arXiv subjects

Yair Bartal

Publications and source records attributed to Yair Bartal.

At least 19 recordsLinked to original sources

arXiv2025

Improved fixed-parameter bounds for Min-Sum-Radii and Diameters $k$-clustering and their fair variants

We provide improved upper and lower bounds for the Min-Sum-Radii (MSR) and Min-Sum-Diameters (MSD) clustering problems with a bounded number of clusters $k$. In particular, we propose an exact MSD algorithm with running-time $n^{O(k)}$. We also provide $(1+ε)$ approximation algorithms for both MSR and MSD with running-times of $O(kn) +(1/ε)^{O(dk)}$ in metrics spaces of doubling dimension $d$. Our algorithms extend to $k$-center, improving upon previous results, and to $α$-MSR, where radii are raised to the $α$ power for $α>1$. For $α$-MSD we prove an exponential time ETH-based lower bound for $α>\log 3$. All algorithms can also be modified to handle outliers. Moreover, we can extend the results to variants that observe fairness constraints, as well as to the general framework of mergeable clustering, which includes many other popular clustering variants. We complement these upper bounds with ETH-based lower bounds for these problems, in particular proving that $n^{O(k)}$ time is tight for MSR and $α$-MSR even in doubling spaces, and that $2^{o(k)}$ bounds are impossible for MSD.

cs.DS↗

arXiv2024

Online Probabilistic Metric Embedding: A General Framework for Bypassing Inherent Bounds

Probabilistic metric embedding into trees is a powerful technique for designing online algorithms. The standard approach is to embed the entire underlying metric into a tree metric and then solve the problem on the latter. The overhead in the competitive ratio depends on the expected distortion of the embedding, which is logarithmic in $n$, the size of the underlying metric. For many online applications, such as online network design problems, it is natural to ask if it is possible to construct such embeddings in an online fashion such that the distortion would be a polylogarithmic function of $k$, the number of terminals. Our first main contribution is answering this question negatively, exhibiting a \emph{lower bound} of $\tildeΩ(\log k \log Φ)$, where $Φ$ is the aspect ratio of the set of terminals, showing that a simple modification of the probabilistic embedding into trees of Bartal (FOCS 1996), which has expected distortion of $O(\log k \log Φ)$, is \emph{nearly-tight}. Unfortunately, this may result in a very bad dependence in terms of $k$, namely, a power of $k$. Our second main contribution is a general framework for bypassing this limitation. We show that for a large class of online problems this online probabilistic embedding can still be used to devise an algorithm with $O(\min\{\log k\log (kλ),\log^3 k\})$ overhead in the competitive ratio, where $k$ is the current number of terminals, and $λ$ is a measure of subadditivity of the cost function, which is at most $r$, the current number of requests. In particular, this implies the first algorithms with competitive ratio $\operatorname{polylog}(k)$ for online subadditive network design (buy-at-bulk network design being a special case), and $\operatorname{polylog}(k,r)$ for online group Steiner forest.

cs.DS↗

arXiv2022

Optimality of the Johnson-Lindenstrauss Dimensionality Reduction for Practical Measures

It is well known that the Johnson-Lindenstrauss dimensionality reduction method is optimal for worst case distortion. While in practice many other methods and heuristics are used, not much is known in terms of bounds on their performance. The question of whether the JL method is optimal for practical measures of distortion was recently raised in BFN19 (NeurIPS'19). They provided upper bounds on its quality for a wide range of practical measures and showed that indeed these are best possible in many cases. Yet, some of the most important cases, including the fundamental case of average distortion were left open. In particular, they show that the JL transform has $1+ε$ average distortion for embedding into $k$-dimensional Euclidean space, where $k=O(1/ε^2)$, and for more general $q$-norms of distortion, $k = O(\max\{1/ε^2,q/ε\})$, whereas tight lower bounds were established only for large values of $q$ via reduction to the worst case. In this paper we prove that these bounds are best possible for any dimensionality reduction method, for any $1 \leq q \leq O(\frac{\log (2ε^2 n)}ε)$ and $ε\geq \frac{1}{\sqrt{n}}$, where $n$ is the size of the subset of Euclidean space. Our results imply that the JL method is optimal for various distortion measures commonly used in practice such as stress, energy and relative error. We prove that if any of these measures is bounded by $ε$ then $k=Ω(1/ε^2)$ for any $ε\geq \frac{1}{\sqrt{n}}$, matching the upper bounds of BFN19 and extending their tightness results for the full range moment analysis. Our results may indicate that the JL dimensionality reduction method should be considered more often in practical applications, and the bounds we provide for its quality should be served as a measure for comparison when evaluating the performance of other methods and heuristics.

cs.DS↗

arXiv2021

Advances in Metric Ramsey Theory and its Applications

Metric Ramsey theory is concerned with finding large well-structured subsets of more complex metric spaces. For finite metric spaces this problem was first studies by Bourgain, Figiel and Milman \cite{bfm}, and studied further in depth by Bartal et. al \cite{BLMN03}. In this paper we provide deterministic constructions for this problem via a novel notion of \emph{metric Ramsey decomposition}. This method yields several more applications, reflecting on some basic results in metric embedding theory. The applications include various results in metric Ramsey theory including the first deterministic construction yielding Ramsey theorems with tight bounds, a well as stronger theorems and properties, implying appropriate distance oracle applications. In addition, this decomposition provides the first deterministic Bourgain-type embedding of finite metric spaces into Euclidean space, and an optimal multi-embedding into ultrametrics, thus improving its applications in approximation and online algorithms. The decomposition presented here, the techniques and its consequences have already been used in recent research in the field of metric embedding for various applications.

cs.DS↗

arXiv2019

Covering Metric Spaces by Few Trees

A {\em tree cover} of a metric space $(X,d)$ is a collection of trees, so that every pair $x,y\in X$ has a low distortion path in one of the trees. If it has the stronger property that every point $x\in X$ has a single tree with low distortion paths to all other points, we call this a {\em Ramsey} tree cover. Tree covers and Ramsey tree covers have been studied by \cite{BLMN03,GKR04,CGMZ05,GHR06,MN07}, and have found several important algorithmic applications, e.g. routing and distance oracles. The union of trees in a tree cover also serves as a special type of spanner, that can be decomposed into a few trees with low distortion paths contained in a single tree; Such spanners for Euclidean pointsets were presented by \cite{ADMSS95}. In this paper we devise efficient algorithms to construct tree covers and Ramsey tree covers for general, planar and doubling metrics. We pay particular attention to the desirable case of distortion close to 1, and study what can be achieved when the number of trees is small. In particular, our work shows a large separation between what can be achieved by tree covers vs. Ramsey tree covers.

cs.DS↗

arXiv2019

Near-linear time approximation schemes for Steiner tree and forest in low-dimensional spaces

We give an algorithm that computes a $(1+ε)$-approximate Steiner forest in near-linear time $n \cdot 2^{(1/ε)^{O(ddim^2)} (\log \log n)^2}$. This is a dramatic improvement upon the best previous result due to Chan et al., who gave a runtime of $n^{2^{O(ddim)}} \cdot 2^{(ddim/ε)^{O(ddim)} \sqrt{\log n}}$. For Steiner tree our methods achieve an even better runtime $n (\log n)^{(1/ε)^{O(ddim^2)}}$ in doubling spaces. For Euclidean space the runtime can be reduced to $2^{(1/ε)^{O(d^2)}} n \log n$, improving upon the result of Arora in fixed dimension $d$.

cs.CG↗

arXiv2018

On Notions of Distortion and an Almost Minimum Spanning Tree with Constant Average Distortion

This paper makes two main contributions: The first is the construction of a near-minimum spanning tree with constant average distortion. The second is a general equivalence theorem relating two refined notions of distortion: scaling distortion and prioritized distortion. Minimum Spanning Trees of weighted graphs are fundamental objects in numerous applications. In particular in distributed networks, the minimum spanning tree of the network is often used to route messages between network nodes. Unfortunately, while being most efficient in the total cost of connecting all nodes, minimum spanning trees fail miserably in the desired property of approximately preserving distances between pairs. While known lower bounds exclude the possibility of the worst case distortion of a tree being small, it was shown in [ABN15] that there exists a spanning tree with constant average distortion. Yet, the weight of such a tree may be significantly larger than that of the MST. In this paper, we show that any weighted undirected graph admits a {\em spanning tree} whose weight is at most $(1+ρ)$ times that of the MST, providing {\em constant average distortion} $O(1/ρ)$. Our result exhibits the best possible tradeoff of this type. This result makes use of a general equivalence theorem relating two recently developed notions of distortion for metric embedding. The first is the notion of scaling distortion, which provides improved distortion for $1-ε$ fractions of the pairs, for all $ε$ simultaneously. A stronger version called coarse scaling distortion, has improved distortion guarantees for the furthest pairs. The second notion is that of prioritized distortion, a property allowing to prioritize the nodes whose associated distortions will be improved. We show that prioritized distortion is essentially equivalent to coarse scaling distortion via a general transformation.

cs.DS↗

arXiv2015

Dimension reduction techniques for $\ell_p$, $1 \le p \le 2$, with applications

For Euclidean space ($\ell_2$), there exists the powerful dimension reduction transform of Johnson and Lindenstrauss, with a host of known applications. Here, we consider the problem of dimension reduction for all $\ell_p$ spaces $1 \le p \le 2$. Although strong lower bounds are known for dimension reduction in $\ell_1$, Ostrovsky and Rabani successfully circumvented these by presenting an $\ell_1$ embedding that maintains fidelity in only a bounded distance range, with applications to clustering and nearest neighbor search. However, their embedding techniques are specific to $\ell_1$ and do not naturally extend to other norms. In this paper, we apply a range of advanced techniques and produce bounded range dimension reduction embeddings for all of $1 \le p \le 2$, thereby demonstrating that the approach initiated by Ostrovsky and Rabani for $\ell_1$ can be extended to a much more general framework. We also obtain improved bounds in terms of the intrinsic dimensionality. As a result we achieve improved bounds for proximity problems including snowflake embeddings and clustering.

cs.CG↗

arXiv2015

Approximate nearest neighbor search for $\ell_p$-spaces ($2 < p < \infty$) via embeddings

While the problem of approximate nearest neighbor search has been well-studied for Euclidean space and $\ell_1$, few non-trivial algorithms are known for $\ell_p$ when ($2 < p < \infty$). In this paper, we revisit this fundamental problem and present approximate nearest-neighbor search algorithms which give the first non-trivial approximation factor guarantees in this setting.

cs.CG↗

arXiv2015

The Traveling Salesman Problem: Low-Dimensionality Implies a Polynomial Time Approximation Scheme

The Traveling Salesman Problem (TSP) is among the most famous NP-hard optimization problems. We design for this problem a randomized polynomial-time algorithm that computes a (1+eps)-approximation to the optimal tour, for any fixed eps>0, in TSP instances that form an arbitrary metric space with bounded intrinsic dimension. The celebrated results of Arora (A-98) and Mitchell (M-99) prove that the above result holds in the special case of TSP in a fixed-dimensional Euclidean space. Thus, our algorithm demonstrates that the algorithmic tractability of metric TSP depends on the dimensionality of the space and not on its specific geometry. This result resolves a problem that has been open since the quasi-polynomial time algorithm of Talwar (T-04).

cs.CC↗

arXiv2013

On the Impossibility of Dimension Reduction for Doubling Subsets of $\ell_p$, $p>2$

A major open problem in the field of metric embedding is the existence of dimension reduction for $n$-point subsets of Euclidean space, such that both distortion and dimension depend only on the {\em doubling constant} of the pointset, and not on its cardinality. In this paper, we negate this possibility for $\ell_p$ spaces with $p>2$. In particular, we introduce an $n$-point subset of $\ell_p$ with doubling constant O(1), and demonstrate that any embedding of the set into $\ell_p^d$ with distortion $D$ must have $D\geΩ\left(\left(\frac{c\log n}{d}\right)^{\frac{1}{2}-\frac{1}{p}}\right)$.

cs.CG↗

arXiv2010

Fast, precise and dynamic distance queries

We present an approximate distance oracle for a point set S with n points and doubling dimension λ. For every ε>0, the oracle supports (1+ε)-approximate distance queries in (universal) constant time, occupies space [ε^{-O(λ)} + 2^{O(λ log λ)}]n, and can be constructed in [2^{O(λ)} log3 n + ε^{-O(λ)} + 2^{O(λ log λ)}]n expected time. This improves upon the best previously known constructions, presented by Har-Peled and Mendel. Furthermore, the oracle can be made fully dynamic with expected O(1) query time and only 2^{O(λ)} log n + ε^{-O(λ)} + 2^{O(λ log λ)} update time. This is the first fully dynamic (1+ε)-distance oracle.

cs.DS↗

arXiv2008

Nearly Tight Low Stretch Spanning Trees

We prove that any graph $G$ with $n$ points has a distribution $\mathcal{T}$ over spanning trees such that for any edge $(u,v)$ the expected stretch $E_{T \sim \mathcal{T}}[d_T(u,v)/d_G(u,v)]$ is bounded by $\tilde{O}(\log n)$. Our result is obtained via a new approach of building ``highways'' between portals and a new strong diameter probabilistic decomposition theorem.

cs.DS↗

arXiv2007

On metric Ramsey-type phenomena

The main question studied in this article may be viewed as a nonlinear analogue of Dvoretzky's theorem in Banach space theory or as part of Ramsey theory in combinatorics. Given a finite metric space on n points, we seek its subspace of largest cardinality which can be embedded with a given distortion in Hilbert space. We provide nearly tight upper and lower bounds on the cardinality of this subspace in terms of n and the desired distortion. Our main theorem states that for any epsilon>0, every n point metric space contains a subset of size at least n^{1-ε} which is embeddable in Hilbert space with O(\frac{\log(1/ε)}ε) distortion. The bound on the distortion is tight up to the log(1/ε) factor. We further include a comprehensive study of various other aspects of this problem.

math.MG↗

arXiv2006

Embedding Metrics into Ultrametrics and Graphs into Spanning Trees with Constant Average Distortion

This paper addresses the basic question of how well can a tree approximate distances of a metric space or a graph. Given a graph, the problem of constructing a spanning tree in a graph which strongly preserves distances in the graph is a fundamental problem in network design. We present scaling distortion embeddings where the distortion scales as a function of $ε$, with the guarantee that for each $ε$ the distortion of a fraction $1-ε$ of all pairs is bounded accordingly. Such a bound implies, in particular, that the \emph{average distortion} and $\ell_q$-distortions are small. Specifically, our embeddings have \emph{constant} average distortion and $O(\sqrt{\log n})$ $\ell_2$-distortion. This follows from the following results: we prove that any metric space embeds into an ultrametric with scaling distortion $O(\sqrt{1/ε})$. For the graph setting we prove that any weighted graph contains a spanning tree with scaling distortion $O(\sqrt{1/ε})$. These bounds are tight even for embedding in arbitrary trees. For probabilistic embedding into spanning trees we prove a scaling distortion of $\tilde{O}(\log^2 (1/ε))$, which implies \emph{constant} $\ell_q$-distortion for every fixed $q<\infty$.

cs.DM↗

arXiv2004

Multi-Embedding of Metric Spaces

Metric embedding has become a common technique in the design of algorithms. Its applicability is often dependent on how high the embedding's distortion is. For example, embedding finite metric space into trees may require linear distortion as a function of its size. Using probabilistic metric embeddings, the bound on the distortion reduces to logarithmic in the size. We make a step in the direction of bypassing the lower bound on the distortion in terms of the size of the metric. We define "multi-embeddings" of metric spaces in which a point is mapped onto a set of points, while keeping the target metric of polynomial size and preserving the distortion of paths. The distortion obtained with such multi-embeddings into ultrametrics is at most O(log Delta loglog Delta) where Delta is the aspect ratio of the metric. In particular, for expander graphs, we are able to obtain constant distortion embeddings into trees in contrast with the Omega(log n) lower bound for all previous notions of embeddings. We demonstrate the algorithmic application of the new embeddings for two optimization problems: group Steiner tree and metrical task systems.

cs.DS↗

arXiv2004

On Metric Ramsey-type Dichotomies

The classical Ramsey theorem, states that every graph contains either a large clique or a large independent set. Here we investigate similar dichotomic phenomena in the context of finite metric spaces. Namely, we prove statements of the form "Every finite metric space contains a large subspace that is nearly quilateral or far from being equilateral". We consider two distinct interpretations for being "far from equilateral". Proximity among metric spaces is quantified through the metric distortion D. We provide tight asymptotic answers for these problems. In particular, we show that a phase transition occurs at D=2.

math.CO↗

arXiv2004

On some low distortion metric Ramsey problems

In this note, we consider the metric Ramsey problem for the normed spaces l_p. Namely, given some 1<=p<=infinity and alpha>=1, and an integer n, we ask for the largest m such that every n-point metric space contains an m-point subspace which embeds into l_p with distortion at most alpha. In [arXiv:math.MG/0406353] it is shown that in the case of l_2, the dependence of $m$ on alpha undergoes a phase transition at alpha=2. Here we consider this problem for other l_p, and specifically the occurrence of a phase transition for p other than 2. It is shown that a phase transition does occur at alpha=2 for every p in the interval [1,2]. For p>2 we are unable to determine the answer, but estimates are provided for the possible location of such a phase transition. We also study the analogous problem for isometric embedding and show that for every 1<p<infinity there are arbitrarily large metric spaces, no four points of which embed isometrically in l_p.

math.MG↗