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### Data Structure Lower Bounds on Random Access to Grammar-Compressed Strings [article]

Shiteng Chen, Elad Verbin, Wei Yu
2012 arXiv   pre-print
All of our lower bounds hold even when the strings are over an alphabet of size 2 and hold even for randomized data structures with 2-sided error.  ...  In this paper we investigate the problem of building a static data structure that represents a string s using space close to its compressed size, and allows fast access to individual characters of s.  ...  Lower Bound for Grammar Random Access In this section we prove the main lower bound for grammar random access. In Section 3.1 we show the main reduction from SD and BLSD.  ...

### Data Structure Lower Bounds on Random Access to Grammar-Compressed Strings [chapter]

2013 Lecture Notes in Computer Science
All of our lower bounds hold even when the strings are over an alphabet of size 2 and hold even for randomized data structures with 2-sided error.  ...  In this paper we investigate the problem of building a static data structure that represents a string s using space close to its compressed size, and allows fast access to individual characters of s.  ...  Lower Bound for Grammar Random Access In this section we prove the main lower bound for grammar random access. In Section 3.1 we show the main reduction from SD and BLSD.  ...

### Optimal Time Random Access to Grammar-Compressed Strings in Small Space [article]

Patrick Hagge Cording
2015 arXiv   pre-print
The random access problem for compressed strings is to build a data structure that efficiently supports accessing the character in position $i$ of a string given in compressed form.  ...  Given a grammar of size $n$ compressing a string of size $N$, we present a data structure using $O(n\Delta \log_\Delta \frac N n \log N)$ bits of space that supports accessing position $i$ in \$O(\log_\  ...  The random access problem on grammar-compressed strings is to build a data structure that efficiently supports accessing S[i].  ...

### Balancing Straight-Line Programs for Strings and Trees [chapter]

Markus Lohrey
2020 Lecture Notes in Computer Science
The talk will explain a recent balancing result according to which a context-free grammar in Chomsky normal form of size m that produces a single string w of length n (such a grammar is also called a straight-line  ...  Similar balancing results can be formulated for various grammar-based tree compression formalism like top DAGs and forest straight-line programs.  ...  We can also obtain an algorithm for the random access problem with access time O(log n/log log n) using a data structure with O(m · log n) words for any > 0; previously this bound was only shown for SLPs  ...

### Finger Search in Grammar-Compressed Strings [article]

Philip Bille, Anders Roy Christiansen, Patrick Hagge Cording, Inge Li Gørtz
2016 arXiv   pre-print
Grammar-based compression, where one replaces a long string by a small context-free grammar that generates the string, is a simple and powerful paradigm that captures many popular compression schemes.  ...  Given a grammar, the random access problem is to compactly represent the grammar while supporting random access, that is, given a position in the original uncompressed string report the character at that  ...  The random access problem is one of the most basic primitives for computation on grammar compressed strings, and solutions to the problem are a key component in a wide range of algorithms and data structures  ...

### Finger Search in Grammar-Compressed Strings

Philip Bille, Anders Roy Christiansen, Patrick Hagge Cording, Inge Li Gørtz
2018 Theory of Computing Systems
Grammar-based compression, where one replaces a long string by a small context-free grammar that generates the string, is a simple and powerful paradigm that captures many popular compression schemes.  ...  Given a grammar, the random access problem is to compactly represent the grammar while supporting random access, that is, given a position in the original uncompressed string report the character at that  ...  The random access problem is one of the most basic primitives for computation on grammar compressed strings, and solutions to the problem are a key component in a wide range of algorithms and data structures  ...

Travis Gagie, Tomohiro I, Giovanni Manzini, Gonzalo Navarro, Hiroshi Sakamoto, Louisa Seelbach Benkner, Yoshimasa Takabatake
2020 arXiv   pre-print
so attractive: the possibility of supporting fast random access.  ...  In a recent paper (SPIRE 2019) we showed how simple pre-processing can dramatically improve those trade-offs, and in this paper we turn our attention to one of the features that make grammar-based compression  ...  Prezza [27] sidestepped Verbin and Yu's lower bound to obtain constant-time random access to T with an O(gn )-space grammar (after Belazzougui et al. [3] achieved that tradeoff with block trees).  ...

### Optimal Rank and Select Queries on Dictionary-Compressed Text [article]

Nicola Prezza
2018 arXiv   pre-print
Recent works showed that random access on S can be supported in optimal O((n/γ)/ n) time within O (γ polylog n ) space.  ...  We also provide matching lower and upper bounds for partial sum and predecessor queries within attractor-bounded space, and extend our lower bounds to encompass navigation of dictionary-compressed tree  ...  Any static data structure taking O(g polylog n) space cannot answer random access queries on S in less than O(log n/ log log n) time.  ...

### A Succinct Grammar Compression [article]

Yasuo Tabei, Yoshimasa Takabatake, Hiroshi Sakamoto
2013 arXiv   pre-print
We solve an open problem related to an optimal encoding of a straight line program (SLP), a canonical form of grammar compression deriving a single string deterministically.  ...  The space is at most 2n log rho(1 + o(1)) bits for rho leq 2sqrtn, while supporting random access to any production rule of an SLP in O(log log n) time.  ...  Such a data structure is called a naming function, and is also necessary for practical grammar compressions.  ...

### A Simple Online Competitive Adaptation of Lempel-Ziv Compression with Efficient Random Access Support

A. Dutta, R. Levi, D. Ron, R. Rubinfeld
2013 2013 Data Compression Conference
We present a simple adaptation of the Lempel Ziv 78' (LZ78) compression scheme (IEEE Transactions on Information Theory, 1978) that supports efficient random access to the input string.  ...  Namely, given query access to the compressed string, it is possible to efficiently recover any symbol of the input string.  ...  We would like to thank Giuseppe Ottaviano for a helpful discussion and two anonymous referees for their constructive comments.  ...

### A Succinct Grammar Compression [chapter]

Yasuo Tabei, Yoshimasa Takabatake, Hiroshi Sakamoto
2013 Lecture Notes in Computer Science
We solve an open problem related to an optimal encoding of a straight line program (SLP), a canonical form of grammar compression deriving a single string deterministically.  ...  The space is at most 2n log ρ(1 + o(1)) bits for ρ ≤ 2 √ n, while supporting random access to any production rule of an SLP in O(log log n) time.  ...  Such a data structure is called a naming function, and is also necessary for practical grammar compressions.  ...

Philip Bille, Gad M. Landau, Rajeev Raman, Kunihiko Sadakane, Srinivasa Rao Satti, Oren Weimann
2013 arXiv   pre-print
To achieve these bounds, we introduce several new techniques and data structures of independent interest, including a predecessor data structure, two "biased" weighted ancestor data structures, and a compact  ...  In this paper, we present a novel grammar representation that allows efficient random access to any character or substring without decompressing the string.  ...  approximate matching grammar compressed strings (giving the bounds of Theorem 2).  ...

### Page 716 of Mathematical Reviews Vol. , Issue 2001A [page]

2001 Mathematical Reviews
Summary: “We investigate a type of lossless source code called a grammar-based code, which, in response to any input data string x over a fixed finite alphabet, selects a context-free grammar G, representing  ...  Redun- dancy bounds for grammar-based codes are established.  ...

### Rank, select and access in grammar-compressed strings [article]

Djamal Belazzougui, Simon J. Puglisi, Yasuo Tabei
2014 arXiv   pre-print
In this paper we describe data structures to support the following operations on a grammar-compressed string: rank_c(S,i) (return the number of occurrences of symbol c before position i in S); select_c  ...  This matches a lower bound stated by Verbin and Yu for strings where N is polynomially related to n.  ...  We note that our upper bound for access matches the lower bound of Verbin and Yu [48] who have shown that for "not-so-compressible" strings -those that have a grammar of length n such that N ≤ n 1+ for  ...

### A simple online competitive adaptation of Lempel-Ziv compression with efficient random access support [article]

Akashnil Dutta, Reut Levi, Dana Ron, Ronitt Rubinfeld
2013 arXiv   pre-print
We present a simple adaptation of the Lempel Ziv 78' (LZ78) compression scheme ( IEEE Transactions on Information Theory, 1978) that supports efficient random access to the input string.  ...  Namely, given query access to the compressed string, it is possible to efficiently recover any symbol of the input string.  ...  We would like to thank Giuseppe Ottaviano for a helpful discussion and two anonymous referees for their constructive comments.  ...
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