arXiv subjects

Amir Carmel

Publications and source records attributed to Amir Carmel.

6 recordsLinked to original sources

arXiv2026

Recovery Beats Storage: Improved Space for Preprocessed 3SUM

The 3SUM problem asks, given sets $A,B,C$ of integers, whether there exist $a\in A$ and $b\in B$ whose sum belongs to $C$. In the preprocessed variant with unknown $C$, one preprocesses sets $A$ and $B$, each of size $n$, and subsequently answers a query specified by subsets $A'\subseteq A$, $B'\subseteq B$ and a target set $C'$ of size $O(n)$, by solving the 3SUM instance $(A',B',C')$. Kirkpatrick, Kuszmaul, Mathialagan, and Vassilevska Williams [ICALP 2026] gave the first algorithm with subquadratic space for this problem, achieving $\tilde{O}(n^{3/2+ε})$ query time using $\tilde{O}(n^{2-2ε/3})$ space, for every $ε\in[0,1/2]$. Their algorithm employs separate mechanisms for heavy and light targets, and for each heavy target it stores explicitly the list of pairs $(a,b)$ summing to it; these lists dominate the space bound. We present a unified construction that uses a single mechanism for all queries. Instead of storing these lists of pairs, we recover them on demand by leveraging the Fiat--Naor data structure [SICOMP 1999] to invert the function $(a,b)\mapsto (a+b\bmod p)$. This simplification improves the space bound to $\tilde{O}(n^{\max(2-ε, 11/6-ε/3)})$, while maintaining the same query time. Moreover, our construction is the first to achieve subquadratic space while supporting adaptively chosen queries.

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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.

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arXiv2026

A Scalable and Unified Framework to Weighted Rank Aggregation

The rank aggregation problem seeks to combine multiple rank orderings of the same set of candidates into a single consensus ordering. Such problems arise in diverse domains, including web search, employment, college admissions, and voting. In this work we focus on the 1-median objective: given a set of m rankings over [n], the goal is to compute a ranking that minimizes the sum of its distances to all input rankings. We study rank aggregation under several classical distance metrics: Ulam distance, Spearman's footrule, Hamming distance, and Kendall-tau, as well as their weighted variants. Our contributions begin with a novel unified framework that identifies a key structural property: it suffices to focus on a small subset of rankings, where the corresponding local one-median provides a good approximation to the global median. This principle extends across these distance measures, yielding a general algorithmic framework for weighted rank aggregation. Building on this, we present a new approximation algorithm for rank aggregation under the Ulam distance that scales in the Massively Parallel Computation (MPC) model. Our algorithm computes a $(2-α)$-approximation, for a constant $α>0$, to the 1-median in a constant number of rounds, using local memory sublinear in n and total memory near-linear in n. We further design new MPC approximation algorithms for Spearman's footrule and for the element-weighted variants of Hamming and Kendall-tau distances. For each metric, we obtain a $(2-ζ)$-approximation, for a constant $ζ>0$, to the 1-median in a constant number of rounds, using local memory sublinear in n and total memory linear or near-linear in n. Moreover, for the Ulam distance, we simplify and strengthen the analysis of Chakraborty et al., obtaining an improved 1.968-approximation that further extends to the weighted setting.

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arXiv2026

Stable coresets: Unleashing the power of uniform sampling

Uniform sampling is a highly efficient method for data summarization. However, its effectiveness in producing coresets for clustering problems is not yet well understood, primarily because it generally does not yield a strong coreset, which is the prevailing notion in the literature. We formulate \emph{stable coresets}, a notion that is intermediate between the standard notions of weak and strong coresets, and effectively combines the broad applicability of strong coresets with highly efficient constructions, through uniform sampling, of weak coresets. Our main result is that a uniform sample of size $O(ε^{-2}\log d)$ yields, with high constant probability, a stable coreset for $1$-median in $\mathbb{R}^d$ under the $\ell_1$ metric. We then leverage the powerful properties of stable coresets to easily derive new coreset constructions, all through uniform sampling, for $\ell_1$ and related metrics, such as Kendall-tau and Jaccard. We also show applications to fair rank aggregation and to approximation algorithms for $k$-median problem in these metric spaces. Our experiments validate the benefits of stable coresets in practice, in terms of both construction time and approximation quality.

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arXiv2025

Fitting Tree Metrics and Ultrametrics in Data Streams

Fitting distances to tree metrics and ultrametrics are two widely used methods in hierarchical clustering, primarily explored within the context of numerical taxonomy. Given a positive distance function $D:\binom{V}{2}\rightarrow\mathbb{R}_{>0}$, the goal is to find a tree (or ultrametric) $T$ including all elements of set $V$ such that the difference between the distances among vertices in $T$ and those specified by $D$ is minimized. In this paper, we initiate the study of ultrametric and tree metric fitting problems in the semi-streaming model, where the distances between pairs of elements from $V$ (with $|V|=n$), defined by the function $D$, can arrive in an arbitrary order. We study these problems under various distance norms: For the $\ell_0$ objective, we provide a single-pass polynomial-time $\tilde{O}(n)$-space $O(1)$ approximation algorithm for ultrametrics and prove that no single-pass exact algorithm exists, even with exponential time. Next, we show that the algorithm for $\ell_0$ implies an $O(Δ/δ)$ approximation for the $\ell_1$ objective, where $Δ$ is the maximum and $δ$ is the minimum absolute difference between distances in the input. This bound matches the best-known approximation for the RAM model using a combinatorial algorithm when $Δ/δ=O(n)$. For the $\ell_\infty$ objective, we provide a complete characterization of the ultrametric fitting problem. We present a single-pass polynomial-time $\tilde{O}(n)$-space 2-approximation algorithm and show that no better than 2-approximation is possible, even with exponential time. We also show that, with an additional pass, it is possible to achieve a polynomial-time exact algorithm for ultrametrics. Finally, we extend the results for all these objectives to tree metrics by using only one additional pass through the stream and without asymptotically increasing the approximation factor.

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arXiv2018

Dynamic all scores matrices for LCS score

The problem of aligning two strings A,B in order to determine their similarity is fundamental in the field of pattern matching. An important concept in this domain is the "all scores matrix" that encodes the local alignment comparison of two strings. Namely, let K denote the all scores matrix containing the alignment score of every substring of B with A, and let J denote the all scores matrix containing the alignment score of every suffix of B with every prefix of A. In this paper we consider the problem of maintaining an all scores matrix where the scoring function is the LCS score, while supporting single character prepend and append operations to A and N. Our algorithms exploit the sparsity parameters L=LCS(A,B) and Delta = |B|-L. For the matrix K we propose an algorithm that supports incremental operations to both ends of A in O(Delta) time. Whilst for the matrix J we propose an algorithm that supports a single type of incremental operation, either a prepend operation to A or an append operation to B, in O(L) time. This structure can also be extended to support both operations simultaneously in O(L log log L) time.

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