Succinct k-mer Sets Using Subset Rank Queries on the Spectral Burrows-Wheeler Transform ^*

Abstract

The k-spectrum of a string is the set of all distinct substrings of length k occurring in the string. This is a lossy but computationally convenient representation of the information in the string, with many applications in high-throughput bioinformatics. In this work, we define the notion of the Spectral Burrows-Wheeler Transform (SBWT), which is a sequence of subsets of the alphabet of the string encoding the k-spectrum of the string. The SBWT is a distillation of the ideas found in the BOSS and Wheeler graph data structures. We explore multiple different approaches to index the SBWT for membership queries on the underlying k-spectrum. We identify subset rank queries as the essential subproblem, and propose four succinct index structures to solve it. One of the approaches essentially leads to the known BOSS data structure, while the other three offer attractive time-space trade-offs and support simpler query algorithms that rely only on fast rank queries. The most general approach involves a novel data structure we call the subset wavelet tree, which we find to be of independent interest. All of the approaches are also amendable to entropy compression, which leads to good space bounds on the sizes of the data structures. Using entropy compression, we show that the SBWT can support membership queries on the k-spectrum of a single string in O(k) time and (n + k)(log σ + 1/ ln 2) + o((n + k)σ) bits of space, where n is the number of distinct substrings of length k in the input and σ is the size of the alphabet. This improves from the time O(k log σ) achieved by the BOSS data structure. We show, via experiments on a range of genomic data sets, that the simplicity of our new indexes translates into large performance gains in practice over prior art.