arXiv subjects

Ioannis Psarros

Publications and source records attributed to Ioannis Psarros.

At least 19 recordsLinked to original sources

arXiv2026

Time Series Decomposition using the Fr\'echet Distance

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 Fr\'echet distances of the time series to a ``Fr\'echet combination'' of the base curves is minimized. Here, a Fr\'echet 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 Fr\'echet 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 Fr\'echet decomposition problem as a Fr\'echet variant of principal component analysis. For the case of a single base curve we develop a $(1+\varepsilon)$-approximation algorithm for the Fr\'echet 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 Fr\'echet decomposition problem for general $k$ when curves come from a fixed candidate set.

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arXiv2026

Near Linear Time Approximation Schemes for Clustering of Partially Doubling Metrics

Given a finite metric space $(X\cup Y, \mathbf{d})$ the $k$-median problem is to find a set of $k$ centers $C\subseteq Y$ that minimizes $\sum_{p\in X} \min_{c\in C} \mathbf{d}(p,c)$. In general metrics, the best polynomial time algorithm computes a $(2+ε)$-approximation for arbitrary $ε>0$ (Cohen-Addad et al. STOC 2025). However, if the metric is doubling, a near linear time $(1+ε)$-approximation algorithm is known (Cohen-Addad et al. J. ACM 2021). 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 $X$ is doubling 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 $Y$ is doubling 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 $(k,\ell)$-median problem under discrete Fréchet distance when $\ell$ is constant. We further 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 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 (Talwar STOC 2004) to our setting. The running time of our algorithms is $2^{2^t} \tilde O(n+m)$ where $t=O(\mathrm{ddim} \log \frac{\mathrm{ddim}}ε)$ and where $\mathrm{ddim}$ is the doubling dimension of $X$ (resp.\ $Y$). The results also extend to the metric facility location problem.

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arXiv2026

Space-Efficient Approximate Spherical Range Counting in High Dimensions

We study the following range searching problem in high-dimensional Euclidean spaces: given a finite set $P\subset \mathbb{R}^d$, where each $p\in P$ 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 $q\in \mathbb{R}^d$ 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 $(1+\varepsilon)r$ away from $q$ may be taken into account, where $\varepsilon>0$ is an input parameter known during preprocessing. We build a data structure with near-linear space usage, and query time in $n^{1-Θ(\varepsilon^4/\log(1/\varepsilon))}+t_q^{\varrho}\cdot n^{1-\varrho}$, for some $\varrho=Θ(\varepsilon^2)$, where $t_q$ is the number of points of $P$ in the ambiguity zone, i.e., at distance between $r$ and $(1+\varepsilon)r$ 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 $\varepsilon>0$) 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.

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arXiv2025

A near-linear time approximation scheme for $(k,\ell)$-median clustering under discrete Fréchet distance

A time series of complexity $m$ is a sequence of $m$ real valued measurements. The discrete Fréchet distance $d_{dF}(x,y)$ is a distance measure between two time series $x$ and $y$ of possibly different complexity. Given a set of $n$ time series represented as $m$-dimensional vectors over the reals, the $(k,\ell)$-median problem under discrete Fréchet distance aims to find a set $C$ of $k$ time series of complexity $\ell$ such that $$\sum_{x\in P} \min_{c\in C} d_{dF}(x,c)$$ is minimized. In this paper, we give the first near-linear time $(1+\varepsilon)$-approximation algorithm for this problem when $\ell$ and $\varepsilon$ are constants but $k$ can be as large as $Ω(n)$. We obtain our result by introducing a new dimension reduction technique for discrete Fréchet distance and then adapt an algorithm of Cohen-Addad et al. (J. ACM 2021) to work on the dimension-reduced input. As a byproduct we also improve the best coreset construction for $(k,\ell)$-median under discrete Fréchet distance (Cohen-Addad et al., SODA 2025) and show that its size can be independent of the number of input time series \emph{ and } their complexity.

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arXiv2025

A Query-Driven Approach to Space-Efficient Range Searching

We initiate a study of a query-driven approach to designing partition trees for range-searching problems. Our model assumes that a data structure is to be built for an unknown query distribution that we can access through a sampling oracle, and must be selected such that it optimizes a meaningful performance parameter on expectation. Our first contribution is to show that a near-linear sample of queries allows the construction of a partition tree with a near-optimal expected number of nodes visited during querying. We enhance this approach by treating node processing as a classification problem, leveraging fast classifiers like shallow neural networks to obtain experimentally efficient query times. Our second contribution is to develop partition trees using sparse geometric separators. Our preprocessing algorithm, based on a sample of queries, builds a balanced tree with nodes associated with separators that minimize query stabs on expectation; this yields both fast processing of each node and a small number of visited nodes, significantly reducing query time.

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arXiv2025

Faster Approximation Algorithms for k-Center via Data Reduction

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 $α$-coreset, which is a small subset $S$ of the dataset $P$ such that any $β$-approximation on $S$ is an $(α+ β)$-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.

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arXiv2024

Fast Approximations and Coresets for (k, l)-Median under Dynamic Time Warping

We present algorithms for the computation of $\varepsilon$-coresets for $k$-median clustering of point sequences in $\mathbb{R}^d$ under the $p$-dynamic time warping (DTW) distance. Coresets under DTW have not been investigated before, and the analysis is not directly accessible to existing methods as DTW is not a metric. The three main ingredients that allow our construction of coresets are the adaptation of the $\varepsilon$-coreset framework of sensitivity sampling, bounds on the VC dimension of approximations to the range spaces of balls under DTW, and new approximation algorithms for the $k$-median problem under DTW. We achieve our results by investigating approximations of DTW that provide a trade-off between the provided accuracy and amenability to known techniques. In particular, we observe that given $n$ curves under DTW, one can directly construct a metric that approximates DTW on this set, permitting the use of the wealth of results on metric spaces for clustering purposes. The resulting approximations are the first with polynomial running time and achieve a very similar approximation factor as state-of-the-art techniques. We apply our results to produce a practical algorithm approximating $(k,\ell)$-median clustering under DTW.

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arXiv2023

Random projections for curves in high dimensions

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 $1\pm ε$. We conclude with applications on clustering.

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arXiv2023

Data-driven soiling detection in PV modules

Soiling is the accumulation of dirt in solar panels which leads to a decreasing trend in solar energy yield and may be the cause of vast revenue losses. The effect of soiling can be reduced by washing the panels, which is, however, a procedure of non-negligible cost. Moreover, soiling monitoring systems are often unreliable or very costly. We study the problem of estimating the soiling ratio in photo-voltaic (PV) modules, i.e., the ratio of the real power output to the power output that would be produced if solar panels were clean. A key advantage of our algorithms is that they estimate soiling, without needing to train on labelled data, i.e., periods of explicitly monitoring the soiling in each park, and without relying on generic analytical formulas which do not take into account the peculiarities of each installation. We consider as input a time series comprising a minimum set of measurements, that are available to most PV park operators. Our experimental evaluation shows that we significantly outperform current state-of-the-art methods for estimating soiling ratio.

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arXiv2022

Approximating Length-Restricted Means under Dynamic Time Warping

We study variants of the mean problem under the $p$-Dynamic Time Warping ($p$-DTW) distance, a popular and robust distance measure for sequential data. In our setting we are given a set of finite point sequences over an arbitrary metric space and we want to compute a mean point sequence of given length that minimizes the sum of $p$-DTW distances, each raised to the $q$\textsuperscript{th} power, between the input sequences and the mean sequence. In general, the problem is $\mathrm{NP}$-hard and known not to be fixed-parameter tractable in the number of sequences. On the positive side, we show that restricting the length of the mean sequence significantly reduces the hardness of the problem. We give an exact algorithm running in polynomial time for constant-length means. We explore various approximation algorithms that provide a trade-off between the approximation factor and the running time. Our approximation algorithms have a running time with only linear dependency on the number of input sequences. In addition, we use our mean algorithms to obtain clustering algorithms with theoretical guarantees.

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arXiv2021

Tight Bounds for Approximate Near Neighbor Searching for Time Series under the Fréchet Distance

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 $k \leq m$, 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 $c\cdot δ$ 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 $0 < \varepsilon \leq 1$ an approximation factor of $(1+\varepsilon)$ can be achieved within the same asymptotic time bounds as the previously best result for $(2+\varepsilon)$. Moreover, we show that an approximation factor of $(2+\varepsilon)$ can be obtained by using preprocessing time and space $O(nm)$, which is linear in the input size, and query time in $O(\frac{1}{\varepsilon})^{k+2}$, where the previously best result used preprocessing time in $n \cdot O(\frac{m}{\varepsilon k})^k$ 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.

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arXiv2021

$(2+ε)$-ANN for time series under the Fréchet distance

We study approximate-near-neighbor data structures for time series under the continuous Fréchet distance. For an attainable approximation factor $c>1$ and a query radius $r$, an approximate-near-neighbor data structure can be used to preprocess $n$ curves in $\mathbb{R}$ (aka time series), each of complexity $m$, to answer queries with a curve of complexity $k$ by either returning a curve that lies within Fréchet distance $cr$, or answering that there exists no curve in the input within distance $r$. In both cases, the answer is correct. Our first data structure achieves a $(5+ε)$ approximation factor, uses space in $n\cdot \mathcal{O}\left({ε^{-1}}\right)^{k} + \mathcal{O}(nm)$ and has query time in $\mathcal{O}\left(k\right)$. Our second data structure achieves a $(2+ε)$ approximation factor, uses space in $n\cdot \mathcal{O}\left(\frac{m}{kε}\right)^{k} + \mathcal{O}(nm)$ and has query time in $\mathcal{O}\left(k\cdot 2^k\right)$. Our third positive result is a probabilistic data structure based on locality-sensitive hashing, which achieves space in $\mathcal{O}(n\log n+nm)$ and query time in $\mathcal{O}(k\log n)$, and which answers queries with an approximation factor in $\mathcal{O}(k)$. All of our data structures make use of the concept of signatures, which were originally introduced for the problem of clustering time series under the Fréchet distance. In addition, we show lower bounds for this problem. Consider any data structure which achieves an approximation factor less than $2$ and which supports curves of arclength up to $L$ and answers the query using only a constant number of probes. We show that under reasonable assumptions on the word size any such data structure needs space in $L^{Ω(k)}$.

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arXiv2020

Products of Euclidean metrics and applications to proximity questions among curves

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.

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arXiv2019

The VC Dimension of Metric Balls under Fréchet and Hausdorff Distances

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 $\mathbb{R}^d$ and the sets $\mathcal{R}$ 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 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.

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arXiv2019

Near neighbor preserving dimension reduction for doubling subsets of $\ell_1$

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 $(\ell_2)$ metric, but much less for the Manhattan $(\ell_1)$ metric. Our primary motivation is the approximate nearest neighbor problem in $\ell_1$. We exploit its reduction to the decision-with-witness version, called approximate \textit{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 $\ell_2$ and $\ell_1$ metrics, as well as for doubling subsets of $\ell_2$. The case that remained open were doubling subsets of $\ell_1$. In this paper, we propose a dimension reduction by means of a \textit{near} neighbor-preserving embedding for doubling subsets of $\ell_1$. 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.

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arXiv2019

Sublinear data structures for short Fréchet queries

We study metric data structures for curves in doubling spaces, such as trajectories of moving objects in Euclidean $\mathbb{R}^d$, where the distance between two curves is measured using the discrete Fréchet distance. We design data structures in an \emph{asymmetric} setting where the input is a curve (or a set of $n$ curves) each of complexity $m$ and the queries are with curves of complexity $k\ll m$. We show that there exist approximate data structures that are independent of the input size $N = d \cdot n \cdot m$ and we study how to maintain them dynamically if the input is given in the stream. Concretely, we study two types of data structures: (i) distance oracles, where the task is to store a compressed version of the input curve, which can be used to answer queries for the distance of a query curve to the input curve, and (ii) nearest-neighbor data structures, where the task is to preprocess a set of input curves to answer queries for the input curve closest to the query curve. In both cases we are interested in approximation. For curves embedded in Euclidean $\mathbb{R}^d$ with constant $d$, our distance oracle uses space in $\mathcal{O}((k \log(ε^{-1}) ε^{-d})^k)$ ($ε$ is the precision parameter). The oracle performs $(1+ε)$-approximate queries in time in $\mathcal{O}(k^2)$ and is deterministic. We show how to maintain this distance oracle in the stream using polylogarithmic additional memory. In the stream, we can dynamically answer distance queries to the portion of the stream seen so far in $\mathcal{O}(k^4 \log^2 m)$ time. We apply our techniques to the second problem, approximate near neighbor (ANN) data structures, and achieve an exponential improvement in the dependency on the complexity of the input curves compared to the state of the art.

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arXiv2017

High-dimensional approximate $r$-nets

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 $ε>0$, the approximation factor is $1+ε$ and the complexity is polynomial in the dimension and subquadratic in the number of points. The algorithm succeeds with high probability. More specifically, the best previously known LSH-based construction of Eppstein et al.\ \cite{EHS15} is improved in terms of complexity by reducing the dependence on $ε$, provided that $ε$ is sufficiently small. Our method does not require LSH but, instead, follows Valiant's \cite{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 $(1+ε)$-approximate $k$th nearest neighbor distance in time subquadratic in the size of the input.

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arXiv2017

Counting Euclidean embeddings of rigid graphs

A graph is called (generically) rigid in $\mathbb{R}^d$ if, for any choice of sufficiently generic edge lengths, it can be embedded in $\mathbb{R}^d$ in a finite number of distinct ways, modulo rigid transformations. Here we deal with the problem of determining the maximum number of planar Euclidean embeddings as a function of the number of the vertices. We obtain polynomial systems which totally capture the structure of a given graph, by exploiting distance geometry theory. Consequently, counting the number of Euclidean embeddings of a given rigid graph, reduces to the problem of counting roots of the corresponding polynomial system.

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