On The Maximal Independent Sets Of k-mers With The Edit Distance

Ma Leran, Chen Ke, Shao Mingfu. Arxiv 2023

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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 .

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