Compressed Data Structures for Population-Scale Positional Burrows–Wheeler Transforms

Abstract

The positional Burrows–Wheeler Transform (PBWT) was presented in 2014 by Durbin as a means to find all maximal haplotype matches in h sequences containing w variation sites in O(hw)-time. This time complexity of finding maximal haplotype matches using the PBWT is a significant improvement over the naïve pattern-matching algorithm that requires O(h2w)-time. Compared to the more famous Burrows-Wheeler Transform (BWT), however, a relatively little amount of attention has been paid to the PBWT. This has resulted in less space-efficient data structures for building and storing the PBWT. Given the increasing size of available haplotype datasets, and the applicability of the PBWT to pangenomics, the time is ripe for identifying efficient data structures that can be constructed for large datasets. Here, we present a comprehensive study of the memory footprint of data structures supporting maximal haplotype matching in conjunction with the PBWT. In particular, we present several data structure components that act as building blocks for constructing six different data structures that store the PBWT in a manner that supports efficiently finding the maximal haplotype matches. We estimate the memory usage of the data structures by bounding the space usage with respect to the input size. In light of this experimental analysis, we implement the solutions that are deemed to be superior with respect to the memory usage and show the performance on haplotype datasets taken from the 1000 Genomes Project data.

μ-PBWT: Enabling the Storage and Use of UK Biobank Data on a Commodity Laptop

“…We have a mismatch at column 3 since P [3] ̸ = col(PBWT) 3 [19]. At this point, we can move to either the last character of the previous run, col(PBWT) 3 [17], or the first character of the next run, col(PBWT) 3 [20], having PA 3 [17] = and PA 3 [20] = 18. If we look at the input matrix M, we have that, up to column 3 excluded, row 17 has a common suffix to row 20 longer than row 18.…”

“…We point to the experimental result for illustration of this fact in section 4. Lastly, we refer the reader to Bonizzoni et al [17] for a more thorough evaluation of the data structures for the PBWT that support different time/space tread-offs for SMEM-finding.…”

“…, given a value of PA k in position i, the Φ function returns the preceding value of PA k in position i − 1. Analogously, we can define the inverse of Φ for all 16,1,10,11,17,9,12,13,14,19,20,2,3,4,18,5,6,7,8] and i = 3, we have that Φ 6 (4) = 3 and Φ −1 6 (3) = 18. Gagie et al in [20] showed that the Φ (or Φ −1 ) function for the BWT can be stored in O(r) words, and evaluated in O(log log w (n/r))-time, where w = Ω(log n).…”