Constant-Time Tree Traversal and Subtree Equality Check for Grammar-Compressed Trees

Lohrey, Markus; Maneth, Sebastian; Reh, Carl Philipp

doi:10.1007/s00453-017-0331-3

Cited by 7 publications

(10 citation statements)

References 17 publications

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“…the discussion at the end of the article [18]). The article [18] further generalizes the method to SL tree grammars provided the grammar is given in a normal form, which structures the derivation into "string-like" parts that are branched off from. It is unlikely that this method generalizes to SL-HR grammars for at least two reasons: first there is a major difference in the grammar formalisms.…”

Section: Delay Of Naive Implementation and Other Approachesmentioning

confidence: 98%

“…We would like to have a data structure which allows to carry out any traversal step in constant time, and which can be computed from the grammar in linear (or polynomial) time. This problem was solved for SLPs by Gasieniec et al [11] and was later extended to SL tree grammars [18]. The idea of these solutions is to generate terminal objects only in the leaves of the derivation tree and to represent the leaves of the derivation tree by certain "left-most derivations".…”

Section: Introductionmentioning

confidence: 99%

“…In fact, also for SL tree grammars the left-most-derivation approach can only be applied after bringing the grammar in a particular ("string like") normal form where each nonterminal uses at most one parameter [19]. It remains open in the paper [18] and completely unclear how to apply the left-most-derivation approach to SL tree grammars with nonterminals using several parameters. This is unfortunate because the normal form causes a blow-up in grammar size that can be bounded by a factor of r, where r is the maximal rank of terminal symbols [10,14,19].…”

Section: Introductionmentioning

confidence: 99%

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Constant delay traversal of grammar-compressed graphs with bounded rank

Maneth¹,

Peternek²

2020

Information and Computation

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We present a pointer-based data structure for constant time traversal of the edges of an edge-labeled (alphabet Σ) directed hypergraph (a graph where edges can be incident to more than two vertices, and the incident vertices are ordered) given as hyperedge-replacement grammar G. It is assumed that the grammar has a fixed rank κ (maximal number of vertices connected to a nonterminal hyperedge) and that each vertex of the represented graph is incident to at most one σ-edge per direction (σ ∈ Σ). Precomputing the data structure needs O(|G||Σ|κrh) space and O(|G||Σ|κrh 2 ) time, where h is the height of the derivation tree of G and r is the maximal rank of a terminal edge occurring in the grammar.

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Section: Delay Of Naive Implementation and Other Approachesmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Constant delay traversal of grammar-compressed graphs with bounded rank

Maneth¹,

Peternek²

2020

Information and Computation

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“…The navigation data structure on FSLPs is based on a navigation data structure on (string) SLPs from [18], which extends the data structure from [12] from one-way to two-way navigation. The data structure represents a position 1 ≤ i ≤ |A| in a variable A by a data structure σ(A, i), that we will call pointer, which is a compact representation of the path in the derivation tree from A to the leaf corresponding to position i.…”

Section: Slp Navigationmentioning

confidence: 99%

Compression by Contracting Straight-Line Programs

Ganardi

2021

Preprint

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In grammar-based compression a string is represented by a context-free grammar, also called a straight-line program (SLP), that generates only that string. We refine a recent balancing result stating that one can transform an SLP of size g in linear time into an equivalent SLP of size O(g) so that the height of the unique derivation tree is O(log N ) where N is the length of the represented string (FOCS 2019). We introduce a new class of balanced SLPs, called contracting SLPs, where for every rule A → β1 . . . β k the string length of every variable βi on the right-hand side is smaller by a constant factor than the string length of A. In particular, the derivation tree of a contracting SLP has the property that every subtree has logarithmic height in its leaf size. We show that a given SLP of size g can be transformed in linear time into an equivalent contracting SLP of size O(g) with rules of constant length. This result is complemented by a lower bound, proving that converting SLPs into so called α-balanced SLPs or AVL-grammars can incur an increase by a factor of Ω(log N ).We present an application to the navigation problem in compressed unranked trees, represented by forest straight-line programs (FSLPs). A linear space data structure by Reh and Sieber (2020) supports navigation steps such as going to the parent, left/right sibling, or to the first/last child in constant time. We extend their solution by the operation of moving to the i-th child in time O(log d) where d is the degree of the current node.Contracting SLPs are also applied to the finger search problem over SLP-compressed strings where one wants to access positions near to a pre-specified finger position, ideally in O(log d) time where d is the distance between the accessed position and the finger. We give a linear space solution for the dynamic variant where one can set the finger in O(log N ) time, and then access symbols or move the finger in time O(log d + log (t) N ) for any constant t where log (t) N is the t-fold logarithm of N . This improves a previous solution by Bille, Christiansen, Cording, and Gørtz (2018) with access/move time O(log d + log log N ).

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“…We can also obtain an algorithm for the random access problem with running time O(log n/ log log n) using O(m · log ǫ n) words, previously this bound was only shown for balanced SSLPs [2]. Section 10 contains a list of further applications of Theorem 10.3, which include the following problems on SSLP-compressed strings: rank and select queries [2], subsequence matching [3], computing Karp-Rabin fingerprints [5], computing runs, squares, and palindromes [22], and real-time traversal [18,31]. In all these applications we either improve existing results or significantly simplify existing proofs by replacing depth(G) by O(log n) in time/space bounds.…”

Section: Introductionmentioning

confidence: 99%

Balancing Straight-Line Programs

Ganardi

Jeż

Lohrey

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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We show 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 O(m), whose unique derivation tree has depth O(log |w|). This solves an open problem in the area of grammar-based compression. Similar results are shown for two formalisms 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 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. In particular, this results applies to standard arithmetic circuits over (noncommutative) semirings.

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Constant-Time Tree Traversal and Subtree Equality Check for Grammar-Compressed Trees

Cited by 7 publications

References 17 publications

Constant delay traversal of grammar-compressed graphs with bounded rank

Constant delay traversal of grammar-compressed graphs with bounded rank

Compression by Contracting Straight-Line Programs

Balancing Straight-Line Programs

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