Space-efficient computation of k-mer dictionaries for large values of k

Abstract

Computing k-mer frequencies in a collection of reads is a common procedure in many genomic applications. Several state-of-the-art k-mer counters rely on hash tables to carry out this task but they are often optimised for small k as a hash table keeping keys explicitly (i.e., k-mer sequences) takes O(Nkw)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(N\\frac{k}{w})$$\\end{document} computer words, N being the number of distinct k-mers and w the computer word size, which is impractical for long values of k. This space usage is an important limitation as analysis of long and accurate HiFi sequencing reads can require larger values of k. We propose Kaarme, a space-efficient hash table for k-mers using O(N+ukw)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(N+u\\frac{k}{w})$$\\end{document} words of space, where u is the number of reads. Our framework exploits the fact that consecutive k-mers overlap by k-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k-1$$\\end{document} symbols. Thus, we only store the last symbol of a k-mer and a pointer within the hash table to a previous one, which we can use to recover the remaining k-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k-1$$\\end{document} symbols. We adapt Kaarme to compute canonical k-mers as well. This variant also uses pointers within the hash table to save space but requires more work to decode the k-mers. Specifically, it takes O(σk)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\\sigma ^{k})$$\\end{document} time in the worst case, σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma$$\\end{document} being the DNA alphabet, but our experiments show this is hardly ever the case. The canonical variant does not improve our theoretical results but greatly reduces space usage in practice while keeping a competitive performance to get the k-mers and their frequencies. We compare canonical Kaarme to a regular hash table storing canonical k-mers explicitly as keys and show that our method uses up to five times less space while being less than 1.5 times slower. We also show that canonical Kaarme uses significantly less memory than state-of-the-art k-mer counters when they do not resort to disk to keep intermediate results.