Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2206.07554

Hierarchical Clustering in Graph Streams: Single-Pass Algorithms and Space Lower Bounds

Sepehr Assadi·Vaggos Chatziafratis·Jakub Łącki·Vahab Mirrokni·Chen Wang
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Hierarchical Clustering (HC) problem consists of building a hierarchy of clusters to represent a given dataset. Motivated by the modern large-scale applications, we study the problem in the \streaming model, in which the memory is heavily limited and only a single or very few passes over the input are allowed. Specifically, we investigate whether a good hierarchical clustering can be obtained, or at least whether we can approximately estimate the value of the optimal hierarchy. To measure the quality of a hierarchy, we use the HC minimization objective introduced by Dasgupta. Assuming that the input is an $n$-vertex weighted graph whose edges arrive in a stream, we derive the following results on space-vs-accuracy tradeoffs: * With $O(n\cdot \text{polylog}\,{n})$ space, we develop a single-pass algorithm, whose approximation ratio matches the currently best offline algorithm. * When the space is more limited, namely, $n^{1-o(1)}$, we prove that no algorithm can even estimate the value of optimum HC tree to within an $o(\frac{\log{n}}{\log\log{n}})$ factor, even when allowed $\text{polylog}{\,{n}}$ passes over the input. * In the most stringent setting of $\text{polylog}\,{n}$ space, we rule out algorithms that can even distinguish between "highly"-vs-"poorly" clusterable graphs, namely, graphs that have an $n^{1/2-o(1)}$ factor gap between their HC objective value. * Finally, we prove that any single-pass streaming algorithm that computes an optimal HC tree requires to store almost the entire input even if allowed exponential time. Our algorithmic results establish a general structural result that proves that cut sparsifiers of input graph can preserve cost of "balanced" HC trees to within a constant factor. Our lower bound results include a new streaming lower bound for a novel problem "One-vs-Many-Expanders", which can be of independent interest.

Explore related subjects

cs.DS

Keep this discovery

BibTeXRIS

Sepehr Assadi, Vaggos Chatziafratis, Jakub Łącki, Vahab Mirrokni, Chen Wang. 2022-06-15. Hierarchical Clustering in Graph Streams: Single-Pass Algorithms and Space Lower Bounds. https://arxiv.org/abs/2206.07554

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗
arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗