprocessing. We build a data structure with near-linear space usage, and query time in n^{1-Θ(ε⁴/log(1/ε))}+t_q^ϱ⋅n^{1-ϱ}, for some ϱ = Θ(ε²), where t_q is the number of points of P in the ambiguity zone, i.e., at distance between r and (1+ε)r from the query q. To the best of our knowledge, this is the first data structure with efficient space usage (subquadratic or near-linear for any ε > 0) and query time that remains sublinear for any sublinear t_q. We supplement our worst-case bounds with a query-driven preprocessing algorithm to build data structures that are well-adapted to the query distribution.

In this paper, we introduce a new data analysis problem that aims to decompose a set of univariate time series into a small set of k base curves of length at most l such that the sum of Fréchet distances of the time series to a "Fréchet combination" of the base curves is minimized. Here, a Fréchet combination allows to combine individually scaled base curves using a k-dimensional traversal. We call the problem of finding a set of optimal base curves the Fréchet decomposition problem and we consider two variants: (a) the base curves can be arbitrary curves of bounded length and (b) the curves come from a given finite set of candidate curves. We think of the Fréchet decomposition problem as a Fréchet variant of principal component analysis. For the case of a single base curve we develop a (1+ε)-approximation algorithm for the Fréchet decomposition problem. Additionally we give an exact algorithm for the projection distance problem that asks to compute the distance of one given time series to a given set of k base curves. This allows us to design an exact algorithm for the Fréchet decomposition problem for general k when curves come from a fixed candidate set.

In the metric k-median problem we are given a finite metric space (X∪ Y, 𝐝) and the objective is to compute a set of k centers C ⊆ Y that minimizes ∑_{p ∈ X} min_{c ∈ C} 𝐝(p,c). In general metric spaces, the best polynomial time algorithm, which is due to Cohen-Addad, Grandoni, Lee, Schwiegelshohn, and Svensson [Vincent Cohen-Addad et al., 2025], computes a (2+ε)-approximation for arbitrary constant ε > 0. However, if the metric space has bounded doubling dimension, a near linear time (1+ε)-approximation algorithm is known due to the work of Cohen-Addad, Feldmann, and Saulpic [Vincent Cohen{-}Addad et al., 2021]. In this paper, we show that the (1+ε)-approximation algorithm can be generalized to the case when either X or Y has bounded doubling dimension (but the other set not). The case when X has bounded doubling dimension is motivated by the assumption that even though X is part of a high-dimensional space, it may be that it is close to a low-dimensional structure. The case when Y has bounded doubling dimension is perhaps more natural. It is motivated by specific clustering problems where the centers are low-dimensional. Specifically, our work in this setting implies the first near linear time approximation algorithm for the (k,𝓁)-median problem under discrete Fréchet distance when 𝓁 is constant. The latter problem is a version of the k-median problem under Fréchet distance when the input consists of time series of z reals and where the centers are time series of 𝓁 reals [Anne Driemel et al., 2016]. Previously, for this problem no (1+ε)-approximation algorithm with running time polynomial in k was known. We also introduce a novel complexity reduction for time series of real values that leads to a similar result for the case of discrete Fréchet distance. In order to solve the case when Y has a bounded doubling dimension, we introduce a form of dimension reduction that replaces points from X by sets of points in Y. To solve the case when X has a bounded doubling dimension, we generalize Talwar’s decomposition [Kunal Talwar, 2004] of doubling metrics to our setting. The running time of our algorithms is 2^{2^t} Õ(n+m) where t = O(ddim log ddim/ε) and where ddim is the doubling dimension of X (resp. Y). The results also extend to the metric (uncapacitated) facility location problem. We believe that our techniques are likely applicable to other problems.

We study efficient algorithms for the Euclidean $k$-Center problem, focusing on the regime of large $k$. We take the approach of data reduction by considering $\alpha$-coreset, which is a small subset $S$ of the dataset $P$ such that any $\beta$-approximation on $S$ is an $(\alpha + \beta)$-approximation on $P$. We give efficient algorithms to construct coresets whose size is $k \cdot o(n)$, which immediately speeds up existing approximation algorithms. Notably, we obtain a near-linear time $O(1)$-approximation when $k = n^c$ for any $0 < c < 1$. We validate the performance of our coresets on real-world datasets with large $k$, and we observe that the coreset speeds up the well-known Gonzalez algorithm by up to $4$ times, while still achieving similar clustering cost. Technically, one of our coreset results is based on a new efficient construction of consistent hashing with competitive parameters. This general tool may be of independent interest for algorithm design in high dimensional Euclidean spaces.

Modern time series analysis requires the ability to handle datasets that are inherently high-dimensional; examples include applications in climatology, where measurements from numerous sensors must be taken into account, or inventory tracking of large shops, where the dimension is defined by the number of tracked items. The standard way to mitigate computational issues arising from the high dimensionality of the data is by applying some dimension reduction technique that preserves the structural properties of the ambient space. The dissimilarity between two time series is often measured by "discrete" notions of distance, e.g. the dynamic time warping or the discrete Fréchet distance. Since all these distance functions are computed directly on the points of a time series, they are sensitive to different sampling rates or gaps. The continuous Fréchet distance offers a popular alternative which aims to alleviate this by taking into account all points on the polygonal curve obtained by linearly interpolating between any two consecutive points in a sequence. We study the ability of random projections à la Johnson and Lindenstrauss to preserve the continuous Fréchet distance of polygonal curves by effectively reducing the dimension. In particular, we show that one can reduce the dimension to O(ε^{-2} log N), where N is the total number of input points while preserving the continuous Fréchet distance between any two determined polygonal curves within a factor of 1± ε. We conclude with applications on clustering.

With the lowering costs of sensors, high-volume and high-velocity data are increasingly being generated and analyzed, especially in IoT domains like energy and smart homes. Consequently, applications that require accurate short-term forecasts and predictions are also steadily increasing. In this paper, we provide an overview of a novel end-to-end platform that provides efficient ingestion, compression, transfer, query processing, and machine learning-based analytics for high-frequency and high-volume time series from IoT. The performance of the platform is evaluated using real-world dataset from RES installations. The results show the importance of high-frequency analytics and the surprisingly positive impact of error bounded lossy compression on machine learning in the form of AutoML. For example, when detecting yaw misalignments in wind turbines, an improvement of 9% in accuracy was observed for AutoML models on lossy compressed data compared to the current industry standard of 10-minute aggregated data. Thus, these small-scale experiments show the potential of the platform, and larger pilots are planned.

We study the c-approximate near neighbor problem under the continuous Fréchet distance: Given a set of n polygonal curves with m vertices, a radius δ > 0, and a parameter k ≤ m, we want to preprocess the curves into a data structure that, given a query curve q with k vertices, either returns an input curve with Fréchet distance at most c · δ to q, or returns that there exists no input curve with Fréchet distance at most δ to q. We focus on the case where the input and the queries are one-dimensional polygonal curves—also called time series—and we give a comprehensive analysis for this case. We obtain new upper bounds that provide different tradeoffs between approximation factor, preprocessing time, and query time. Our data structures improve upon the state of the art in several ways. We show that for any 0 < ∊ ≤ 1 an approximation factor of (1 + ∊) can be achieved within the same asymptotic time bounds as the previously best result for (2 + ∊). Moreover, we show that an approximation factor of (2 + ∊) can be obtained by using preprocessing time and space O(nm), which is linear in the input size, and query time in , where the previously best result used preprocessing time in and query time in O(1)k. We complement our upper bounds with matching conditional lower bounds based on the Orthogonal Vectors Hypothesis. Interestingly, some of our lower bounds already hold for any super-constant value of k. This is achieved by proving hardness of a one-sided sparse version of the Orthogonal Vectors problem as an intermediate problem, which we believe to be of independent interest.

Abstract The Vapnik–Chervonenkis dimension provides a notion of complexity for systems of sets. If the VC dimension is small, then knowing this can drastically simplify fundamental computational tasks such as classification, range counting, and density estimation through the use of sampling bounds. We analyze set systems where the ground set X is a set of polygonal curves in $$\mathbb {R}^d$$ R d and the sets $$\mathcal {R}$$ R are metric balls defined by curve similarity metrics, such as the Fréchet distance and the Hausdorff distance, as well as their discrete counterparts. We derive upper and lower bounds on the VC dimension that imply useful sampling bounds in the setting that the number of curves is large, but the complexity of the individual curves is small. Our upper and lower bounds are either near-quadratic or near-linear in the complexity of the curves that define the ranges and they are logarithmic in the complexity of the curves that define the ground set.

Randomized dimensionality reduction has been recognized as one of the fundamental techniques in handling high-dimensional data. Starting with the celebrated Johnson-Lindenstrauss Lemma, such reductions have been studied in depth for the Euclidean (l_2) metric, but much less for the Manhattan (l_1) metric. Our primary motivation is the approximate nearest neighbor problem in l_1. We exploit its reduction to the decision-with-witness version, called approximate near neighbor, which incurs a roughly logarithmic overhead. In 2007, Indyk and Naor, in the context of approximate nearest neighbors, introduced the notion of nearest neighbor-preserving embeddings. These are randomized embeddings between two metric spaces with guaranteed bounded distortion only for the distances between a query point and a point set. Such embeddings are known to exist for both l_2 and l_1 metrics, as well as for doubling subsets of l_2. The case that remained open were doubling subsets of l_1. In this paper, we propose a dimension reduction by means of a near neighbor-preserving embedding for doubling subsets of l_1. Our approach is to represent the pointset with a carefully chosen covering set, then randomly project the latter. We study two types of covering sets: c-approximate r-nets and randomly shifted grids, and we discuss the tradeoff between them in terms of preprocessing time and target dimension. We employ Cauchy variables: certain concentration bounds derived should be of independent interest.

The problem of Approximate Nearest Neighbor (ANN) search is fundamental in computer science and has benefited from significant progress in the past couple of decades. However, most work has been devoted to pointsets whereas complex shapes have not been sufficiently treated. Here, we focus on distance functions between discretized curves in Euclidean space: they appear in a wide range of applications, from road segments to time-series in general dimension. For $\ell_p$-products of Euclidean metrics, for any $p$, we design simple and efficient data structures for ANN, based on randomized projections, which are of independent interest. They serve to solve proximity problems under a notion of distance between discretized curves, which generalizes both discrete Fréchet and Dynamic Time Warping distances. These are the most popular and practical approaches to comparing such curves. We offer the first data structures and query algorithms for ANN with arbitrarily good approximation factor, at the expense of increasing space usage and preprocessing time over existing methods. Query time complexity is comparable or significantly improved by our algorithms, our algorithm is especially efficient when the length of the curves is bounded.

Approximate nearest neighbor search (ϵ-ANN) in high dimensions has been mainly addressed by Locality Sensitive Hashing (LSH), which has complexity with polynomial dependence in dimension, sublinear query time, but subquadratic space requirement. We introduce a new “low-quality” embedding for metric spaces requiring that, for some query, there exists an approximate nearest neighbor among the pre-images of its k > 1 approximate nearest neighbors in the target space. In Euclidean spaces, we employ random projections to a dimension inversely proportional to k . Our approach extends to the decision problem with witness of checking whether there exists an approximate near neighbor; this also implies a solution for ϵ-ANN. After dimension reduction, we store points in a uniform grid of side length ϵ /√ d ′ , where d ′ is the reduced dimension. Given a query, we explore cells intersecting the unit ball around the query. This data structure requires linear space and query time in O ( d n ρ ), ρ ≈ 1-ϵ 2 i>/log(1ϵ), where n denotes input cardinality and d space dimension. Bounds are improved for doubling subsets via r -nets. We present our implementation for ϵ-ANN in C++ and experiments for d ≤ 960, n ≤ 10 6 , using synthetic and real datasets, which confirm the theoretical analysis and, typically, yield better practical performance. We compare to FALCONN, the state-of-the-art implementation of multi-probe LSH: our prototype software is essentially comparable in terms of preprocessing, query time, and storage usage.

The construction of r-nets offers a powerful tool in computational and metric geometry. We focus on high- dimensional spaces and present a new randomized algorithm which efficiently computes approximate r-nets with respect to Euclidean distance. For any fixed ∊ > 0, the approximation factor is 1 + ∊ and the complexity is polynomial in the dimension and subquadratic in the number of points. The algorithm succeeds with high probability. Specifically, we improve upon the best previously known (LSH- based) construction of Eppstein et al. [EHS15] in terms of complexity, by reducing the dependence on ∊, provided that ∊ is sufficiently small. Our method does not require LSH but, instead, follows Valiant's [Val15] approach in designing a sequence of reductions of our problem to other problems in different spaces, under Euclidean distance or inner product, for which r-nets are computed efficiently and the error can be controlled. Our result immediately implies efficient solutions to a number of geometric problems in high dimension, such as finding the (1 + ∊)-approximate kth nearest neighbor distance in time subquadratic in the size of the input.

The approximate nearest neighbor problem (epsilon-ANN) in Euclidean settings is a fundamental question, which has been addressed by two main approaches: Data-dependent space partitioning techniques perform well when the dimension is relatively low, but are affected by the curse of dimensionality. On the other hand, locality sensitive hashing has polynomial dependence in the dimension, sublinear query time with an exponent inversely proportional to (1+epsilon)^2, and subquadratic space requirement. We generalize the Johnson-Lindenstrauss Lemma to define "low-quality" mappings to a Euclidean space of significantly lower dimension, such that they satisfy a requirement weaker than approximately preserving all distances or even preserving the nearest neighbor. This mapping guarantees, with high probability, that an approximate nearest neighbor lies among the k approximate nearest neighbors in the projected space. These can be efficiently retrieved while using only linear storage by a data structure, such as BBD-trees. Our overall algorithm, given n points in dimension d, achieves space usage in O(dn), preprocessing time in O(dn log n), and query time in O(d n^{rho} log n), where rho is proportional to 1 - 1/loglog n, for fixed epsilon in (0, 1). The dimension reduction is larger if one assumes that point sets possess some structure, namely bounded expansion rate. We implement our method and present experimental results in up to 500 dimensions and 10^6 points, which show that the practical performance is better than predicted by the theoretical analysis. In addition, we compare our approach with E2LSH.