The wavelet matrix: An efficient wavelet tree for large alphabets

Claude, Francisco; Navarro, Gonzalo; Ordóñez, Alberto

doi:10.1016/j.is.2014.06.002

Cited by 59 publications

(75 citation statements)

References 62 publications

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“…Zero-order compression is also obtained, with faster time in practice, by retaining the CM representation but using a tree with Huffman [19] shape instead of a balanced one, which gives n(H 0 (S) + 1)(1 + o(1)) + O(σ log n) bits. The results are called WTH (Huffmanshaped WT) or WMH (Huffman-shaped WM [13]). …”

Section: Sequence Representationsmentioning

confidence: 99%

Grammar Compressed Sequences with Rank/Select Support

Navarro

Ordóñez

2014

String Processing and Information Retrieval

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Abstract. Sequence representations supporting not only direct access to their symbols, but also rank/select operations, are a fundamental building block in many compressed data structures. In several recent applications, the need to represent highly repetitive sequences arises, where statistical compression is ineffective. We introduce grammar-based representations for repetitive sequences, which use up to 10% of the space needed by representations based on statistical compression, and support direct access and rank/select operations within tens of microseconds.

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Section: Sequence Representationsmentioning

confidence: 99%

Grammar Compressed Sequences with Rank/Select Support

Navarro

Ordóñez

2014

String Processing and Information Retrieval

Self Cite

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“…There is an alternative implementation of the wavelet tree called wavelet matrix (Claude et al, 2015) that was specifically proposed in the literature to account for big alphabets. Given an alphabet its size can be extended to match the next power of two, yielding a complete binary tree for the wavelet tree representation.…”

Section: Big Alphabets and The Wavelet Matrixmentioning

confidence: 99%

“…Existing experimental analyses about wavelet trees focus mostly on compression characteristics (Claude et al, 2015). Moreover, they do not consider the time required to build the structure because from the compression point of view the preprocessing time is not the most relevant parameter.…”

Section: Performance Testsmentioning

confidence: 99%

Wavelet Trees for Competitive Programming

CASTRO¹,

Lehmann²,

Pérez³

et al. 2016

IOI

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Abstract.The wavelet tree is a data structure to succinctly represent sequences of elements over a fixed but potentially large alphabet. It is a very versatile data structure which exhibits interesting properties even when its compression capabilities are not considered, efficiently supporting several queries. Although the wavelet tree was proposed more than a decade ago, it has not yet been widely used by the competitive programming community. This paper tries to fill the gap by showing how this data structure can be used in classical competitive programming problems, discussing some implementation details, and presenting a performance analysis focused in a competitive programming setting.

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“…We implemented the kd-tree and the z-order-based KDW-tree in Java. The KDW-tree was built on a practical variant of the wavelet tree called the wavelet matrix [12]. The coordinates that occur in each dataset are sorted in ascending order for each dimension and compressed in sdarray [22], a compressed indexable dictionary, which was used to convert points and query regions from general space to rank space.…”

Section: Settingsmentioning

confidence: 99%

Faster Linear-space Orthogonal Range Searching in Arbitrary Dimensions

Okajima

Maruyama

2014

2015 Proceedings of the Seventeenth Workshop on Algorithm Engineering and Experiments (ALENEX)

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We consider the problem of multi-dimensional orthogonal range searching in linear space for any d dimensions. The kd-tree achieves O(n (d−1)/d ) query time for range counting, which is optimal among bounding-box tree structures, and it has been considered to be the best complexity bound in practice for four decades, while the non-overlapping krange achieves O(n ϵ ) query time in theory. Several twodimensional data structures have better query times than the kd-tree, but have never been generalized to higher dimensions in linear space. In this paper, we propose a new succinct data structure, called the KDW-tree, which requires less space partitioning than the kd-tree and achieves O(n (d−2)/d log n) time for range counting. This is the first succinct data structure that has a lower time complexity than the kd-tree in arbitrary dimensions. In experiments, our data structure significantly outperformed the kd-tree using linear space both for range counting and sum queries in low dimensions for high selectivity.

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The wavelet matrix: An efficient wavelet tree for large alphabets

Cited by 59 publications

References 62 publications

Grammar Compressed Sequences with Rank/Select Support

Grammar Compressed Sequences with Rank/Select Support

Wavelet Trees for Competitive Programming

Faster Linear-space Orthogonal Range Searching in Arbitrary Dimensions

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