arXiv subjects

Pinki Pradhan

Publications and source records attributed to Pinki Pradhan.

3 recordsLinked to original sources

arXiv2026

Improved Algorithms for Clustering with Noisy Distance Oracles

Bateni et al. has recently introduced the weak-strong distance oracle model to study clustering problems in settings with limited distance information. Given query access to the strong-oracle and weak-oracle in the weak-strong oracle model, the authors design approximation algorithms for $k$-means and $k$-center clustering problems. In this work, we design algorithms with improved guarantees for $k$-means and $k$-center clustering problems in the weak-strong oracle model. The $k$-means++ algorithm is routinely used to solve $k$-means in settings where complete distance information is available. One of the main contributions of this work is to show that $k$-means++ algorithm can be adapted to work in the weak-strong oracle model using only a small number of strong-oracle queries, which is the critical resource in this model. In particular, our $k$-means++ based algorithm gives a constant approximation for $k$-means and uses $O(k^2 \log^2{n})$ strong-oracle queries. This improves on the algorithm of Bateni et al. that uses $O(k^2 \log^4n \log^2 \log n)$ strong-oracle queries for a constant factor approximation of $k$-means. For the $k$-center problem, we give a simple ball-carving based $6(1 + \epsilon)$-approximation algorithm that uses $O(k^3 \log^2{n} \log{\frac{\log{n}}{\epsilon}})$ strong-oracle queries. This is an improvement over the $14(1 + \epsilon)$-approximation algorithm of Bateni et al. that uses $O(k^2 \log^4{n} \log^2{\frac{\log{n}}{\epsilon}})$ strong-oracle queries. To show the effectiveness of our algorithms, we perform empirical evaluations on real-world datasets and show that our algorithms significantly outperform the algorithms of Bateni et al.

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arXiv2025

Distribution Testing Meets Sum Estimation

We study the problem of estimating the sum of $n$ elements, each with weight $w(i)$, in a structured universe. Our goal is to estimate $W = \sum_{i=1}^n w(i)$ within a $(1 \pm \epsilon)$ factor using a sublinear number of samples, assuming weights are non-increasing, i.e., $w(1) \geq w(2) \geq \dots \geq w(n)$. The sum estimation problem is well-studied under different access models to the universe $U$. However, to the best of our knowledge, nothing is known about the sum estimation problem using non-adaptive conditional sampling. In this work, we explore the sum estimation problem using non-adaptive conditional weighted and non-adaptive conditional uniform samples, assuming that the underlying distribution ($D(i)=w(i)/W$) is monotone. We also extend our approach to to the case where the underlying distribution of $U$ is unimodal. Additionally, we consider support size estimation when $w(i) = 0$ or $w(i) \geq W/n$, using hybrid sampling (both weighted and uniform) to access $U$. We propose an algorithm to estimate $W$ under the non-increasing weight assumption, using $O(\frac{1}{\epsilon^3} \log{n} + \frac{1}{\epsilon^6})$ non-adaptive weighted conditional samples and $O(\frac{1}{\epsilon^3} \log{n})$ uniform conditional samples. Our algorithm matches the $\Omega(\log{n})$ lower bound by \cite{ACK15}. For unimodal distributions, the sample complexity remains similar, with an additional $O(\log{n})$ evaluation queries to locate the minimum weighted point in the domain. For estimating the support size $k$ of $U$, where weights are either $0$ or at least $W/n$, our algorithm uses $O\big( \frac{\log^3(n/\epsilon)}{\epsilon^8} \cdot \log^4 \frac{\log(n/\epsilon)}{\epsilon} \big)$ uniform samples and $O\big( \frac{\log(n/\epsilon)}{\epsilon^2} \cdot \log \frac{\log(n/\epsilon)}{\epsilon} \big)$ weighted samples to output $\hat{k}$ satisfying $k - 2\epsilon n \leq \hat{k} \leq k + \epsilon n$.

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arXiv2025

Improved Sublinear-time Moment Estimation using Weighted Sampling

In this work we study the {\it moment estimation} problem using weighted sampling. Given sample access to a set $A$ with $n$ weighted elements, and a parameter $t>0$, we estimate the $t$-th moment of $A$ given as $S_t=\sum_{a\in A} w(a)^t$. For t=1, this is the sum estimation problem. The moment estimation problem along with a number of its variants have been extensively studied in streaming, sublinear and distributed communication models. Despite being well studied, we don't yet have a complete understanding of the sample complexity of the moment estimation problem in the sublinear model and in this work, we make progress on this front. On the algorithmic side, our upper bounds match the known upper bounds for the problem for $t>1$. To the best of our knowledge, no sublinear algorithms were known for this problem for $0 1/2$ and show that no sublinear algorithms exist for $t\leq 1/2$. We prove a $\Omega(\frac{n^{1-1/t}\ln 1/\delta}{\epsilon^2})$ lower bound for moment estimation for $t>1$, and show optimal sample complexity bound $\Theta(\frac{n^{1-1/t}\ln 1/\delta}{\epsilon^2})$ for moment estimation for $t\geq 2$. Hence, we obtain a complete understanding of the sample complexity for moment estimation using proportional sampling for $t\geq 2$. We also study the moment estimation problem in the beyond worst-case analysis paradigm and identify a new {\it moment-density} parameter of the input that characterizes the sample complexity of the problem using proportional sampling and derive tight sample complexity bounds with respect to that parameter. We also study the moment estimation problem in the hybrid sampling framework in which one is given additional access to a uniform sampling oracle and show that hybrid sampling framework does not provide any additional gain over the proportional sampling oracle in the worst case.

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