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Given the popularity and elegance of k-mer-based tools, finding a space-efficient way to represent a set of k-mers is important for improving the scalability of bioinformatics analyses. One popular approach is to convert the set of k-mers into the more compact set of unitigs. We generalize this approach and formulate it as the problem of finding a smallest spectrum-preserving string set (SPSS) representation. We show that this problem is equivalent to finding a smallest path cover in a compacted de Bruijn graph. Using this reduction, we prove a lower bound on the size of the optimal SPSS and propose a greedy method called UST (Unitig-STitch) that results in a smaller representation than unitigs and is nearly optimal with respect to our lower bound. We demonstrate the usefulness of the SPSS formulation with two applications of UST. The first one is a compression algorithm, UST-Compress, which, we show, can store a set of k-mers by using an order-of-magnitude less disk space than other lossless compression tools. The second one is an exact static k-mer membership index, UST-FM, which, we show, improves index size by 10%-44% compared with other state-of-the-art low-memory indices.

Given the popularity and elegance of k-mer based tools, finding a space-efficient way to represent a set of k-mers is important for improving the scalability of bioinformatics analyses. One popular approach is to convert the set of k-mers into the more compact set of unitigs. We generalize this approach and formulate it as the problem of finding a smallest spectrum-preserving string set (SPSS) representation. We show that this problem is equivalent to finding a smallest path cover in a compacted de Bruijn graph. Using this reduction, we prove a lower bound on the size of the optimal SPSS and propose a greedy method called UST that results in a smaller representation than unitigs and is nearly optimal with respect to our lower bound. We demonstrate the usefulness of the SPSS formulation with two applications of UST. The first one is a compression algorithm, UST-Compress, which we show can store a set of k-mers using an order-of-magnitude less disk space than other lossless compression tools. The second one is an exact static k-mer membership index, UST-FM, which we show improves index size by 10-44% compared to other state-of-the-art low memory indices. Our tool is publicly available at: https://github.com/medvedevgroup/UST/.

K-mer based methods have become prevalent in many areas of bioinformatics. In applications such as database search, they often work with large multi-terabyte-sized datasets. Storing such large datasets is a detriment to tool developers, tool users, and reproducibility efforts. General purpose compressors like gzip, or those designed for read data, are sub-optimal because they do not take into account the specific redundancy pattern in k-mer sets. In our earlier work (Rahman and Medvedev, RECOMB 2020), we presented an algorithm UST-Compress that uses a spectrum-preserving string set representation to compress a set of k-mers to disk. In this paper, we present two improved methods for disk compression of k-mer sets, called ESS-Compress and ESS-Tip-Compress. They use a more relaxed notion of string set representation to further remove redundancy from the representation of UST-Compress. We explore their behavior both theoretically and on real data. We show that they improve the compression sizes achieved by UST-Compress by up to 27 percent, across a breadth of datasets. We also derive lower bounds on how well this type of compression strategy can hope to do.

Recent assemblies by the T2T and VGP consortia have achieved significant accuracy but required a tremendous amount of effort and resources. More typical assembly efforts, on the other hand, still suffer both from misassemblies (joining sequences that should not be adjacent) and from under-assemblies (not joining sequences that should be adjacent). To better understand the common algorithm-driven causes of these limitations, we investigated the unitig algorithm, which is a core algorithm at the heart of most assemblers. We prove that, contrary to popular belief, even when there are no sequencing errors, unitigs are not always safe (i.e. they are not guaranteed to be substrings of the sequenced genome). We also prove that the unitigs of a bidirected de Bruijn graph are different from those of a doubled de Bruijn graph and, contrary to our expectations, result in under-assembly. Using experimental simulations, we then confirm that these two artifacts exist not only in theory but also in the output of widely used assemblers. To the best of our knowledge, this paper is the first to theoretically predict the existence of these assembler artifacts and confirm and measure the extent of their occurrence in practice.

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