An improved lower bound for one-dimensional online unit clustering

Jun Kawahara; Koji M. Kobayashi

Home/Archive/arXiv

arXiv · 1502.02422

An improved lower bound for one-dimensional online unit clustering

Jun Kawahara·Koji M. Kobayashi

View original source ↗Read source document ↗Export metadata ↓

Abstract

The online unit clustering problem was proposed by Chan and Zarrabi-Zadeh (WAOA2007 and Theory of Computing Systems 45(3), 2009), which is defined as follows: "Points" are given online in the $d$-dimensional Euclidean space one by one. An algorithm creates a "cluster," which is a $d$-dimensional rectangle. The initial length of each edge of a cluster is 0. An algorithm can extend an edge until it reaches unit length independently of other dimensions. The task of an algorithm is to cover a new given point either by creating a new cluster and assigning it to the point, or by extending edges of an existing cluster created in past times. The goal is to minimize the total number of created clusters. Chan and Zarrabi-Zadeh proposed some method to obtain a competitive algorithm for the $d$-dimensional case using an algorithm for the one-dimensional case, and thus the one-dimensional case has been extensively studied including some variants of the unit clustering problem. In this paper, we show a lower bound of $13/8 = 1.625$ on the competitive ratio of any deterministic online algorithm for the one-dimensional unit clustering, improving the previous lower bound $8/5 (=1.6)$ presented by Epstein and van Stee (WAOA2007 and ACM Transactions on Algorithms 7(1), 2010). Note that Ehmsen and Larsen (SWAT2010 and Theoretical Computer Science, 500, 2013) showed the current best upper bound of $5/3$, and conjectured that the exact competitive ratio in the one-dimensional case may be $13/8$.

Explore related subjects

cs.DS cs.CG

Keep this discovery

BibTeX RIS

Jun Kawahara, Koji M. Kobayashi. 2015-02-09. An improved lower bound for one-dimensional online unit clustering. https://arxiv.org/abs/1502.02422

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

A deterministic $(1+\varepsilon)^n$ approximation for the permanent of a nonnegative matrix

For every fixed $0<\varepsilon\le1$, we give a deterministic strongly polynomial algorithm that, given a nonnegative matrix $A\in\mathbb{R}_{\ge0}^{n\times n}$, returns $Q$ satisfying $\operatorname{per} A\le Q\le(1+\varepsilon)^n\operatorname{per} A$.

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↗