Wavelet Trees Meet Suffix Trees

Babenko, Maxim A.; Gawrychowski, Paweł; Kociumaka, Tomasz; Starikovskaya, Tatiana

doi:10.1137/1.9781611973730.39

Cited by 50 publications

(78 citation statements)

References 33 publications

(65 reference statements)

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“…• We parallelize our new algorithms, obtaining the fastest parallel WT-construction algorithms on medium-sized workstations of up to 32 cores. 1 • In particular, this results in the first practical parallel algorithms for wavelet matrices. A final (theoretical) contribution of this paper is that we show that the wavelet tree and the wavelet matrix are equivalent, in the sense that every algorithm that can compute the former can also compute the latter in the same time with only (n + σ)(1 + o(1)) + (σ + 2) lg n bits of additional space.…”

Section: Our Contributionsmentioning

confidence: 91%

Simple, Fast and Lightweight Parallel Wavelet Tree Construction

Fischer

Kurpicz

Löbel

2018

2018 Proceedings of the Twentieth Workshop on Algorithm Engineering and Experiments (ALENEX)

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The wavelet tree (Grossi et al. [SODA, 2003]) and wavelet matrix (Claude et al. [Inf. Syst., 47:15-32, 2015]) are compact indices for texts over an alphabet [0, σ) that support rank, select and access queries in O(lg σ) time. We first present new practical sequential and parallel algorithms for wavelet tree construction. Their unifying characteristics is that they construct the wavelet tree bottom-up, i. e., they compute the last level first. We also show that this bottom-up construction can easily be adapted to wavelet matrices. In practice, our best sequential algorithm is up to twice as fast as the currently fastest sequential wavelet tree construction algorithm (Shun [DCC, 2015]), simultaneously saving a factor of 2 in space. This scales up to 32 cores, where we are about equally fast as the currently fastest parallel wavelet tree construction algorithm (Labeit et al. [DCC, 2016]), but still use only about 75 % of the space. An additional theoretical result shows how to adapt any wavelet tree construction algorithm to the wavelet matrix in the same (asymptotic) time, using only little extra space.

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Section: Our Contributionsmentioning

confidence: 91%

Simple, Fast and Lightweight Parallel Wavelet Tree Construction

Fischer

Kurpicz

Löbel

2018

2018 Proceedings of the Twentieth Workshop on Algorithm Engineering and Experiments (ALENEX)

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“…We partition L into sublists of elements not exceeding p (L ≤ p ) and larger than p (L > p ) for some pivot value p, recurse on L ≤ p and L > p , and retrieve FirstOcc L from FirstOcc L ≤ p and FirstOcc L > p . This approach is similar to efficient wavelet tree construction for sequences over a small universe; see [3,22].…”

Section: Efficient Element Location In Packed Listsmentioning

confidence: 99%

Efficient Indexes for Jumbled Pattern Matching with Constant-Sized Alphabet

2016

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We introduce efficient indexes for a problem in non-standard stringology: jumbled pattern matching. An index is a data structure constructed for a text of length n over an alphabet of size σ that can answer queries asking if the text contains a fragment which is jumbled (Abelian) equivalent to a pattern, specified by its so-called Parikh vector. We denote the length of the pattern by m. Moosa and Rahman (J Discrete Algorithms 10:5-9, 2012) gave an index for the case of binary alphabets with O n 2 (log n) 2 -time construction in the word-RAM model. Several earlier papers stated as an open problem the existence of an efficient solution for larger alphabets. In this paper we develop an index for any constant-sized alphabet. The construction involves a trade-off parameter, which in particular lets us achieve the following complexities: O(n 2−δ ) space and O(m (2σ −1)δ ) query time for any 0 < δ < 1, or O n 2 (log log n) 2 log n space and polylogarithmic, o(log 2σ −1 m), query time. The construction time in both cases is subquadratic: O n 2 (log log n) 2 log n in the word-RAM model (using bit-parallelism). Our construction algorithms are randomized (Las Vegas, running time w.h.p.), which is due to the usage of perfect hashing. On the other hand, all queries are answered deterministically. A preliminary version of this work appeared at ESA 2013 (Kociumaka et al. in Algorithms, ESA 2013. LNCS, vol 8125. Springer, Berlin, pp. 625-636, 2013 time construction of the index with O(n 2−δ ) space and O(m (2σ −1)δ ) query time, which was not present in the preliminary version. We also extend the index so that the position of the leftmost occurrence of the query pattern is provided at no additional cost in the complexity; this required rather nontrivial changes in the construction algorithm.

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“…Thus, the total size of a wavelet tree is O 2 b + nb log n machine words, which is O nb log n if b ≤ log n. As shown recently, a wavelet tree can be constructed efficiently from the packed representation of W . Theorem 2.5 ( [3,37]). Given the packed representation of a string W of length n over [0 .…”

Section: Wavelet Treesmentioning

confidence: 99%

“…, where X denotes the string-reversal operation. 3 In the word RAM model with word size w = Ω(log n), reversing any O(log n)-bit string takes O(1) time after O(n δ )-time (δ < 1) preprocessing. Thus, W [1 .…”

Section: The Nonperiodic Casementioning

confidence: 99%

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String synchronizing sets: sublinear-time BWT construction and optimal LCE data structure

Kempa

Kociumaka

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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Burrows-Wheeler transform (BWT) is an invertible text transformation that, given a text T of length n, permutes its symbols according to the lexicographic order of suffixes of T . BWT is one of the most heavily studied algorithms in data compression with numerous applications in indexing, sequence analysis, and bioinformatics. Its construction is a bottleneck in many scenarios, and settling the complexity of this task is one of the most important unsolved problems in sequence analysis that has remained open for 25 years. Given a binary string of length n, occupying O(n/ log n) machine words, the BWT construction algorithm due to Hon et al. (SIAM J. Comput., 2009) runs in O(n) time and O(n/ log n) space. Recent advancements (Belazzougui, STOC 2014, and Munro et al., SODA 2017) focus on removing the alphabet-size dependency in the time complexity, but they still require Ω(n) time. Despite the clearly suboptimal running time, the existing techniques appear to have reached their limits.In this paper, we propose the first algorithm that breaks the O(n)-time barrier for BWT construction. Given a binary string of length n, our procedure builds the Burrows-Wheeler transform in O(n/ √ log n) time and O(n/ log n) space. We complement this result with a conditional lower bound proving that any further progress in the time complexity of BWT construction would yield faster algorithms for the very well studied problem of counting inversions: it would improve the state-of-the-art O(m √ log m)-time solution by Chan and Pǎtraşcu (SODA 2010). Our algorithm is based on a novel concept of string synchronizing sets, which is of independent interest. As one of the applications, we show that this technique lets us design a data structure of the optimal size O(n/ log n) that answers Longest Common Extension queries (LCE queries) in O(1) time and, furthermore, can be deterministically constructed in the optimal O(n/ log n) time.

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Wavelet Trees Meet Suffix Trees

Cited by 50 publications

References 33 publications

Simple, Fast and Lightweight Parallel Wavelet Tree Construction

Simple, Fast and Lightweight Parallel Wavelet Tree Construction

Efficient Indexes for Jumbled Pattern Matching with Constant-Sized Alphabet

String synchronizing sets: sublinear-time BWT construction and optimal LCE data structure

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