Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1412.5302

Optimal-Depth Sorting Networks

Daniel Bundala·Michael Codish·Luís Cruz-Filipe·Peter Schneider-Kamp·Jakub Závodný
View original source ↗Read source document ↗Export metadata ↓

Abstract

We solve a 40-year-old open problem on the depth optimality of sorting networks. In 1973, Donald E. Knuth detailed, in Volume 3 of "The Art of Computer Programming", sorting networks of the smallest depth known at the time for n =< 16 inputs, quoting optimality for n =< 8. In 1989, Parberry proved the optimality of the networks with 9 =< n =< 10 inputs. In this article, we present a general technique for obtaining such optimality results, and use it to prove the optimality of the remaining open cases of 11 =< n =< 16 inputs. We show how to exploit symmetry to construct a small set of two-layer networks on n inputs such that if there is a sorting network on n inputs of a given depth, then there is one whose first layers are in this set. For each network in the resulting set, we construct a propositional formula whose satisfiability is necessary for the existence of a sorting network of a given depth. Using an off-the-shelf SAT solver we show that the sorting networks listed by Knuth are optimal. For n =< 10 inputs, our algorithm is orders of magnitude faster than the prior ones.

Explore related subjects

cs.DScs.DM

Keep this discovery

BibTeXRIS

Daniel Bundala, Michael Codish, Luís Cruz-Filipe, Peter Schneider-Kamp, Jakub Závodný. 2014-12-17. Optimal-Depth Sorting Networks. https://doi.org/10.1016/j.jcss.2016.09.004

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↗