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Author: search.filters.author.Belazzougui, DjamalSubject: search.filters.subject.Maximal repeat

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    Fully-Functional Bidirectional Burrows-Wheeler Indexes and Infinite-Order De Bruijn Graphs
    (Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, 2019-06-18) Belazzougui, Djamal; Cunial, Fabio
    Given a string T on an alphabet of size σ, we describe a bidirectional Burrows-Wheeler index that takes O(|T| log σ) bits of space, and that supports the addition and removal of one character, on the left or right side of any substring of T, in constant time. Previously known data structures that used the same space allowed constant-time addition to any substring of T, but they could support removal only from specific substrings of T. We also describe an index that supports bidirectional addition and removal in O(log log |T|) time, and that takes a number of words proportional to the number of left and right extensions of the maximal repeats of T. We use such fully-functional indexes to implement bidirectional, frequency-aware, variable-order de Bruijn graphs with no upper bound on their order, and supporting natural criteria for increasing and decreasing the order during traversal.
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    Fast Label Extraction in the CDAWG
    (Springer, 2017-09-06) Belazzougui, Djamal; Cunial, Fabio
    The compact directed acyclic word graph (CDAWG) of a string T of length n takes space proportional just to the number e of right extensions of the maximal repeats of T, and it is thus an appealing index for highly repetitive datasets, like collections of genomes from similar species, in which e grows significantly more slowly than n. We reduce from O(m log ⁡log⁡ n) to O(m) the time needed to count the number of occurrences of a pattern of length m, using an existing data structure that takes an amount of space proportional to the size of the CDAWG. This implies a reduction from O(m log log n+occ) to O(m+occ) in the time needed to locate all the occocc occurrences of the pattern. We also reduce from O(k log ⁡log ⁡n) to O(k) the time needed to read the k characters of the label of an edge of the suffix tree of T, and we reduce from O(m log ⁡log ⁡n) to O(m) the time needed to compute the matching statistics between a query of length m and T, using an existing representation of the suffix tree based on the CDAWG. All such improvements derive from extracting the label of a vertex or of an arc of the CDAWG using a straight-line program induced by the reversed CDAWG.
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    A Framework for Space-Efficient String Kernels
    (Springer, 2017-02-17) Belazzougui, Djamal; Cunial, Fabio
    String kernels are typically used to compare genome-scale sequences whose length makes alignment impractical, yet their computation is based on data structures that are either space-inefficient, or incur large slowdowns. We show that a number of exact kernels on pairs of strings of total length n, like the k-mer kernel, the substrings kernels, a number of length-weighted kernels, the minimal absent words kernel, and kernels with Markovian corrections, can all be computed in O(nd) time and in o(n) bits of space in addition to the input, using just a rangeDistinct data structure on the Burrows–Wheeler transform of the input strings that takes O(d) time per element in its output. The same bounds hold for a number of measures of compositional complexity based on multiple values of k, like the k-mer profile and the k-th order empirical entropy, and for calibrating the value of k using the data. All such algorithms become O(n) using a suitable implementation of the rangeDistinct data structure, and by concatenating them to a suitable BWT construction algorithm, we can compute all the mentioned kernels and complexity measures, directly from the input strings, in O(n) time and in O(n log ⁡σ) bits of space in addition to the input, where σ is the size of the alphabet. Using similar data structures, we also show how to build a compact representation of the variable-length Markov chain of a string T of length n, that takes just 3n log ⁡σ+o(n log ⁡σ) bits of space, and that can be learnt in randomized O(n) time using O(n log ⁡σ) bits of space in addition to the input. Such model can then be used to assign a probability to a query string S of length m in O(m) time and in 2m+o(m) bits of additional space, thus providing an alternative, compositional measure of the similarity between S and T that does not require alignment.