On the Maximal Independent Sets of $k$-mers with the Edit Distance

In computational biology, $k$-mers and edit distance are fundamental concepts. However, little is known about the metric space of all $k$-mers equipped with the edit distance. In this work, we explore the structure of the $k$-mer space by studying its maximal independent sets (MISs). An MIS is a sparse sketch of all $k$-mers with nice theoretical properties, and therefore admits critical applications in clustering, indexing, hashing, and sketching large-scale sequencing data, particularly those with high error-rates. Finding an MIS is a challenging problem, as the size of a $k$-mer space grows geometrically with respect to $k$. We propose three algorithms for this problem. The first and the most intuitive one uses a greedy strategy. The second method implements two techniques to avoid redundant comparisons by taking advantage of the locality-property of the $k$-mer space and the estimated bounds on the edit distance. The last algorithm avoids expensive calculations of the edit distance by translating the edit distance into the shortest path in a specifically designed graph. These algorithms are implemented and the calculated MISs of $k$-mer spaces and their statistical properties are reported and analyzed for $k$ up to 15. Source code is freely available at https://github.com/Shao-Group/kmerspace .