On the Complexity of Optimal Grammar-Based Compression

Arpe, Jan; Reischuk, Rüdiger

doi:10.1109/dcc.2006.59

Cited by 9 publications

(8 citation statements)

References 14 publications

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“…A similar question has been considered by Charikar et al [2] and Arpe and Reischuk [3], who show that it is hard to approximate the size of the smallest grammar generating a given word.…”

Section: Introductionmentioning

confidence: 88%

Lower bounds for context-free grammars

Filmus

2011

Information Processing Letters

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“…A similar question has been considered by Charikar et al [2] and Arpe and Reischuk [3], who show that it is hard to approximate the size of the smallest grammar generating a given word.…”

Section: Introductionmentioning

confidence: 88%

Lower bounds for context-free grammars

Filmus

2011

Information Processing Letters

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“…Consequently, since the motivation for the approximation algorithms mentioned above is of a rather practical kind (i. e., string compression in real-world scenarios), this theoretical foundation falls apart (in particular, note that an unbounded alphabet is also necessary for the inapproximability result of [33,34]). One reason for this situation is probably that in [41], it is claimed that the hardness for alphabets of size 3 follows from [10], but a closer look into [10] does not confirm this (we elaborate on this claim in Section 2.4). Consequently, the NP-hardness of the smallest grammar problem for fixed alphabets is essentially open (for well over 30 years, taking [9,10] as the first reference, which investigates hardness and complexity questions).…”

Section: The Smallest Grammar Problemmentioning

confidence: 98%

“…As defined in Section 2.2, the rule-size also takes the number of rules into account. In fact, the literature on grammar-based compression is inconsistent with respect to which kind of size is used, e. g., in [6,8,15,33,34,34,41], the size of a grammar coincides with our definition | • |, while in [4,[53][54][55], the rule-size is used. The rule-size seems to be mainly motivated by the question of how a grammar is encoded as a single string, which, in any reasonable way, requires an additional symbol per rule.…”

Section: Theorem 4 Sgp Opt Is Apx-hard Even For Alphabets Of Size 24mentioning

confidence: 99%

On the Complexity of the Smallest Grammar Problem over Fixed Alphabets

et al. 2020

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In the smallest grammar problem, we are given a word w and we want to compute a preferably small context-free grammar G for the singleton language {w} (where the size of a grammar is the sum of the sizes of its rules, and the size of a rule is measured by the length of its right side). It is known that, for unbounded alphabets, the decision variant of this problem is NP-hard and the optimisation variant does not allow a polynomial-time approximation scheme, unless P = NP. We settle the long-standing open problem whether these hardness results also hold for the more realistic case of a constant-size alphabet. More precisely, it is shown that the smallest grammar problem remains NP-complete (and its optimisation version is APX-hard), even if the alphabet is fixed and has size of at least 17. The corresponding reduction is robust in the sense that it also works for an alternative size-measure of grammars that is commonly used in the literature (i. e., a size measure also taking the number of rules into account), and it also allows to conclude that even computing the number of rules required by a smallest grammar is a hard problem. On the other hand, if the number of nonterminals (or, equivalently, the number of rules) is bounded by a constant, then the smallest grammar problem can be solved in polynomial time, which is shown by encoding it as a problem on graphs with interval structure. However, treating the number of rules as a parameter (in terms of parameterised complexity) yields W[1]-hardness. Furthermore, we present an $\mathcal {O}(3^{\mid {w}\mid })$ O ( 3 ∣ w ∣ ) exact exponential-time algorithm, based on dynamic programming. These three main questions are also investigated for 1-level grammars, i. e., grammars for which only the start rule contains nonterminals on the right side; thus, investigating the impact of the “hierarchical depth” of grammars on the complexity of the smallest grammar problem. In this regard, we obtain for 1-level grammars similar, but slightly stronger results.

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“…It is an open problem whether there is a polynomial time algorithm that computes for a binary string w 2 ¹0; 1º an SLP A such that eval.A/ D w and jAj D opt.w/. Some partial results can be found in [8].…”

Section: Grammar-based Compressionmentioning

confidence: 98%

Algorithmics on SLP-compressed strings: A survey

Lohrey

2012

Groups - Complexity - Cryptology

122

105

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Results on algorithmic problems on strings that are given in a compressed form via straight-line programs are surveyed. A straight-line program is a context-free grammar that generates exactly one string. In this way, exponential compression rates can be achieved. Among others, we study pattern matching for compressed strings, membership problems for compressed strings in various kinds of formal languages, and the problem of querying compressed strings. Applications in combinatorial group theory and computational topology and to the solution of word equations are discussed as well. Finally, extensions to compressed trees and pictures are considered.

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On the Complexity of Optimal Grammar-Based Compression

Cited by 9 publications

References 14 publications

Lower bounds for context-free grammars

Lower bounds for context-free grammars

On the Complexity of the Smallest Grammar Problem over Fixed Alphabets

Algorithmics on SLP-compressed strings: A survey

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