arXiv subjects

Artur Jez

Publications and source records attributed to Artur Jez.

6 recordsLinked to original sources

arXiv2019

The smallest grammar problem revisited

In a seminal paper of Charikar et al. on the smallest grammar problem, the authors derive upper and lower bounds on the approximation ratios for several grammar-based compressors, but in all cases there is a gap between the lower and upper bound. Here the gaps for $\mathsf{LZ78}$ and $\mathsf{BISECTION}$ are closed by showing that the approximation ratio of $\mathsf{LZ78}$ is $Θ( (n/\log n)^{2/3})$, whereas the approximation ratio of $\mathsf{BISECTION}$ is $Θ(\sqrt{n/\log n})$. In addition, the lower bound for $\mathsf{RePair}$ is improved from $Ω(\sqrt{\log n})$ to $Ω(\log n/\log\log n)$. Finally, results of Arpe and Reischuk relating grammar-based compression for arbitrary alphabets and binary alphabets are improved.

cs.DS↗

arXiv2017

Approximation ratio of RePair

In a seminal paper of Charikar et al.~on the smallest grammar problem, the authors derive upper and lower bounds on the approximation ratios for several grammar-based compressors. Here we improve the lower bound for the famous {\sf RePair} algorithm from $Ω(\sqrt{\log n})$ to $Ω(\log n/\log\log n)$. The family of words used in our proof is defined over a binary alphabet, while the lower bound from Charikar et al. needs an alphabet of logarithmic size in the length of the provided words.

cs.DS↗

arXiv2015

Constructing small tree grammars and small circuits for formulas

It is shown that every tree of size $n$ over a fixed set of $σ$ different ranked symbols can be decomposed (in linear time as well as in logspace) into $O\big(\frac{n}{\log_σn}\big) = O\big(\frac{n \log σ}{\log n}\big)$ many hierarchically defined pieces. Formally, such a hierarchical decomposition has the form of a straight-line linear context-free tree grammar of size $O\big(\frac{n}{\log_σn}\big)$, which can be used as a compressed representation of the input tree. This generalizes an analogous result for strings. Previous grammar-based tree compressors were not analyzed for the worst-case size of the computed grammar, except for the top dag of Bille et al., for which only the weaker upper bound of $O\big(\frac{n}{\log_σ^{0.19} n}\big)$ (which was very recently improved to $O\big(\frac{n \cdot \log \log_σn}{\log_σn}\big)$ by Hübschle-Schneider and Raman) for unranked and unlabelled trees has been derived. The main result is used to show that every arithmetical formula of size $n$, in which only $m \leq n$ different variables occur, can be transformed (in linear time as well as in logspace) into an arithmetical circuit of size $O\big(\frac{n \cdot \log m}{\log n}\big)$ and depth $O(\log n)$. This refines a classical result of Brent from 1974, according to which an arithmetical formula of size $n$ can be transformed into a logarithmic depth circuit of size $O(n)$.

cs.DS↗

arXiv2009

Online validation of the pi and pi' failure functions

Let pi_w denote the failure function of the Morris-Pratt algorithm for a word w. In this paper we study the following problem: given an integer array A[1..n], is there a word w over arbitrary alphabet such that A[i]=pi_w[i] for all i? Moreover, what is the minimum required cardinality of the alphabet? We give a real time linear algorithm for this problem in the unit-cost RAM model with Θ(log n) bits word size. Our algorithm returns a word w over minimal alphabet such that pi_w = A as well and uses just o(n) words of memory. Then we consider function pi' instead of pi and give an online O(n log n) algorithm for this case. This is the first polynomial algorithm for online version of this problem.

cs.DS↗

arXiv2008

Generalized Whac-a-Mole

We consider online competitive algorithms for the problem of collecting weighted items from a dynamic set S, when items are added to or deleted from S over time. The objective is to maximize the total weight of collected items. We study the general version, as well as variants with various restrictions, including the following: the uniform case, when all items have the same weight, the decremental sets, when all items are present at the beginning and only deletion operations are allowed, and dynamic queues, where the dynamic set is ordered and only its prefixes can be deleted (with no restriction on insertions). The dynamic queue case is a generalization of bounded-delay packet scheduling (also referred to as buffer management). We present several upper and lower bounds on the competitive ratio for these variants.

cs.DS↗

arXiv2007

Generalized Cauchy identities, trees and multidimensional Brownian motions. Part II: Combinatorial differential calculus

We present an analogue of the differential calculus in which the role of polynomials is played by certain ordered sets and trees. Our combinatorial calculus has all nice features of the usual calculus and has an advantage that the elements of the considered ordered sets might carry some additional information. In this way an analytic proof of generalized Cauchy identities from the previous work of the second author can be directly reformulated in our new language of the combinatorial calculus; furthermore the additional information carried by vertices determines uniquely the bijections presented in Part I of this series.

math.CO↗