A Randomized Parallel Algorithm for Efficiently Finding Near-Optimal Universal Hitting Sets

Abstract

As the volume of next generation sequencing data increases, an urgent need for algorithms to efficiently process the data arises. Universal hitting sets (UHS) were recently introduced as an alternative to the central idea of minimizers in sequence analysis with the hopes that they could more efficiently address common tasks such as computing hash functions for read overlap, sparse suffix arrays, and Bloom filters. A UHS is a set of $k$-mers that hit every sequence of length $L$, and can thus serve as indices to $L$-long sequences. Unfortunately, methods for computing small UHSs are not yet practical for real-world sequencing instances due to their serial and deterministic nature, which leads to long runtimes and high memory demands when handling typical values of $k$ (e.g. $k>13$). To address this bottleneck, we present two algorithmic innovations to significantly decrease runtime while keeping memory usage low: (i) we leverage advanced theoretical and architectural techniques to parallelize and decrease memory usage in calculating $k$-mer hitting numbers; and (ii) we build upon techniques from randomized Set Cover to select universal $k$-mers much faster. We implemented these innovations in PASHA, the first randomized parallel algorithm for generating nearoptimal UHSs, which newly handles $k>13$. We demonstrate empirically that PASHA produces sets only slightly larger than those of serial deterministic algorithms; moreover, the set size is provably guaranteed to be within a small constant factor of the optimal size. PASHA's runtime and memory-usage improvements are orders of magnitude faster than the current best algorithms. We expect our newly-practical construction of UHSs to be adopted in many high-throughput sequence analysis pipelines.

Keywords: Parallelization; Randomization; Universal hitting sets.

Fig. 1.

Runtimes (left) and UHS sizes (divided by 10⁴, right) for values of $k=10$ (A, B), 11 (C, D), 12 (E, F), and 13 (G, H) and $20\le L\le 200$ for the different methods. Note that the y-axes for runtimes are in logarithmic scale.

Fig. 2.

Runtimes (A) and UHS sizes (divided by 10⁶) (B) for $14\le k\le 16$ and $L=100$ for PASHA. Note that the y-axis for runtime is in logarithmic scale.

Fig. 3.

Mean approximate expected density (A), and density on the human reference genome (B) for different methods, for $5\le k\le 16$ and $L=100$. Error bars represent one standard deviation from the mean across 10 random sequences of length 10⁶. Density is the fraction of selected $k$-mer positions over the number of $k$-mers in the sequence.