arXiv subjects

Nir Petruschka

Publications and source records attributed to Nir Petruschka.

6 recordsLinked to original sources

arXiv2026

Fast Metric Decompositions in High Dimension

Metric decompositions are a fundamental tool in the design of algorithms involving distances. We study fast algorithms for sampling from probabilistic metric decompositions of $n$-point sets in $\ell_\infty$ and $\ell_2$ spaces of high dimension $d$. For $\ell_\infty$, we design a padded-decomposition algorithm that runs in time $\tilde{O}(nd^2)$, which is near-linear in $n$, and achieves padding parameter $\tilde{O}(\log n)$. Our algorithm constructs a new sparse neighborhood cover that is based on geometric properties of $\ell_\infty$ [Indyk, JCSS'01], and utilizes recent reductions between covers and decompositions [Conroy and Filtser, STOC'25]. For $\ell_2$, we design a separating-decomposition algorithm that achieves near optimal separation $\tilde{O}(\sqrt{\log n})$ in almost-linear time $n^{1+o(1)}$. Our bounds improve over known algorithms with similar running time by a factor $Ω(\sqrt{\log n})$, and the techniques have additional applications to spanners and nearest-neighbor search.

cs.DS↗

arXiv2026

Towards Lower Bounds for Geometric Spanners in High Dimension

We study the stretch--size tradeoff for geometric spanners in high-dimensional $\ell_p$ spaces. Our main contribution is a simple proof of a lower bound shown by Har-Peled, Indyk, and Sidiropoulos [SODA 2013]: Every $2$-hop $t$-spanner of the pointset $\{0,1\}^d$ under $\ell_2$ norm has at least $(2^d)^{1+Ω(1/t^2)}$ edges. Our proof further extends this result to spanners with Steiner vertices. In addition, we establish a connection between bounded-hop spanners and general spanners, as follows. If every subset $Y$ of an $n$-point metric has a $t$-spanner with at most $μ|Y|$ edges, then the metric has an $O(t)$-hop $O(t)$-spanner of size $O(n(μ+\log n))$. Consequently, hop-restricted spanner lower bounds for a metric imply lower bounds without hop restriction for one of its subsets.

cs.CG↗

arXiv2026

Optimal Stable Coresets for Geometric Median via Uniform Sampling

The geometric median problem asks to find a point in $\mathbb{R}^d$ that minimizes the sum of Euclidean distances to an input set. It is a classical problem in computational geometry and appears as a subroutine in numerous optimization tasks, many of which require the solution to satisfy additional structural constraints. A common approach to reduce the input size is to construct a coreset, which is a small weighted subset that faithfully represents the input for a specific optimization problem. Strong coresets preserve the cost of every candidate solution but require linear time to construct; weak coresets admit sublinear construction, in fact by uniform sampling, but only preserve near-optimal solutions, which is insufficient when the solution is constrained. To address this, we focus instead on the recently introduced intermediate notion of a \emph{stable coreset}, which simultaneously handles all constrained variants. Currently, there is a large gap between the known sample sizes for stable and weak coresets. Our main result is that a uniform sample of size $O(ε^{-2} \log \tfrac{1}ε)$ is a stable $(ε, O(ε))$-coreset for the geometric median, with high constant probability, and this bound is tight up to the logarithmic factor. Our analysis adapts recent machinery of Carmel and Krauthgamer (ICLR 2026) for constructing stable coresets, which incurs an $O(\log d)$ factor. We show an iterative argument that progressively reduces the sample size, and eliminates this dependence on the dimension $d$. At a high level, this approach resembles the technique of iterative size reduction, which is applicable for strong coresets but not for weak coresets.

cs.DS↗

arXiv2026

Fast Nearest Neighbor Search for $\ell_p$ Metrics

The Nearest Neighbor Search (NNS) problem asks to design a data structure that preprocesses an $n$-point dataset $X$ lying in a metric space $\mathcal{M}$, so that given a query point $q \in \mathcal{M}$, one can quickly return a point of $X$ minimizing the distance to $q$. The efficiency of such a data structure is evaluated primarily by the amount of space it uses and the time required to answer a query. We focus on the fast query-time regime, which is crucial for modern large-scale applications, where datasets are massive and queries must be processed online, and is often modeled by query time $\text{poly}(d \log n)$. Our main result is such a randomized data structure for NNS in $\ell_p$ spaces, $p>2$, that achieves $p^{O(1) + \log\log p}$ approximation with fast query time and $\text{poly}(dn)$ space. Our data structure improves, or is incomparable to, the state-of-the-art for the fast query-time regime from [Bartal and Gottlieb, TCS 2019] and [Krauthgamer, Petruschka and Sapir, FOCS 2025].

cs.DS↗

arXiv2025

The Power of Recursive Embeddings for $\ell_p$ Metrics

Metric embedding is a powerful tool used extensively in mathematics and computer science. We devise a new method of using metric embeddings recursively, which turns out to be particularly effective in $\ell_p$ spaces, $p>2$, yielding state-of-the-art results for Lipschitz decomposition, for Nearest Neighbor Search, and for embedding into $\ell_2$. In a nutshell, our method composes metric embeddings by viewing them as reductions between problems, and thereby obtains a new reduction that is substantially more effective than the known reduction that employs a single embedding. We in fact apply this method recursively, oftentimes using double recursion, which further amplifies the gap from a single embedding.

cs.CG↗

arXiv2025

Lipschitz Decompositions of Finite $\ell_{p}$ Metrics

Lipschitz decomposition is a useful tool in the design of efficient algorithms involving metric spaces. While many bounds are known for different families of finite metrics, the optimal parameters for $n$-point subsets of $\ell_p$, for $p > 2$, remained open, see e.g. [Naor, SODA 2017]. We make significant progress on this question and establish the bound $β=O(\log^{1-1/p} n)$. Building on prior work, we demonstrate applications of this result to two problems, high-dimensional geometric spanners and distance labeling schemes. In addition, we sharpen a related decomposition bound for $1<p<2$, due to Filtser and Neiman [Algorithmica 2022].

cs.CG↗