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K-mer-based analysis of genomic data is ubiquitous, but the presence of repetitive k-mers continues to pose problems for the accuracy of many methods. For example, the Mash tool (Ondov et al. 2016) can accurately estimate the substitution rate between two low-repetitive sequences from their k-mer sketches; however, it is inaccurate on repetitive sequences such as the centromere of a human chromosome. Follow-up work by Blanca et al. (2021) has attempted to model how mutations affect k-mer sets based on strong assumptions that the sequence is non-repetitive and that mutations do not create spurious k-mer matches. However, the theoretical foundations for extending an estimator like Mash to work in the presence of repeat sequences have been lacking. In this work, we relax the non-repetitive assumption and propose a novel estimator for the mutation rate. We derive theoretical bounds on our estimator’s bias. Our experiments show that it remains accurate for repetitive genomic sequences, such as the alpha satellite higher order repeats in centromeres. We demonstrate our estimator’s robustness across diverse datasets and various ranges of the substitution rate and k-mer size. Finally, we show how sketching can be used to avoid dealing with large k-mer sets while retaining accuracy. Our software is available at https://github.com/medvedevgroup/Repeat-Aware_Substitution_Rate_Estimator. 

The evolutionary history of a tumor, inferred from single-cell sequencing data, is typically represented as a tree in which each subtree corresponds to a clade of cells seeded by a specific set of mutations. Traditional methods typically identify a single most likely tree for downstream analyses, such as detecting driver mutations, studying mutation co-occurrence patterns and identifying common evolutionary trajectories. However, the reliability of such inferred trees, particularly their topology, clade composition, and mutational placements, often remains uncertain. To quantify this uncertainty, the concept of a Bi-partition Function was recently introduced, providing a probabilistic measure of how reliably a mutation seeds a given clade of cells. The single available algorithm for estimating the Bi-partition Function relies on simplifying assumptions and uses sampling for limited exploration of the tree-space. In this paper, we introduce the first exact algorithm for computing the Bi-partition Function. Our algorithm scales linearly with the number of mutations but exhibits super-exponential complexity with respect to the number of cells. Despite this complexity, it establishes crucial ground truth values, essential for accurately benchmarking and validating approximate methods. Additionally, we present a GPU-accelerated version of the available sampling-based algorithm, significantly boosting the computational performance through large-scale parallelization, enabling more accurate Bi-partition Function estimates via deeper exploration of the tree spaces. We compare our methods on synthetic datasets, demonstrating that especially when the number of mutations sufficiently exceed the number of cells, our GPU-accelerated sampling algorithm closely approximates the exact ground truth values.