Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1506.03676

From Independence to Expansion and Back Again

Tobias Christiani·Rasmus Pagh·Mikkel Thorup
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider the following fundamental problems: (1) Constructing $k$-independent hash functions with a space-time tradeoff close to Siegel's lower bound. (2) Constructing representations of unbalanced expander graphs having small size and allowing fast computation of the neighbor function. It is not hard to show that these problems are intimately connected in the sense that a good solution to one of them leads to a good solution to the other one. In this paper we exploit this connection to present efficient, recursive constructions of $k$-independent hash functions (and hence expanders with a small representation). While the previously most efficient construction (Thorup, FOCS 2013) needed time quasipolynomial in Siegel's lower bound, our time bound is just a logarithmic factor from the lower bound.

Explore related subjects

cs.DS

Keep this discovery

BibTeXRIS

Tobias Christiani, Rasmus Pagh, Mikkel Thorup. 2015-06-11. From Independence to Expansion and Back Again. https://arxiv.org/abs/1506.03676

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↗