Succinct Representations of Ordinal Trees

Raman, Rajeev; Rao, S. Srinivasa

doi:10.1007/978-3-642-40273-9_20

Cited by 18 publications

(17 citation statements)

References 42 publications

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“…This result has been extended to labelled trees, where (log σ) · n + 2n + o(n) bits suffice when σ is the number of node labels [17]. See [41] for a survey.…”

Section: Introductionmentioning

confidence: 98%

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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“…This result has been extended to labelled trees, where (log σ) · n + 2n + o(n) bits suffice when σ is the number of node labels [17]. See [41] for a survey.…”

Section: Introductionmentioning

confidence: 98%

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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“…4 We have thus shown: Note that γ n is a worst-case bound. The size of the encoding can be less for other values of n 0 .…”

Section: (Along With the Operations Mentioned In §2)mentioning

confidence: 82%

Encoding range minima and range top-2 queries

Davoodi

Navarro

Raman

et al. 2014

Phil. Trans. R. Soc. A.

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We consider the problem of encoding range minimum queries (RMQs): given an array A [1.. n ] of distinct totally ordered values, to pre-process A and create a data structure that can answer the query RMQ( i , j ), which returns the index containing the smallest element in A [ i .. j ], without access to the array A at query time. We give a data structure whose space usage is 2 n + o ( n ) bits, which is asymptotically optimal for worst-case data, and answers RMQs in O (1) worst-case time. This matches the previous result of Fischer and Heun, but is obtained in a more natural way. Furthermore, our result can encode the RMQs of a random array A in 1.919 n + o ( n ) bits in expectation, which is not known to hold for Fischer and Heun’s result. We then generalize our result to the encoding range top-2 query (RT2Q) problem, which is like the encoding RMQ problem except that the query RT2Q( i , j ) returns the indices of both the smallest and second smallest elements of A [ i .. j ]. We introduce a data structure using 3.272 n + o ( n ) bits that answers RT2Qs in constant time, and also give lower bounds on the effective entropy of the RT2Q problem.

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“…A balanced parentheses (BP) representation (there are others [34]) of the topology of an ordinal tree T of t nodes is a binary sequence (or bitvector) P [1, 2t] built as…”

Section: Succinct Tree Representationsmentioning

confidence: 99%

“…GCSTs use grammar compression on the parentheses sequence that represents the suffix tree topology [34], which inherits the repetitiveness of the text collection. While Lempel-Ziv compression is stronger, it does not support easy access to the sequence.…”

Section: Introductionmentioning

confidence: 99%

Faster Repetition-Aware Compressed Suffix Trees Based on Block Trees

Cáceres

Navarro

2019

String Processing and Information Retrieval

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Suffix trees are a fundamental data structure in stringology, but their space usage, though linear, is an important problem for its applications. We design and implement a new compressed suffix tree targeted to highly repetitive texts, such as large genomic collections of the same species. Our suffix tree builds on Block Trees, a recent Lempel-Ziv-bounded data structure that captures the repetitiveness of its input. We use Block Trees to compress the topology of the suffix tree, and augment the Block Tree nodes with data that speeds up suffix tree navigation. Our compressed suffix tree is slightly larger than previous repetition-aware suffix trees based on grammars, but outperforms them in time, often by orders of magnitude. The component that represents the tree topology achieves a speed comparable to that of general-purpose compressed trees, while using 2.3-10 times less space, and might be of interest in other scenarios.

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Succinct Representations of Ordinal Trees

Abstract: Abstract. We survey succinct representations of ordinal, or rooted, ordered trees. Succinct representations use space that is close to the appropriate information-theoretic minimum, but support operations on the tree rapidly, usually in O(1) time.

Cited by 18 publications

References 42 publications

Constructing small tree grammars and small circuits for formulas

Constructing small tree grammars and small circuits for formulas

Encoding range minima and range top-2 queries

Faster Repetition-Aware Compressed Suffix Trees Based on Block Trees

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