Improved Periodicity Mining in Time Series Databases

Time series data represents information about real world phenomena and periodicity mining explores the interesting periodic behavior that is inherent in the data. Periodicity mining has numerous applications such as in weather forecasting, stock market prediction and analysis, pattern recognition, etc. Recently, the suffix tree, a powerful data structure that efficiently solves many strings related problems has been used to gather information about repeated substrings in the text and then perform periodicity mining. However, periodicity mining deals with large amounts of data which makes it difficult to perform mining in main memory due to the space constraints of the suffix tree. Thus, we first propose the use of the Compressed Suffix Tree (CST) for space efficient periodicity mining in very large datasets. Given the time-space trade-off that comes with any practical usage of the CST, we provide a comprehensive empirical analysis on the practical usage of CSTs and traditional suffix trees for periodicity mining.;Noise is an inherent part of practical time series data, and it is important to mine periods in spite of the noise. This leads to the problem of approximate periodicity mining. Existing algorithms have dealt with the noise introduced between the occurrences of the periodic pattern, but not the noise introduced in the structure of the pattern itself. We present a taxonomy for approximate periodicity and then propose an algorithm that performs periodicity mining in the presence of noise introduced simultaneously in both the structure of the pattern and between the periodic occurrences of the pattern

Composite repetition-aware data structures

In highly repetitive strings, like collections of genomes from the same species, distinct measures of repetition all grow sublinearly in the length of the text, and indexes targeted to such strings typically depend only on one of these measures. We describe two data structures whose size depends on multiple measures of repetition at once, and that provide competitive tradeoffs between the time for counting and reporting all the exact occurrences of a pattern, and the space taken by the structure. The key component of our constructions is the run-length encoded BWT (RLBWT), which takes space proportional to the number of BWT runs: rather than augmenting RLBWT with suffix array samples, we combine it with data structures from LZ77 indexes, which take space proportional to the number of LZ77 factors, and with the compact directed acyclic word graph (CDAWG), which takes space proportional to the number of extensions of maximal repeats. The combination of CDAWG and RLBWT enables also a new representation of the suffix tree, whose size depends again on the number of extensions of maximal repeats, and that is powerful enough to support matching statistics and constant-space traversal.Comment: (the name of the third co-author was inadvertently omitted from previous version

Optimal-Time Text Indexing in BWT-runs Bounded Space

Indexing highly repetitive texts --- such as genomic databases, software repositories and versioned text collections --- has become an important problem since the turn of the millennium. A relevant compressibility measure for repetitive texts is $r$, the number of runs in their Burrows-Wheeler Transform (BWT). One of the earliest indexes for repetitive collections, the Run-Length FM-index, used $O(r)$ space and was able to efficiently count the number of occurrences of a pattern of length $m$ in the text (in loglogarithmic time per pattern symbol, with current techniques). However, it was unable to locate the positions of those occurrences efficiently within a space bounded in terms of $r$. Since then, a number of other indexes with space bounded by other measures of repetitiveness --- the number of phrases in the Lempel-Ziv parse, the size of the smallest grammar generating the text, the size of the smallest automaton recognizing the text factors --- have been proposed for efficiently locating, but not directly counting, the occurrences of a pattern. In this paper we close this long-standing problem, showing how to extend the Run-Length FM-index so that it can locate the $occ$ occurrences efficiently within $O(r)$ space (in loglogarithmic time each), and reaching optimal time $O(m+occ)$ within $O(r\log(n/r))$ space, on a RAM machine of $w=\Omega(\log n)$ bits. Within $O(r\log (n/r))$ space, our index can also count in optimal time $O(m)$. Raising the space to $O(r w\log_\sigma(n/r))$, we support count and locate in $O(m\log(\sigma)/w)$ and $O(m\log(\sigma)/w+occ)$ time, which is optimal in the packed setting and had not been obtained before in compressed space. We also describe a structure using $O(r\log(n/r))$ space that replaces the text and extracts any text substring of length $\ell$ in almost-optimal time $O(\log(n/r)+\ell\log(\sigma)/w)$. (...continues...

Novel Results on the Number of Runs of the Burrows-Wheeler-Transform

The Burrows-Wheeler-Transform (BWT), a reversible string transformation, is one of the fundamental components of many current data structures in string processing. It is central in data compression, as well as in efficient query algorithms for sequence data, such as webpages, genomic and other biological sequences, or indeed any textual data. The BWT lends itself well to compression because its number of equal-letter-runs (usually referred to as $r$) is often considerably lower than that of the original string; in particular, it is well suited for strings with many repeated factors. In fact, much attention has been paid to the $r$ parameter as measure of repetitiveness, especially to evaluate the performance in terms of both space and time of compressed indexing data structures. In this paper, we investigate $\rho(v)$, the ratio of $r$ and of the number of runs of the BWT of the reverse of $v$. Kempa and Kociumaka [FOCS 2020] gave the first non-trivial upper bound as $\rho(v) = O(\log^2(n))$, for any string $v$ of length $n$. However, nothing is known about the tightness of this upper bound. We present infinite families of binary strings for which $\rho(v) = \Theta(\log n)$ holds, thus giving the first non-trivial lower bound on $\rho(n)$, the maximum over all strings of length $n$. Our results suggest that $r$ is not an ideal measure of the repetitiveness of the string, since the number of repeated factors is invariant between the string and its reverse. We believe that there is a more intricate relationship between the number of runs of the BWT and the string's combinatorial properties.Comment: 14 pages, 2 figue