arXiv subjects

Markus Lohrey

Publications and source records attributed to Markus Lohrey.

At least 37 records · Page 2Linked to original sources

arXiv2022

Low-Latency Sliding Window Algorithms for Formal Languages

Low-latency sliding window algorithms for regular and context-free languages are studied, where latency refers to the worst-case time spent for a single window update or query. For every regular language $L$ it is shown that there exists a constant-latency solution that supports adding and removing symbols independently on both ends of the window (the so-called two-way variable-size model). We prove that this result extends to all visibly pushdown languages. For deterministic 1-counter languages we present a $\mathcal{O}(\log n)$ latency sliding window algorithm for the two-way variable-size model where $n$ refers to the window size. We complement these results with a conditional lower bound: there exists a fixed real-time deterministic context-free language $L$ such that, assuming the OMV (online matrix vector multiplication) conjecture, there is no sliding window algorithm for $L$ with latency $n^{1/2-ε}$ for any $ε>0$, even in the most restricted sliding window model (one-way fixed-size model). The above mentioned results all refer to the unit-cost RAM model with logarithmic word size. For regular languages we also present a refined picture using word sizes $\mathcal{O}(1)$, $\mathcal{O}(\log\log n)$, and $\mathcal{O}(\log n)$.

cs.FL↗

arXiv2022

Membership Problems in Finite Groups

We show that the subset sum problem, the knapsack problem and the rational subset membership problem for permutation groups are NP-complete. Concerning the knapsack problem we obtain NP-completeness for every fixed $n \geq 3$, where $n$ is the number of permutations in the knapsack equation. In other words: membership in products of three cyclic permutation groups is NP-complete. This sharpens a result of Luks, which states NP-completeness of the membership problem for products of three abelian permutation groups. We also consider the context-free membership problem in permutation groups and prove that it is PSPACE-complete but NP-complete for a restricted class of context-free grammars where acyclic derivation trees must have constant Horton-Strahler number. Our upper bounds hold for black box groups. The results for context-free membership problems in permutation groups yield new complexity bounds for various intersection non-emptiness problems for DFAs and a single context-free grammar.

math.GR↗

arXiv2021

Complexity of word problems for HNN-extensions

The computational complexity of the word problem in HNN-extension of groups is studied. HNN-extension is a fundamental construction in combinatorial group theory. It is shown that the word problem for an ascending HNN-extension of a group H is logspace reducible to the so-called compressed word problem for H. The main result of the paper states that the word problem for an HNN-extension of a hyperbolic group H with cyclic associated subgroups can be solved in polynomial time. This result can be easily extended to fundamental groups of graphs of groups with hyperbolic vertex groups and cyclic edge groups.

math.GR↗

arXiv2020

Balancing Straight-Line Programs

It is shown that a context-free grammar of size $m$ that produces a single string $w$ (such a grammar is also called a string straight-line program) can be transformed in linear time into a context-free grammar for $w$ of size $\mathcal{O}(m)$, whose unique derivation tree has depth $\mathcal{O}(\log |w|)$. This solves an open problem in the area of grammar-based compression. Similar results are shown for two formalism for grammar-based tree compression: top dags and forest straight-line programs. These balancing results are all deduced from a single meta theorem stating that the depth of an algebraic circuit over an algebra with a certain finite base property can be reduced to $\mathcal{O}(\log n)$ with the cost of a constant multiplicative size increase. Here, $n$ refers to the size of the unfolding (or unravelling) of the circuit.

cs.DS↗

arXiv2020

Groups with ALOGTIME-hard word problems and PSPACE-complete compressed word problems

We give lower bounds on the complexity of the word problem of certain non-solvable groups: for a large class of non-solvable infinite groups, including in particular free groups, Grigorchuk's group and Thompson's groups, we prove that their word problem is $\mathsf{NC}^1$-hard. For some of these groups (including Grigorchuk's group and Thompson's groups) we prove that the compressed word problem (which is equivalent to the circuit evaluation problem) is $\mathsf{PSPACE}$-complete.

math.GR↗

arXiv2020

A Comparison of Empirical Tree Entropies

Whereas for strings, higher-order empirical entropy is the standard entropy measure, several different notions of empirical entropy for trees have been proposed in the past, notably label entropy, degree entropy, conditional versions of the latter two, and empirical entropy of trees (here, called label-shape entropy). In this paper, we carry out a systematic comparison of these entropy measures. We underpin our theoretical investigations by experimental results with real XML data.

cs.IT↗

arXiv2020

Entropy Bounds for Grammar-Based Tree Compressors

The definition of $k^{th}$-order empirical entropy of strings is extended to node labelled binary trees. A suitable binary encoding of tree straight-line programs (that have been used for grammar-based tree compression before) is shown to yield binary tree encodings of size bounded by the $k^{th}$-order empirical entropy plus some lower order terms. This generalizes recent results for grammar-based string compression to grammar-based tree compression.

cs.DS↗

arXiv2019

Sliding window property testing for regular languages

We study the problem of recognizing regular languages in a variant of the streaming model of computation, called the sliding window model. In this model, we are given a size of the sliding window $n$ and a stream of symbols. At each time instant, we must decide whether the suffix of length $n$ of the current stream ("the active window") belongs to a given regular language. Recent works showed that the space complexity of an optimal deterministic sliding window algorithm for this problem is either constant, logarithmic or linear in the window size $n$ and provided natural language theoretic characterizations of the space complexity classes. Subsequently, those results were extended to randomized algorithms to show that any such algorithm admits either constant, double logarithmic, logarithmic or linear space complexity. In this work, we make an important step forward and combine the sliding window model with the property testing setting, which results in ultra-efficient algorithms for all regular languages. Informally, a sliding window property tester must accept the active window if it belongs to the language and reject it if it is far from the language. We consider deterministic and randomized sliding window property testers with one-sided and two-sided errors. In particular, we show that for any regular language, there is a deterministic sliding window property tester that uses logarithmic space and a randomized sliding window property tester with two-sided error that uses constant space.

cs.DS↗

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↗

arXiv2019

The power word problem

In this work we introduce a new succinct variant of the word problem in a finitely generated group $G$, which we call the power word problem: the input word may contain powers $p^x$, where $p$ is a finite word over generators of $G$ and $x$ is a binary encoded integer. The power word problem is a restriction of the compressed word problem, where the input word is represented by a straight-line program (i.e., an algebraic circuit over $G$). The main result of the paper states that the power word problem for a finitely generated free group $F$ is AC$^0$-Turing-reducible to the word problem for $F$. Moreover, the following hardness result is shown: For a wreath product $G \wr \mathbb{Z}$, where $G$ is either free of rank at least two or finite non-solvable, the power word problem is complete for coNP. This contrasts with the situation where $G$ is abelian: then the power word problem is shown to be in TC$^0$.

math.GR↗

arXiv2019

Knapsack in hyperbolic groups

Recently knapsack problems have been generalized from the integers to arbitrary finitely generated groups. The knapsack problem for a finitely generated group $G$ is the following decision problem: given a tuple $(g, g_1, \ldots, g_k)$ of elements of $G$, are there natural numbers $n_1, \ldots, n_k \in \mathbb{N}$ such that $g = g_1^{n_1} \cdots g_k^{n_k}$ holds in $G$? Myasnikov, Nikolaev, and Ushakov proved that for every (Gromov-)hyperbolic group, the knapsack problem can be solved in polynomial time. In this paper, the precise complexity of the knapsack problem for hyperbolic group is determined: for every hyperbolic group $G$, the knapsack problem belongs to the complexity class $\mathsf{LogCFL}$, and it is $\mathsf{LogCFL}$-complete if $G$ contains a free group of rank two. Moreover, it is shown that for every hyperbolic group $G$ and every tuple $(g, g_1, \ldots, g_k)$ of elements of $G$ the set of all $(n_1, \ldots, n_k) \in \mathbb{N}^k$ such that $g = g_1^{n_1} \cdots g_k^{n_k}$ in $G$ is semilinear and a semilinear representation where all integers are of size polynomial in the total geodesic length of the $g, g_1, \ldots, g_k$ can be computed. Groups with this property are also called knapsack-tame. This enables us to show that knapsack can be solved in $\mathsf{LogCFL}$ for every group that belongs to the closure of hyperbolic groups under free products and direct products with $\mathbb{Z}$.

math.GR↗

arXiv2018

The Complexity of Bisimulation and Simulation on Finite Systems

In this paper the computational complexity of the (bi)simulation problem over restricted graph classes is studied. For trees given as pointer structures or terms the (bi)simulation problem is complete for logarithmic space or NC$^1$, respectively. This solves an open problem from Balcázar, Gabarró, and Sántha. Furthermore, if only one of the input graphs is required to be a tree, the bisimulation (simulation) problem is contained in AC$^1$ (LogCFL). In contrast, it is also shown that the simulation problem is P-complete already for graphs of bounded path-width.

cs.LO↗

arXiv2018

Approximation of smallest linear tree grammar

A simple linear-time algorithm for constructing a linear context-free tree grammar of size O(rg + r g log (n/r g))for a given input tree T of size n is presented, where g is the size of a minimal linear context-free tree grammar for T, and r is the maximal rank of symbols in T (which is a constant in many applications). This is the first example of a grammar-based tree compression algorithm with a good, i.e. logarithmic in terms of the size of the input tree, approximation ratio. The analysis of the algorithm uses an extension of the recompression technique from strings to trees.

cs.DS↗

arXiv2018

Randomized sliding window algorithms for regular languages

A sliding window algorithm receives a stream of symbols and has to output at each time instant a certain value which only depends on the last $n$ symbols. If the algorithm is randomized, then at each time instant it produces an incorrect output with probability at most $ε$, which is a constant error bound. This work proposes a more relaxed definition of correctness which is parameterized by the error bound $ε$ and the failure ratio $ϕ$: A randomized sliding window algorithm is required to err with probability at most $ε$ at a portion of $1-ϕ$ of all time instants of an input stream. This work continues the investigation of sliding window algorithms for regular languages. In previous works a trichotomy theorem was shown for deterministic algorithms: the optimal space complexity is either constant, logarithmic or linear in the window size. The main results of this paper concerns three natural settings (randomized algorithms with failure ratio zero and randomized/deterministic algorithms with bounded failure ratio) and provide natural language theoretic characterizations of the space complexity classes.

cs.FL↗

arXiv2018

Grammar-based Compression of Unranked Trees

We introduce forest straight-line programs (FSLPs) as a compressed representation of unranked ordered node-labelled trees. FSLPs are based on the operations of forest algebra and generalize tree straight-line programs. We compare the succinctness of FSLPs with two other compression schemes for unranked trees: top dags and tree straight-line programs of first-child/next sibling encodings. Efficient translations between these formalisms are provided. Finally, we show that equality of unranked trees in the setting where certain symbols are associative or commutative can be tested in polynomial time. This generalizes previous results for testing isomorphism of compressed unordered ranked trees.

cs.DS↗

arXiv2018

Automata theory on sliding windows

In a recent paper we analyzed the space complexity of streaming algorithms whose goal is to decide membership of a sliding window to a fixed language. For the class of regular languages we proved a space trichotomy theorem: for every regular language the optimal space bound is either constant, logarithmic or linear. In this paper we continue this line of research: We present natural characterizations for the constant and logarithmic space classes and establish tight relationships to the concept of language growth. We also analyze the space complexity with respect to automata size and prove almost matching lower and upper bounds. Finally, we consider the decision problem whether a language given by a DFA/NFA admits a sliding window algorithm using logarithmic/constant space.

cs.FL↗

arXiv2017

Optimal top dag compression

It is shown that for a given ordered node-labelled tree of size $n$ and with $s$ many different node labels, one can construct in linear time a top dag of height $O(\log n)$ and size $O(n / \log_σn) \cap O(d \cdot \log n)$, where $σ= \max\{ 2, s\}$ and $d$ is the size of the minimal dag. The size bound $O(n / \log_σn)$ is optimal and improves on previous bounds.

cs.DS↗

arXiv2017

A universal tree balancing theorem

We present a general framework for balancing expressions (terms) in form of so called tree straight-line programs. The latter can be seen as circuits over the free term algebra extended by contexts (terms with a hole) and the operations which insert terms/contexts into contexts. It is shown that for every term one can compute in DLOGTIME-uniform TC$^0$ a tree straight-line program of logarithmic depth and size $O(n/\log n)$. This allows reducing the term evaluation problem over an arbitrary algebra $\mathcal{A}$ to the term evaluation problem over a derived two-sorted algebra $\mathcal{F}(\mathcal{A})$. Several applications are presented: (i) an alternative proof for a recent result by Krebs, Limaye and Ludwig on the expression evaluation problem is given, (ii) it is shown that expressions for an arbitrary (possibly non-commutative) semiring can be transformed in DLOGTIME-uniform TC$^0$ into equivalent circuits of logarithmic depth and size $O(n/\log n)$, and (iii) a corresponding result for regular expressions is shown.

cs.CC↗