Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1411.5127

PivotCompress: Compression by Sorting

Oscar Stiffelman
View original source ↗Read source document ↗Export metadata ↓

Abstract

Sorted data is usually easier to compress than unsorted permutations of the same data. This motivates a simple compression scheme: specify the sorted permutation of the data along with a representation of the sorted data compressed recursively. The sorted permutation can be specified by recording the decisions made by quicksort. If the size of the data is known, then the quicksort decisions describe the data at a rate that is nearly as efficient as the minimal prefix-free code for the distribution, which is bounded by the entropy of the distribution. This is possible even though the distribution is unknown ahead of time. Used in this way, quicksort acts as a universal code in that it is asymptotically optimal for any stationary source. The Shannon entropy is a lower bound when describing stochastic, independent symbols. However, it is possible to encode non-uniform, finite strings below the entropy of the sample distribution by also encoding symbol counts because the values in the sequence are no longer independent once the counts are known. The key insight is that sparse quicksort comparison vectors can also be compressed to achieve an even lower rate when data is highly non-uniform while incurring only a modest penalty when data is random.

Explore related subjects

cs.DS

Keep this discovery

BibTeXRIS

Oscar Stiffelman. 2014-11-21. PivotCompress: Compression by Sorting. https://arxiv.org/abs/1411.5127

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↗