arXiv subjects

Jianing Lou

Publications and source records attributed to Jianing Lou.

7 recordsLinked to original sources

arXiv2026

Round-efficient Fully-scalable MPC algorithms for k-Means

We study Euclidean $k$-Means under the Massively Parallel Computation (MPC) model, focusing on the \emph{fully-scalable} setting. Our main result is a fully-scalable $O((\log n/\log\log n)^2)$-approximation in $O(1)$ rounds. Previously, fully-scalable algorithms for $k$-Means either run in super-constant $O(\log\log n \cdot \log\log\log n)$ rounds, albeit with a better $O(1)$-approximation [Cohen-Addad et al., SODA'26], or suffer from bicriteria guarantees [Bhaskara and Wijewardena, ICML'18; Czumaj et al., ICALP'24]. Our algorithm also gives an $O(\log n/\log\log n)$-approximation for $k$-Median, which improves a recent $O(\log n)$-approximation [Goranci et al., SODA'26], and this $o(\log n)$ ratio breaks the fundamental barrier of tree embedding methods used therein. Our main technical contribution is a new variant of the MP algorithm [Mettu and Plaxton, SICOMP'03] that works for general metrics, whose new guarantee is the Lagrangian Multiplier Preserving (LMP) property, which, importantly, holds even under arbitrary distance distortions. Allowing distance distortion is crucial for efficient MPC implementations and useful for efficient algorithm design in general, whereas preserving the LMP property under distance distortion is known to be a significant technical challenge. As a byproduct of our techniques, we also obtain an $O(1)$-approximation to the optimal \emph{value} in $O(1)$ rounds, which conceptually suggests that achieving a true $O(1)$-approximation (for the solution) in $O(1)$ rounds may be a sensible goal for future study.

cs.DS↗

arXiv2025

Fully Dynamic Euclidean k-Means

We consider the Euclidean $k$-means clustering problem in a dynamic setting, where we have to explicitly maintain a solution (a set of $k$ centers) $S \subseteq \mathbb{R}^d$ subject to point insertions/deletions in $\mathbb{R}^d$. We present a dynamic algorithm for Euclidean $k$-means with $\mathrm{poly}(1/\epsilon)$-approximation ratio, $\tilde{O}(k^{\epsilon})$ update time, and $\tilde{O}(1)$ recourse, for any $\epsilon \in (0,1)$, even when $d$ and $k$ are both part of the input. This is the first algorithm to achieve a constant ratio with $o(k)$ update time for this problem, whereas the previous $O(1)$-approximation runs in $\tilde O(k)$ update time [Bhattacharya, Costa, Farokhnejad; STOC'25]. In fact, previous algorithms cannot go beyond $O(k)$ update time precisely because they are designed for general metrics where an $\Omega(k)$ lower bound is known. We break this $O(k)$ barrier by devising new fundamental data structures to utilize Euclidean properties: a structure that (implicitly) maintains a clustering subject to both center and data point updates, and a range query structure that can evaluate a mergeable function over any metric ball range given as a query. To obtain these structures, we devise the first consistent hashing scheme [Czumaj, Jiang, Krauthgamer, Vesel{\'{y}}, Yang; FOCS'22] that achieves $\tilde O(n^{\epsilon})$ running time per point evaluation with competitive parameters. Our final algorithm exploits the framework of [Bhattacharya, Costa, Farokhnejad; STOC'25] for general metrics. The key change is to redesign several critical subroutines so that they reduce to our new Euclidean data structures, replacing the general-metric implementations that are unlikely to run efficiently even when Euclidean properties are provided.

cs.DS↗

arXiv2025

Local Search for Clustering in Almost-linear Time

We propose the first \emph{local search} algorithm for Euclidean clustering that attains an $O(1)$-approximation in almost-linear time. Specifically, for Euclidean $k$-Means, our algorithm achieves an $O(c)$-approximation in $\tilde{O}(n^{1 + 1 / c})$ time, for any constant $c \ge 1$, maintaining the same running time as the previous (non-local-search-based) approach [la Tour and Saulpic, arXiv'2407.11217] while improving the approximation factor from $O(c^{6})$ to $O(c)$. The algorithm generalizes to any metric space with sparse spanners, delivering efficient constant approximation in $\ell_p$ metrics, doubling metrics, Jaccard metrics, etc. This generality derives from our main technical contribution: a local search algorithm on general graphs that obtains an $O(1)$-approximation in almost-linear time. We establish this through a new $1$-swap local search framework featuring a novel swap selection rule. At a high level, this rule ``scores'' every possible swap, based on both its modification to the clustering and its improvement to the clustering objective, and then selects those high-scoring swaps. To implement this, we design a new data structure for maintaining approximate nearest neighbors with amortized guarantees tailored to our framework.

cs.DS↗

arXiv2025

Coresets for Robust Clustering via Black-box Reductions to Vanilla Case

We devise $ε$-coresets for robust $(k,z)$-Clustering with $m$ outliers through black-box reductions to vanilla case. Given an $ε$-coreset construction for vanilla clustering with size $N$, we construct coresets of size $N\cdot \mathrm{poly}\log(kmε^{-1}) + O_z\left(\min\{kmε^{-1}, mε^{-2z}\log^z(kmε^{-1}) \}\right)$ for various metric spaces, where $O_z$ hides $2^{O(z\log z)}$ factors. This increases the size of the vanilla coreset by a small multiplicative factor of $\mathrm{poly}\log(kmε^{-1})$, and the additive term is up to a $(ε^{-1}\log (km))^{O(z)}$ factor to the size of the optimal robust coreset. Plugging in vanilla coreset results of [Cohen-Addad et al., STOC'21], we obtain the first coresets for $(k,z)$-Clustering with $m$ outliers with size near-linear in $k$ while previous results have size at least $Ω(k^2)$ [Huang et al., ICLR'23; Huang et al., SODA'25]. Technically, we establish two conditions under which a vanilla coreset is as well a robust coreset. The first condition requires the dataset to satisfy special structures - it can be broken into "dense" parts with bounded diameter. We combine this with a new bounded-diameter decomposition that has only $O_z(km ε^{-1})$ non-dense points to obtain the $O_z(km ε^{-1})$ additive bound. Another condition requires the vanilla coreset to possess an extra size-preserving property. We further give a black-box reduction that turns a vanilla coreset to the one satisfying the said size-preserving property, leading to the alternative $O_z(mε^{-2z}\log^{z}(kmε^{-1}))$ additive bound. We also implement our reductions in the dynamic streaming setting and obtain the first streaming algorithms for $k$-Median and $k$-Means with $m$ outliers, using space $\tilde{O}(k+m)\cdot\mathrm{poly}(dε^{-1}\logΔ)$ for inputs on the grid $[Δ]^d$.

cs.DS↗

arXiv2024

Coresets for Kernel Clustering

We devise coresets for kernel $k$-Means with a general kernel, and use them to obtain new, more efficient, algorithms. Kernel $k$-Means has superior clustering capability compared to classical $k$-Means, particularly when clusters are non-linearly separable, but it also introduces significant computational challenges. We address this computational issue by constructing a coreset, which is a reduced dataset that accurately preserves the clustering costs. Our main result is a coreset for kernel $k$-Means that works for a general kernel and has size $\mathrm{poly}(kε^{-1})$. Our new coreset both generalizes and greatly improves all previous results; moreover, it can be constructed in time near-linear in $n$. This result immediately implies new algorithms for kernel $k$-Means, such as a $(1+ε)$-approximation in time near-linear in $n$, and a streaming algorithm using space and update time $\mathrm{poly}(k ε^{-1} \log n)$. We validate our coreset on various datasets with different kernels. Our coreset performs consistently well, achieving small errors while using very few points. We show that our coresets can speed up kernel $k$-Means++ (the kernelized version of the widely used $k$-Means++ algorithm), and we further use this faster kernel $k$-Means++ for spectral clustering. In both applications, we achieve significant speedup and a better asymptotic growth while the error is comparable to baselines that do not use coresets.

cs.DS↗

arXiv2023

The Power of Uniform Sampling for $k$-Median

We study the power of uniform sampling for $k$-Median in various metric spaces. We relate the query complexity for approximating $k$-Median, to a key parameter of the dataset, called the balancedness $β\in (0, 1]$ (with $1$ being perfectly balanced). We show that any algorithm must make $Ω(1 / β)$ queries to the point set in order to achieve $O(1)$-approximation for $k$-Median. This particularly implies existing constructions of coresets, a popular data reduction technique, cannot be query-efficient. On the other hand, we show a simple uniform sample of $\mathrm{poly}(k ε^{-1} β^{-1})$ points suffices for $(1 + ε)$-approximation for $k$-Median for various metric spaces, which nearly matches the lower bound. We conduct experiments to verify that in many real datasets, the balancedness parameter is usually well bounded, and that the uniform sampling performs consistently well even for the case with moderately large balancedness, which justifies that uniform sampling is indeed a viable approach for solving $k$-Median.

cs.DS↗

arXiv2022

Near-optimal Coresets for Robust Clustering

We consider robust clustering problems in $\mathbb{R}^d$, specifically $k$-clustering problems (e.g., $k$-Median and $k$-Means with $m$ outliers, where the cost for a given center set $C \subset \mathbb{R}^d$ aggregates the distances from $C$ to all but the furthest $m$ data points, instead of all points as in classical clustering. We focus on the $ε$-coreset for robust clustering, a small proxy of the dataset that preserves the clustering cost within $ε$-relative error for all center sets. Our main result is an $ε$-coreset of size $O(m + \mathrm{poly}(k ε^{-1}))$ that can be constructed in near-linear time. This significantly improves previous results, which either suffers an exponential dependence on $(m + k)$ [Feldman and Schulman, SODA'12], or has a weaker bi-criteria guarantee [Huang et al., FOCS'18]. Furthermore, we show this dependence in $m$ is nearly-optimal, and the fact that it is isolated from other factors may be crucial for dealing with large number of outliers. We construct our coresets by adapting to the outlier setting a recent framework [Braverman et al., FOCS'22] which was designed for capacity-constrained clustering, overcoming a new challenge that the participating terms in the cost, particularly the excluded $m$ outlier points, are dependent on the center set $C$. We validate our coresets on various datasets, and we observe a superior size-accuracy tradeoff compared with popular baselines including uniform sampling and sensitivity sampling. We also achieve a significant speedup of existing approximation algorithms for robust clustering using our coresets.

cs.DS↗