Fully Online Grammar Compression in Constant Space

We present novel variants of fully online LCA (FOLCA), a fully online grammar compression that builds a straight line program (SLP) and directly encodes it into a succinct representation in an online manner. FOLCA enables a direct encoding of an SLP into a succinct representation that is asymptotically equivalent to an information theoretic lower bound for representing an SLP (Maruyama et al., SPIRE'13). The compression of FOLCA takes linear time proportional to the length of an input text and its working space depends only on the size of the SLP, which enables us to apply FOLCA to large-scale repetitive texts. Recent repetitive texts, however, include some noise. For example, current sequencing technology has significant error rates, which embeds noise into genome sequences. For such noisy repetitive texts, FOLCA working in the SLP size consumes a large amount of memory. We present two variants of FOLCA working in constant space by leveraging the idea behind stream mining techniques. Experiments using 100 human genomes corresponding to about 300GB from the 1000 human genomes project revealed the applicability of our method to large-scale, noisy repetitive texts.Comment: This is an extended version of a proceeding accepted to Data Compression Conference (DCC), 201

Rank, select and access in grammar-compressed strings

Given a string $S$ of length $N$ on a fixed alphabet of $\sigma$ symbols, a grammar compressor produces a context-free grammar $G$ of size $n$ that generates $S$ and only $S$. In this paper we describe data structures to support the following operations on a grammar-compressed string: \mbox{rank}_c(S,i) (return the number of occurrences of symbol $c$ before position $i$ in $S$); \mbox{select}_c(S,i) (return the position of the $i$th occurrence of $c$ in $S$); and \mbox{access}(S,i,j) (return substring $S[i,j]$). For rank and select we describe data structures of size $O(n\sigma\log N)$ bits that support the two operations in $O(\log N)$ time. We propose another structure that uses $O(n\sigma\log (N/n)(\log N)^{1+\epsilon})$ bits and that supports the two queries in $O(\log N/\log\log N)$, where $\epsilon>0$ is an arbitrary constant. To our knowledge, we are the first to study the asymptotic complexity of rank and select in the grammar-compressed setting, and we provide a hardness result showing that significantly improving the bounds we achieve would imply a major breakthrough on a hard graph-theoretical problem. Our main result for access is a method that requires $O(n\log N)$ bits of space and $O(\log N+m/\log_\sigma N)$ time to extract $m=j-i+1$ consecutive symbols from $S$. Alternatively, we can achieve $O(\log N/\log\log N+m/\log_\sigma N)$ query time using $O(n\log (N/n)(\log N)^{1+\epsilon})$ bits of space. This matches a lower bound stated by Verbin and Yu for strings where $N$ is polynomially related to $n$.Comment: 16 page

siEDM: an efficient string index and search algorithm for edit distance with moves

Although several self-indexes for highly repetitive text collections exist, developing an index and search algorithm with editing operations remains a challenge. Edit distance with moves (EDM) is a string-to-string distance measure that includes substring moves in addition to ordinal editing operations to turn one string into another. Although the problem of computing EDM is intractable, it has a wide range of potential applications, especially in approximate string retrieval. Despite the importance of computing EDM, there has been no efficient method for indexing and searching large text collections based on the EDM measure. We propose the first algorithm, named string index for edit distance with moves (siEDM), for indexing and searching strings with EDM. The siEDM algorithm builds an index structure by leveraging the idea behind the edit sensitive parsing (ESP), an efficient algorithm enabling approximately computing EDM with guarantees of upper and lower bounds for the exact EDM. siEDM efficiently prunes the space for searching query strings by the proposed method, which enables fast query searches with the same guarantee as ESP. We experimentally tested the ability of siEDM to index and search strings on benchmark datasets, and we showed siEDM's efficiency.Comment: 23 page