Approximation of smallest linear tree grammar

Jeż, Artur; Lohrey, Markus

doi:10.1016/j.ic.2016.09.007

Cited by 20 publications

(17 citation statements)

References 35 publications

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“…Note that for the size of a TSLP we do not count nodes of right-hand sides that are labelled with a parameter. To justify this, we use the following internal representation of valid patterns (which is also used in [28]): For every non-parameter node v of a tree, with children v 1 , . .…”

Section: Trees and Tree Straight-line Programsmentioning

confidence: 99%

“…For strings, such a result was obtained in [29] using the fact that every string of length n over a constant size alphabet has an SLP of size O n log n . It would be interesting to know the worst-case output size of the grammar-based tree compressor from [28]. It works in linear time and produces a TSLP that is only by a factor O(log n) larger than an optimal TSLP.…”

Section: Future Workmentioning

confidence: 99%

“…Whereas dags only allow to share repeated subtrees, TSLPs can also share repeated internal tree patterns. Several grammar-based tree compressors were developed in [2,8,11,28,37], where the work from [2] is based on another type of tree grammars (elementary ordered tree grammars). The algorithm from [28] achieves an approximation ratio of O(log n) (for a constant set of node labels).…”

Section: Introductionmentioning

confidence: 99%

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Constructing small tree grammars and small circuits for formulas

Ganardi

Hucke

Je³

et al. 2017

Journal of Computer and System Sciences

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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 n log σ n = O n log σ log n many hierarchically defined pieces. Formally, such a hierarchical decomposition has the form of a straight-line linear context-free tree grammar of size O n log σ n , 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. [6], for which only the weaker upper bound of O n log 0.19 σ n (which was very recently improved to O n·log log σ n log σ n [23]) 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 ≤ n different variables occur, can be transformed (in linear time as well as in logspace) into an arithmetical circuit of size O n·log m log n 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). A short version of this paper appeared as [24]. * The fourth and fifth author are supported by the DFG research project QUANT-KOMP (Lo 748/10-1). †

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Section: Trees and Tree Straight-line Programsmentioning

confidence: 99%

Section: Future Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Constructing small tree grammars and small circuits for formulas

Ganardi

Hucke

Je³

et al. 2017

Journal of Computer and System Sciences

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“…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%

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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“…Motivated by applications where large tree structures occur, like XML processing, SLPs have been extended to node-labelled ranked ordered trees [4,5,12,15,18]. In those papers, straight-line linear context-free tree grammars are used.…”

Section: Introductionmentioning

confidence: 99%