Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2003.06742

Four-Dimensional Dominance Range Reporting in Linear Space

Yakov Nekrich
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we study the four-dimensional dominance range reporting problem and present data structures with linear or almost-linear space usage. Our results can be also used to answer four-dimensional queries that are bounded on five sides. The first data structure presented in this paper uses linear space and answers queries in $O(\log^{1+\varepsilon}n + k\log^{\varepsilon} n)$ time, where $k$ is the number of reported points, $n$ is the number of points in the data structure, and $\varepsilon$ is an arbitrarily small positive constant. Our second data structure uses $O(n \log^{\varepsilon} n)$ space and answers queries in $O(\log n+k)$ time. These are the first data structures for this problem that use linear (resp. $O(n\log^{\varepsilon} n)$) space and answer queries in poly-logarithmic time. For comparison the fastest previously known linear-space or $O(n\log^{\varepsilon} n)$-space data structure supports queries in $O(n^{\varepsilon} + k)$ time (Bentley and Mauer, 1980). Our results can be generalized to $d\ge 4$ dimensions. For example, we can answer $d$-dimensional dominance range reporting queries in $O(\log\log n (\log n/\log\log n)^{d-3} + k)$ time using $O(n\log^{d-4+\varepsilon}n)$ space. Compared to the fastest previously known result (Chan, 2013), our data structure reduces the space usage by $O(\log n)$ without increasing the query time.

Explore related subjects

cs.DScs.CG

Keep this discovery

BibTeXRIS

Yakov Nekrich. 2020-03-15. Four-Dimensional Dominance Range Reporting in Linear Space. https://arxiv.org/abs/2003.06742

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↗