Linear-time String Indexing and Analysis in Small Space

Abstract

The field of succinct data structures has flourished over the last 16 years. Starting from the compressed suffix array by Grossi and Vitter (STOC 2000) and the FM-index by Ferragina and Manzini (FOCS 2000), a number of generalizations and applications of string indexes based on the Burrows-Wheeler transform (BWT) have been developed, all taking an amount of space that is close to the input size in bits. In many large-scale applications, the construction of the index and its usage need to be considered as one unit of computation. For example, one can compare two genomes by building a common index for their concatenation, and by detecting common substructures by querying the index. Efficient string indexing and analysis in small space lies also at the core of a number of primitives in the data-intensive field of high-throughput DNA sequencing.We report the following advances in string indexing and analysis. We show that the BWT of a string T ∈ {1, . . . , σ} n can be built in deterministic O(n) time using just O(n log σ) bits of space, where σ ≤ n. Deterministic linear time is achieved by exploiting a new partial rank data structure that supports queries in constant time, and that might have independent interest. Within the same time and space budget, we can build an index based on the BWT that allows one to enumerate all the internal nodes of the suffix tree of T . Many fundamental string analysis problems, such as maximal repeats, maximal unique matches, and string kernels, can be mapped to such enumeration, and can thus be solved in deterministic O(n) time and in O(n log σ) bits of space from the input string, by tailoring the enumeration algorithm to some problem-specific computations.We also show how to build many of the existing indexes based on the BWT, such as the compressed suffix array, the compressed suffix tree, and the bidirectional BWT index, in randomized O(n) time and in O(n log σ) bits of space. The previously fastest construction algorithms for BWT, compressed suffix array and compressed suffix tree, which used O(n log σ) bits of space, took O(n log log σ) time for the first two structures, and O(n log ǫ n) time for the third, where ǫ is any positive constant smaller than one. Contrary to the state of the art, our bidirectional BWT index supports every operation in constant time per element in its output.

iGenomics: Comprehensive DNA sequence analysis on your Smartphone

“…iGenomics uses a version of QuickSort, a divide-and-conquer sorting algorithm, because on average it takes O( n log n ) time for n objects to be sorted. Although there are now some more efficient BWT construction algorithms [ 31 ], given that iGenomics is targeted towards relatively small genomes (<100,000 bp), the amount of time for BWT sorting is negligible compared to the time to align the reads. Finally, to obtain the BWT from the sorted array, the final character of each row in the matrix is copied into a string with the first character copied having the first position, the second character copied having the second position, and so forth.…”

Comparative genome analysis using sample-specific string detection in accurate long reads

“…We will also propose a heuristic version of the algorithm that solves a relaxed variant of Problem 1 in linear-time O ( n ). All these complexities are on top of the FMD-index construction [25], which in our case can be done in O ( m ) time and space [5].…”

The genome atlas: navigating a new era of reference genomes

“…time as they are sequenced [39]. As long read sequencing replaces the current technology, the advantages of TGA will increase further, because indexing reads will be quicker [40] and the number of blueprint update steps will decrease.…”