SPRISS: approximating frequent k-mers by sampling reads, and applications

Abstract

Motivation

The extraction of k-mers is a fundamental component in many complex analyses of large next-generation sequencing datasets, including reads classification in genomics and the characterization of RNA-seq datasets. The extraction of all k-mers and their frequencies is extremely demanding in terms of running time and memory, owing to the size of the data and to the exponential number of k-mers to be considered. However, in several applications, only frequent k-mers, which are k-mers appearing in a relatively high proportion of the data, are required by the analysis.

Results

In this work, we present SPRISS, a new efficient algorithm to approximate frequent k-mers and their frequencies in next-generation sequencing data. SPRISS uses a simple yet powerful reads sampling scheme, which allows to extract a representative subset of the dataset that can be used, in combination with any k-mer counting algorithm, to perform downstream analyses in a fraction of the time required by the analysis of the whole data, while obtaining comparable answers. Our extensive experimental evaluation demonstrates the efficiency and accuracy of SPRISS in approximating frequent k-mers, and shows that it can be used in various scenarios, such as the comparison of metagenomic datasets, the identification of discriminative k-mers, and SNP (single nucleotide polymorphism) genotyping, to extract insights in a fraction of the time required by the analysis of the whole dataset.

Availability and implementation

SPRISS [a preliminary version (Santoro et al., 2021) of this work was presented at RECOMB 2021] is available at https://github.com/VandinLab/SPRISS.

Supplementary information

Supplementary data are available at Bioinformatics online.

1 Introduction

The study of substrings of length k, or k-mers, is a fundamental task in the analysis of large next-generation sequencing datasets. The extraction of k-mers, and of the frequencies with which they appear in a dataset of reads, is a crucial step in several applications, including the comparison of datasets and reads classification in metagenomics (Wood and Salzberg, 2014), the characterization of variation in RNA-seq data (Audoux et al., 2017), the analysis of structural changes in genomes (Liu et al., 2017; Li and Waterman, 2003), RNA-seq quantification (Patro et al., 2014; Zhang and Wang, 2014), fast search-by-sequence over large high-throughput sequencing repositories (Solomon and Kingsford, 2016), genome comparison (Sims et al., 2009) and error correction for genome assembly (Kelley et al., 2010; Salmela et al., 2016).

k-mers and their frequencies can be obtained with a linear scan of a dataset. However, due to the massive size of the modern datasets and the exponential growth of the k-mers number (with respect to k), the extraction of k-mers is an extremely computationally intensive task, both in terms of running time and memory (Elworth et al., 2020), and several algorithms have been proposed to reduce the running time and memory requirements (see Section 1.2). Nonetheless, the extraction of all k-mers and their frequencies from a reads dataset is still highly demanding in terms of time and memory [e.g. KMC 3 (Kokot et al., 2017), one of the currently best performing tools for k-mer counting, requires more than 2.5 hours, 34 GB of memory and 500 GB of space on disk on a sequence of 729 Gbases (Kokot et al., 2017), and from our experiments more than 30 minutes, 300 GB of memory and 97 GB of disk space for counting k-mers from Mo17 dataset (Using k = 31, 32 workers, and maximum RAM of 350 GB. See Supplementary Table S2 for the size of Mo17.)].

While some applications, such as error correction (Kelley et al., 2010; Salmela et al., 2016) or reads classification (Wood and Salzberg, 2014), require to identify all k-mers, even the ones that appear only once or few times in a dataset, other analyses, such as the comparison of abundances in metagenomic datasets (Benoit et al., 2016; Danovaro et al., 2017; Dickson et al., 2017; Pellegrina et al., 2020) or the discovery of k-mers discriminating between two datasets (Liu et al., 2017; Ounit et al., 2015), hinge on the identification of frequent k-mers, which are k-mers appearing with a (relatively) high frequency in a dataset. For the latter analyses, tools capable of efficiently extracting frequent k-mers only would be extremely beneficial and much more efficient than tools reporting all k-mers (given that a large fraction of k-mers appear with extremely low frequency). However, the efficient identification of frequent k-mers and their frequencies is still relatively unexplored (see Section 1.2).

A natural approach to speed-up the identification of frequent k-mers is to analyze only a sample of the data, since frequent k-mers appear with high probability in a sample, while unfrequent k-mers appear with lower probability. A major challenge in sampling approaches is how to rigorously relate the results obtained analyzing the sample and the results that would be obtained analyzing the whole dataset. Tackling such challenge requires to identify a minimum sample size which guarantees that the results on the sample well represent the results to be obtained on the whole dataset. An additional challenge in the use of sampling for the identification of frequent k-mers is due to the fact that, for values of k of interest in modern applications (e.g. $k\in [20,60]$), even the most frequent k-mers appear in a relatively low portion of the data (e.g. 10⁻⁷–10⁻⁵). The net effect is that the application of standard sampling techniques to rigorously approximate frequent k-mers results in sample sizes larger than the initial dataset.

1.1 Our contributions

In this work, we study the problem of approximating frequent k-mers in a dataset of reads. In this regard, our contributions are:

• We propose $\mathrm{SPRISS}$, SamPling Reads algorIthm to eStimate frequent k-merS (https://vec.wikipedia.org/wiki/Spriss). $\mathrm{SPRISS}$ is based on a simple yet powerful read sampling approach, which renders $\mathrm{SPRISS}$ very flexible and suitable to be used in combination with any k-mer counter. In fact, the read sampling scheme of $\mathrm{SPRISS}$ returns a subset of a dataset of reads, which can be used to obtain representative results for down-stream analyses based on frequent k-mers.

• We prove that $\mathrm{SPRISS}$ provides rigorous guarantees on the quality of the approximation of the frequent k-mers. In this regard, our main technical contribution is the derivation of the sample size required by $\mathrm{SPRISS}$, obtained through the study of the pseudodimension (Pollard, 1984), a key concept from statistical learning theory, of k-mers in reads.

• We show on several real datasets that $\mathrm{SPRISS}$ approximates frequent k-mers with high accuracy, while requiring a fraction of the time needed by approaches that analyze all k-mers in a dataset.

• We show the benefits of using the approximation of frequent k-mers obtained by $\mathrm{SPRISS}$ in three applications: the comparison of metagenomic datasets, the extraction of discriminative k-mers and SNP genotyping. In all these applications, $\mathrm{SPRISS}$ significantly speeds up the analysis, while providing the same insights obtained by the analysis of the whole data.

1.2 Related works

The problem of exactly counting k-mers in datasets has been extensively studied, with several methods proposed for its solution (Audano and Vannberg, 2014; Kokot et al., 2017; Kurtz et al., 2008; Marçais and Kingsford, 2011; Melsted and Pritchard, 2011; Pandey et al., 2017; Rizk et al., 2013; Roy et al., 2014). Such methods are typically highly demanding in terms of time and memory when analyzing large high-throughput sequencing datasets (Elworth et al., 2020). For this reason, many methods have been recently developed to compute approximations of the k-mers abundances to reduce the computational cost of the task (e.g. Chikhi and Medvedev, 2014; Melsted and Halldórsson, 2014; Mohamadi et al., 2017; Pandey et al., 2017; Sivadasan et al., 2016; Zhang et al., 2014). However, such methods do not provide guarantees on the accuracy of their approximations that are simultaneously valid for all (or the most frequent) k-mers. In recent years, other problems closely related to the task of counting k-mers have been studied, including how to efficiently index (Harris and Medvedev, 2020; Marchet et al., 2020a,b; Pandey et al., 2018), represent (Almodaresi et al., 2018; Chikhi et al., 2014; Dadi et al., 2018; Guo et al., 2021; Holley and Melsted, 2020; Marchet et al., 2019; Rahman and Medvedev, 2020), query (Bradley et al., 2019; Marchet et al., 2021; Solomon and Kingsford, 2016, 2018; Sun et al., 2018; Yu et al., 2018) and store (Hernaez et al., 2019; Hosseini et al., 2016; Numanagić et al., 2016; Rahman et al., 2021) the massive collections of sequences or of k-mers that are extracted from the data. See also Chikhi et al. (2021) for a unified presentation of methods to store and query a set of k-mers.

A natural approach to reduce computational demands is to analyze a small sample instead of the entire dataset. To this end, methods that perform a downsampling of massive datasets have been recently proposed (Brown et al., 2012; Coleman et al., 2019; Wedemeyer et al., 2017). These methods focus on discarding reads of the datasets that are very similar to the reads already included in the sample, computing approximate similarity measures as each read is considered. Such measures (i.e. the Jaccard similarity) are designed to maximize the diversity of the content of the reads in the sample. This approach is well suited for applications where rare k-mers are important, but they are less relevant for analyses, of interest to this work, where the most frequent k-mers carry the major part of the information. Furthermore, these methods have a heuristic nature, and do not provide guarantees on the relation between the accuracy of the analysis performed on the sample w.r.t. the analysis performed on the entire dataset. SAKEIMA (Pellegrina et al., 2020) is the first sampling method that provides an approximation of the set of frequent k-mers (together with their estimated frequencies) with rigorous guarantees, based on counting only a subset of all occurrences of k-mers, chosen at random. SAKEIMA performs a full scan of the entire dataset, in a streaming fashion, and processes each k-mer occurrence according to the outcome of its random choices. $\mathrm{SPRISS}$, the algorithm we present in this work, is instead the first sampling algorithm to approximate frequent k-mers (and their frequencies), with rigorous guarantees, by sampling reads from the dataset. In fact, $\mathrm{SPRISS}$ does not require to receive in input and to scan the entire dataset, but, instead, it needs in input only a small sample of reads drawn from the dataset, sample that may be obtained, for example, at the time of the physical creation of the whole dataset. While the sampling strategy of SAKEIMA could be analyzed using the concept of VC dimension (Vapnik, 1998), the reads-sampling strategy of $\mathrm{SPRISS}$ requires the more sophisticated concept of pseudodimension (Pollard, 1984), for its analysis.

In this work, we consider the use of $\mathrm{SPRISS}$ to speed up the computation of the Bray-Curtis distance between metagenomic datasets, the identification of discriminative k-mers and the SNP genotyping process. Computational tools for these problems have been recently proposed (Benoit et al., 2016; Saavedra et al., 2020; Sun and Medvedev, 2018). These tools are based on exact k-mer counting strategies, and the approach we propose with $\mathrm{SPRISS}$ could be applied to such strategies as well.

2 Preliminaries

Let Σ be an alphabet of σ symbols. A dataset $\mathcal{D}=\{r_1,\dots ,r_n\}$ is a bag of $\left|\mathcal{D}\right|=n$  reads, where, for $i\in \{1,\dots ,n\}$, a read r_i is a string of length n_i built from Σ. For a given integer k, a k-mer K is a string of length k on Σ, that is $K\in {\Sigma }^k$. Given a k-mer K, a read r_i of $\mathcal{D}$ and a position $j\in \{0,\dots ,n_i-k\}$, we define the indicator function ${\varphi }_{r_i,K}(j)$ to be 1 if K appears in r_i at position j, that is $K[h]=r_i[j+h] \forall h\in \{0,\dots ,k-1\}$, while ${\varphi }_{r_i,K}(j)$ is 0 otherwise. The size $t_{\mathcal{D},k}$ of the multiset of k-mers that appear in $\mathcal{D}$ is $t_{\mathcal{D},k}=\sum _{r_i\in \mathcal{D}}(n_i-k+1)$. The average size of the multiset of k-mers that appear in a read of $\mathcal{D}$ is $g_{\mathcal{D},k}=t_{\mathcal{D},k}/n$, while the maximum value of such quantity is $g_{\max ,\mathcal{D},k}={\max }_{r_i\in \mathcal{D}}(n_i-k+1)$. The support $o_{\mathcal{D}}(K)$ of k-mer K in dataset $\mathcal{D}$ is the number of distinct positions of $\mathcal{D}$ where k-mer K appears, that is $o_{\mathcal{D}}(K)=\sum _{r_i\in \mathcal{D}}\sum _{j=0}^{n_i-k}{\varphi }_{r_i,K}(j)$. The frequency $f_{\mathcal{D}}(K)$ of a k-mer K in $\mathcal{D}$ is the fraction of all positions in $\mathcal{D}$ where K appears, that is $f_{\mathcal{D}}(K)=o_{\mathcal{D}}(K)/t_{\mathcal{D},k}$.

The task of finding frequent k-mers (FKs) is defined as follows: given a dataset $\mathcal{D}$, a positive integer k and a minimum frequency threshold $\theta \in (0,1]$, find the set $FK(\mathcal{D},k,\theta )$ of all the k-mers whose frequency in $\mathcal{D}$ is at least θ, and their frequencies, that is $FK(\mathcal{D},k,\theta )=\{(K,f_{\mathcal{D}}(K)):K\in {\Sigma }^k,f_{\mathcal{D}}(K)\ge \theta \}$.

The set of frequent k-mers can be computed by scanning the dataset and counting the number of occurrences for each k-mers. However, when dealing with a massive dataset $\mathcal{D}$, the exact computation of the set $FK(\mathcal{D},k,\theta )$ requires large amount of time and memory. For this reason, one could instead focus on finding an approximation of $FK(\mathcal{D},k,\theta )$ with rigorous guarantees on its quality. In this work, we consider the following approximation, introduced in (Pellegrina et al., 2020).

Definition 1.

Given a dataset $\mathcal{D}$, a positive integer k, a frequency threshold $\theta \in (0,1]$, and an accuracy parameter $\epsilon \in (0,\theta )$, an ε-approximation $\mathcal{C}=\{(K,f_K):K\in {\Sigma }^k,f_K\in [0,1]\}$  of $FK(\mathcal{D},k,\theta )$  is a set of pairs $(K,f_K)$  with the following properties:

• $\mathcal{C}$  contains a pair $(K,f_K)$  for every $(K,f_{\mathcal{D}}(K))\in FK(\mathcal{D},k,\theta )$;

• $\mathcal{C}$  contains no pair $(K,f_K)$  such that $f_{\mathcal{D}}(K)<\theta -\epsilon$;

• for every $(K,f_K)\in \mathcal{C}$, it holds $|f_{\mathcal{D}}(K)-f_K|\le \epsilon /2$.

Intuitively, the approximation $\mathcal{C}$ contains no false negatives (i.e. all the frequent k-mers in $FK(\mathcal{D},k,\theta )$ are in C) and no k-mer whose frequency in $\mathcal{D}$ is much smaller than θ. In addition, the frequencies in $\mathcal{C}$ are good approximations of the actual frequencies in $\mathcal{D}$, i.e. within a small error $\epsilon /2$.

Definition 2.

Given a dataset $\mathcal{D}$  of n reads, we define a reads sample S of $\mathcal{D}$  as a bag of m reads, sampled independently and uniformly at random, with replacement, from the bag of reads in $\mathcal{D}$.

A natural way to compute an approximation of the set of frequent k-mers is by processing a sample, i.e. a small portion of the dataset $\mathcal{D}$, instead of the whole dataset. While previous work (Pellegrina et al., 2020) considered samples obtained by drawing k-mers independently from $\mathcal{D}$, we consider samples obtained by drawing entire reads. Note that the development of an efficient scheme to effectively approximate the frequency of all frequent k-mers by sampling reads is highly non-trivial, due to dependencies among k-mers appearing in the same read. As explained in Section 1.1, our approach has several advantages, including the vfact that it can be combined with any efficient k-mer counting procedure, and that it can be used to extract a representative subset of the data on which to conduct down-stream analyses obtaining, in a fraction of the time required to process the whole dataset, the same insights. Such representative subsets could be stored and used for exploratory analyses, with a gain in terms of space and time requirements compared to using the whole dataset. In addition, note that $\mathrm{SPRISS}$ can approximate both canonical and non-canonical k-mers.

3 Method and algorithm

In this section, we develop and analyze our algorithm $\mathrm{SPRISS}$, the first efficient algorithm to approximate frequent k-mers by read sampling.

Let $\mathcal{D}$ be a bag of n reads. We define $I_{\ell }=\{i_1,i_2,\dots ,i_{\ell }\}$ as a bag of $\ell$ indexes of reads of $\mathcal{D}$ chosen uniformly at random, with replacement, from the set $\{1,\dots ,n\}$. Then we define an $\ell$-reads sample $S_{\ell }$ as a collection of m bags of $\ell$ reads $S_{\ell }=\{I_{\ell ,1},\dots ,I_{\ell ,m}\}$. Let k be a positive integer. Define the domain X as the set of bags of $\ell$ indexes of reads of $\mathcal{D}$. Then define the family of real-valued functions $\mathcal{F}=\{f_{K,\ell },\forall K\in {\Sigma }^k\}$ where, for every $I_{\ell }\in X$ and for every $f_{K,\ell }\in \mathcal{F}$, we have $f_{K,\ell }(I_{\ell })=\min (1,o_{I_{\ell }}(K))/(\ell g_{\mathcal{D},k})$, where $o_{I_{\ell }}(K)=\sum _{i\in I_{\ell }}\sum _{j=0}^{n_i-k}{\varphi }_{r_i,K}(j)$ counts the number of occurrences of K in all the $\ell$ reads of $I_{\ell }$. Therefore, $f_{K,\ell }(I_{\ell })\in \{0,\frac{1}{\ell g_{\mathcal{D},k}}\} \forall f_{K,\ell }$ and $\forall I_{\ell }$. Note that, for a given bag $I_{\ell }$, the functions $f_{K,\ell }$ have value equal to $1/\ell g_{\mathcal{D},k}$ even if K appears more than once in all the $\ell$ reads of $I_{\ell }$, thus ignoring multiple occurrences of K in the bag. We define the frequency $f_{S_{\ell }}(K)$ of a k-mer K obtained from the sample $S_{\ell }$ of bags of reads as $f_{S_{\ell }}(K)=\frac{1}{m}\sum _{I_{\ell ,i}\in S_{\ell }}{o}_{I_{\ell }}(K)/(\ell g_{\mathcal{D},k})$, which is an unbiased estimator of $f_{\mathcal{D}}(K)$ (i.e. $\mathbb{E}[f_{S_{\ell }}(K)]=f_{\mathcal{D}}(K)$). While the unbiased estimate $f_{S_{\ell }}(K)$ is the frequency reported by $\mathrm{SPRISS}$ as the estimated frequency of a k-mer K, $\mathrm{SPRISS}$ selects the k-mers to produce in output using a different estimate, namely ${\widehat{f}}_{S_{\ell }}(K)=\frac{1}{m}\sum _{I_{\ell ,i}\in S_{\ell }}{f}_{K,\ell }(I_{\ell ,i})$, which is a ‘biased’ version of $f_{S_{\ell }}(K)$ since multiple occurrences of K in a bag are ignored. For the technical motivation to use the biased frequency ${\widehat{f}}_{S_{\ell }}(K)$, see the analysis in Supplementary Section S3.

Our algorithm $\mathrm{SPRISS}$ (Algorithm 1) starts by computing the number m of bags of $\ell$ reads as in Equation (1), based on the input parameters $k,\theta ,\delta ,\epsilon ,\ell$ and on the characteristics ($g_{\mathcal{D},k},g_{\max ,\mathcal{D},k},\sigma$) of dataset $\mathcal{D}$. It then draws a sample S of exactly $m\ell$ reads, uniformly and independently at random, with replacement, from $\mathcal{D}$. Next, it computes for each k-mer K the number of occurrences $o_S(K)$ of K in sample S, using any exact k-mers counting algorithm. We denote the call of this method by exact_counting(S, k), which returns a collection T of pairs $(K,o_S(K))$. The sample S is then randomly partitioned into m bags, where each bag contains exactly $\ell$ reads. For each k-mer K, $\mathrm{SPRISS}$ computes the biased frequency ${\widehat{f}}_{S_{\ell }}(K)$ and the unbiased frequency $f_{S_{\ell }}(K)$, reporting in output only k-mers with biased frequency at least $\theta -\epsilon /2$. Note that, the estimated frequency of a k-mer K reported in output is always given by the unbiased frequency $f_{S_{\ell }}(K)$.

$\mathrm{SPRISS}$ (Algorithm 1) is motivated by our main technical result, Proposition 1, which establishes a rigorous relation between the number m of bags of $\ell$ reads and the guarantees obtained by approximating the frequency $f_{\mathcal{D}}(K)$ of a k-mer K with its (biased) estimate ${\widehat{f}}_{S_{\ell }}(K)$ (the full analysis is in Supplementary Section S3—see Supplementary Proposition S13).

Proposition 1.

 Let k and $\ell$  be two positive integers. Consider a sample $S_{\ell }$  of m bags of $\ell$  reads from $\mathcal{D}$. For fixed frequency threshold $\theta \in (0,1]$, error parameter $\epsilon \in (0,\theta )$  and confidence parameter $\delta \in (0,1)$, if

$$m\ge \frac{2}{{\epsilon }^2}{\left(\frac{1}{\ell g_{\mathcal{D},k}}\right)}^2\left(\lfloor \hspace{0.17em}{\text{log}}_2\min (2\ell g_{\max ,\mathcal{D},k},{\sigma }^k)\rfloor +\ln \left(\frac{1}{\delta }\right)\right)$$ (1)

then, with probability at least $1-\delta$:

1. for any k-mer $K\in FK(\mathcal{D},k,\theta )$  such that $f_{\mathcal{D}}(A)\ge \tilde{\theta }=\frac{g_{\max ,\mathcal{D},k}}{g_{\mathcal{D},k}}(1-{(1-\ell g_{\mathcal{D},k}\theta )}^{1/\ell })$  it holds ${\widehat{f}}_{S_{\ell }}(K)\ge \theta -\epsilon /2$;

2. for any k-mer K with ${\widehat{f}}_{S_{\ell }}(K)\ge \theta -\epsilon /2$  it holds $f_{\mathcal{D}}(K)\ge \theta -\epsilon$;

3. for any k-mer $K\in FK(\mathcal{D},k,\theta )$  it holds $f_{\mathcal{D}}(K)\ge {\widehat{f}}_{S_{\ell }}(K)-\epsilon /2$;

4. for any k-mer K with $\ell g_{\mathcal{D},k}({\widehat{f}}_{S_{\ell }}(K)+\epsilon /2)\le 1$  it holds $f_{\mathcal{D}}(K)\le \frac{g_{\max ,\mathcal{D},k}}{g_{\mathcal{D},k}}(1-{(1-\ell g_{\mathcal{D},k}({\widehat{f}}_{S_{\ell }}(K)+\epsilon /2))}^{(1/\ell )})$.

$\mathrm{SPRISS}$ builds on Proposition 1, and returns the approximation of $FK(\mathcal{D},k,\theta )$ defined by the set $A=\{(K,f_{S_{\ell }}(K)):{\widehat{f}}_{S_{\ell }}(K)\ge \theta -\epsilon /2\}$. Therefore, with probability at least $1-\delta$ the output of $\mathrm{SPRISS}$ provides the guarantees stated in Proposition 1. Note that, given a sample $S_{\ell }$ of m bags of $\ell$ reads from $\mathcal{D}$, with m satisfying the condition of Proposition 1, the set A is almost an ε-approximation of $FK(\mathcal{D},k,\theta )$: Proposition 1 ensures that all k-mers in A have frequency $f_{\mathcal{D}}(K)\ge \theta -\epsilon$ with probability at least $1-\delta$, but it does not guarantee that all k-mers with frequency $\in [\theta ,\tilde{\theta })$ will be in output. However, we show in Section 4.2 that, in practice, almost all of them are reported in output by $\mathrm{SPRISS}$. Furthermore, we remark that it is possible to obtain different guarantees on the approximation computed by $\mathrm{SPRISS}$ by modifying the criteria used to report k-mers in output; for example, in some applications, perfect recall may be particularly important. To this aim, we note that by reporting all k-mers with upper bound $\ge \theta$ (where the upper bound to $f_{\mathcal{D}}(K)$ is given by (iv) in Proposition 1), we obtain that all frequent k-mers are in the approximation, with relaxed guarantees on the precision (i.e. some k-mers with frequency $<\theta -\epsilon$ may be in the output). Moreover, in applications in which obtaining tight confidence intervals on all exact frequencies $f_{\mathcal{D}}(K)$ is important, an approximation scheme based on using multiple values of $\ell$, analogous to the one described in Section 3.3 of Pellegrina et al. (2020), is directly applicable to $\mathrm{SPRISS}$.

Algorithm 1:  $\mathrm{SPRISS}(\mathcal{D},k,\theta ,\delta ,\epsilon ,\ell )$.

Data: $\mathcal{D}$, k, $\theta \in (0,1],\hspace{0.17em}\delta \in (0,1),\hspace{0.17em}\epsilon \in (0,\theta )$, integer $\ell \ge 1$

Result: Approximation A of $FK(\mathcal{D},k,\theta )$ with probability at least $1-\delta$  

$$m\leftarrow \lceil \frac{2}{{\epsilon }^2}{\left(\frac{1}{\ell g_{\mathcal{D},k}}\right)}^2\left(\lfloor \hspace{0.17em}{\text{log}}_2\min (2\ell g_{\max ,\mathcal{D},k},{\sigma }^k)\rfloor +\ln \left(\frac{1}{\delta }\right)\right)\rceil ;$$

$S\leftarrow$ sample of exactly $m\ell$ reads drawn from $\mathcal{D}$;

$$T\leftarrow \mathtt{exact}\_\mathtt{counting}(S,k);$$

$S_{\ell }\leftarrow$ random partition of S into m bags of $\ell$ reads each;

$$A\leftarrow \varnothing ;$$

for all the  $(K,o_S(K))\in T$  do

$S_K\leftarrow$ number of bags of $S_{\ell }$ where K appears; 

$${\widehat{f}}_{S_{\ell }}\left(K\right)\leftarrow S_K/(m\ell g_{\mathcal{D},k});$$

 

$$f_{S_{\ell }}\left(K\right)\leftarrow o_S\left(K\right)/(m\ell g_{\mathcal{D},k});$$

if  ${\widehat{f}}_{S_{\ell }}(K)\ge \theta -\epsilon /2$  then  $A\leftarrow A\cup (K,f_{S_{\ell }}(K))$

return A;

In practice, in Algorithm 1, the partition of S into m bags and the computation of S_K could be highly demanding in terms of running time and space, since one has to compute and store, for each k-mer K, the exact number S_K of bags where K appears at least once among all reads of the bag. We now describe a much more efficient approach to approximate the values S_K, without the need to explicitly compute the bags. The number of reads in a given bag where K appears is well approximated by a Poisson distribution $\mathit{Poisson}(R[K]/m)$, where $R[K]$ is the number of reads of S where k-mer K appears at least once. Therefore, the number S_K of bags where K appears at least once is approximated by a binomial distribution $\mathit{Binomial}(m,1-e^{-R[K]/m})$. Thus, one can avoid to explicitly create the bags and to exactly count S_K, by replacing line $‘{\widehat{f}}_{S_{\ell }}\left(K\right)\leftarrow S_K/\left(m\ell g_{\mathcal{D},k}\right)’$ with $‘{\widehat{f}}_{S_{\ell }}\left(K\right)\leftarrow \mathit{Binomial}(m,1-e^{-R\left[K\right]/m})/\left(m\ell g_{\mathcal{D},k}\right)’$. Corollary 5.11 of Mitzenmacher and Upfal (2017) guarantees that, by using this Poisson distribution to approximate S_K, the output of $\mathrm{SPRISS}$ satisfies the properties of Proposition 1 with probability at least $1-2\delta$. This leads to the replacement of $‘\ln (1/\delta )’$ with $\mathrm{‘}\ln (2/\delta )’$ in the computation of m.

However, the approach described above requires to compute, for each k-mer K, the number of reads $R[K]$ of S where K appears at least once. We believe such computation can be obtained with minimal effort within the implementation of most k-mer counters, but we now describe a simple way to approximate $R[K]$. Since most k-mers appear at most once in a read, the number of reads $R[K]$ where a k-mer K appears is well approximated by the number of occurrences $T[K]$ of K in the sample S. Thus, instead of using $\mathrm{“}{\widehat{f}}_{S_{\ell }}(K)\leftarrow \mathit{Binomial}(m,1-e^{-R[K]/m})/(m\ell g_{\mathcal{D},k})\mathrm{”}$ we can replace it with $\mathrm{‘}{\widehat{f}}_{S_{\ell }}(K)\leftarrow \mathit{Binomial}(m,1-e^{-T[K]/m})/(m\ell g_{\mathcal{D},k})\mathrm{’}$, which only requires the counts $T[K]$ obtained from the exact counting procedure exact_counting(S, k) (see Algorithm S2 in Supplementary Material). Note that approximating $R[K]$ with $T[K]$ leads to overestimating the frequencies of few k-mers who reside in very repetitive sequences, e.g. k-mers composed by the same k consecutive nucleotides, for which $T[K]\gg R[K]$. However, since the majority of k-mers reside in non-repetitive sequences, we can assume $R[K]\approx T[K]$.

4 Experimental evaluation

In this section, we present the results of our experimental evaluation. In particular:

• We assess the performance of $\mathrm{SPRISS}$ in approximating the set of frequent k-mers from a dataset of reads. In particular, we evaluate the accuracy of estimated frequencies and false negatives in the approximation, and compare $\mathrm{SPRISS}$ with the state-of-the-art sampling algorithm SAKEIMA (Pellegrina et al., 2020) in terms of sample size and running time.

• We evaluate $\mathrm{SPRISS}$’s performance for the comparison of metagenomic datasets. We use $\mathrm{SPRISS}$’s approximations to estimate abundance-based distances (e.g. the Bray-Curtis distance) between metagenomic datasets, and show that the estimated distances can be used to obtain informative clusterings of metagenomic datasets from the Sorcerer II Global Ocean Sampling Expedition (Rusch et al., 2007) (https://www.imicrobe.us) in a fraction of the time required by the exact distances computation (i.e. based on exact k-mers frequencies).

• We test $\mathrm{SPRISS}$ to discover discriminative k-mers between pairs of datasets. We show that $\mathrm{SPRISS}$ identifies almost all discriminative k-mers from pairs of metagenomic datasets rom (Liu et al., 2017) and the Human Microbiome Project (HMP) (https://hmpdacc.org/HMASM/), with a significant speed-up compared to standard approaches.

• We evaluate $\mathrm{SPRISS}$ for approximate SNP genotyping, by combining the sampling scheme of $\mathrm{SPRISS}$ with previously proposed genotyping algorithms. We show that we achieve accurate approximations of the most common performance measures (precision, sensitivity and F-measure), obtaining a significant speed-up of the genotyping process.

4.1 Implementation, datasets, parameters and environment

We implemented $\mathrm{SPRISS}$ as a combination of C++ scripts, which perform the reads sampling and save the sample on a file, and as a modification of KMC 3 (Kokot et al., 2017) (available at https://github.com/refresh-bio/KMC), a fast and efficient counting k-mers algorithm. We used KMC 3 with the default option to count canonical k-mers. Note that our flexible sampling technique can be combined with any k-mer counting algorithm. [See Supplementary Material for results, e.g. Supplementary Figure S1, obtained using Jellyfish v. 2.3 (available at https://github.com/gmarcais/Jellyfish) as k-mer counter in $\mathrm{SPRISS}$.] We use the variant of $\mathrm{SPRISS}$ that employs the Poisson approximation for computing S_K (see end of Section 3). $\mathrm{SPRISS}$ implementation, information about how to retrieve the data used in this work, and scripts for reproducing all results are publicity available (available at https://github.com/VandinLab/SPRISS). We compared $\mathrm{SPRISS}$ with the exact k-mer counter KMC and with SAKEIMA (Pellegrina et al., 2020) (available at https://github.com/VandinLab/SAKEIMA), the state-of-the-art sampling-based algorithm for approximating frequent k-mers. In all experiments we fix $\delta =0.1$ and $\epsilon =\theta -2/t_{\mathcal{D},k}$. If not stated otherwise, we considered k = 31 and $\ell =\lfloor 0.9/(\theta g_{\mathcal{D},k})\rfloor$ in our experiments. For SAKEIMA, as suggested in Pellegrina et al. (2020) we set the number g_SK of k-mers in a bag to be $g_{SK}=\lfloor 0.9/\theta \rfloor$. We remark that a bag of reads of $\mathrm{SPRISS}$ contains the same (expected) number of k-mers positions of a bag of SAKEIMA; this guarantees that both algorithms provide outputs with the same guarantees, thus making the comparison between the two methods fair. To assess $\mathrm{SPRISS}$ in approximating frequent k-mers, we considered six large metagenomic datasets from HMP, each with $\approx {10}^8$ reads and average read length $\approx 100$ (see Supplementary Table S1). For the evaluation of $\mathrm{SPRISS}$ in comparing metagenomic datasets, we also used 37 small metagenomic datasets from the Sorcerer II Global Ocean Sampling Expedition (Rusch et al., 2007), each with $\approx {10}^4-{10}^5$ reads and average read length $\approx 1000$ (see Supplementary Table S4). For the assessment of $\mathrm{SPRISS}$ in the discovery of discriminative k-mers we used two large datasets from (Liu et al., 2017), B73 and Mo17, each with $\approx 4\times {10}^8$ reads and average read length $=250$ (see Supplementary Table S2), and we also experimented with the HMP datasets. To evaluate the benefits of using $\mathrm{SPRISS}$ for SNP genotyping, we used an Illumina WGS dataset from NA12878, with $\approx 1.55\times {10}^9$ reads and average read length $=148$ (see Supplementary Table S3), available from the Genome In A Bottle (GIAB) consortium (Zook et al., 2014). All experiments have been performed on a machine with 512 GB of RAM and 2 Intel(R) Xeon(R) CPU E5-2698 v3 at 2.3 GHz, with one worker, if not stated otherwise. All reported results are averages over five runs.

4.2 Approximation of frequent k-mers

In this section, we first assess the quality of the approximation of $FK(\mathcal{D},k,\theta )$ provided by $\mathrm{SPRISS}$, and then compare $\mathrm{SPRISS}$ with SAKEIMA.

We use $\mathrm{SPRISS}$ to extract approximations of frequent k-mers on six datasets from HMP for values of the minimum frequency threshold $\theta \in \{2.5\times {10}^{-8},5\times {10}^{-8},7.5\times {10}^{-8},{10}^{-7}\}$. The output of $\mathrm{SPRISS}$ satisfied the guarantees from Proposition 1 for all five runs of every combination of dataset and θ. In all cases the estimated frequencies provided by $\mathrm{SPRISS}$ are close to the exact ones (see Fig. 1a for an example). In fact, the average (across all reported k-mers) absolute deviation of the estimated frequency w.r.t. the true frequency is always small, i.e. one order of magnitude smaller than θ (Fig. 1b), and the maximum deviation is very small as well (Supplementary Fig. S2B). In addition, even if the values of $\tilde{\theta }$ [see (i) in Proposition 1] are always between $4.15˟{10}^{-6}$ and $1.81˟{10}^{-5},\hspace{0.17em}\mathrm{SPRISS}$ results in a very low false negative rate (i.e. fraction of k-mers of $FK(\mathcal{D},k,\theta )$ not reported by $\mathrm{SPRISS}$), which is always been below 0.012 in our experiments.

Fig. 1.

(a) k-mers exact frequency and frequency estimated by $\mathrm{SPRISS}$ for dataset SRS024075 and $\theta =2.5˟{10}^{-8}$. (b) Average deviations between exact frequencies and frequencies estimated by $\mathrm{SPRISS}$ ($\mathrm{SP}$) and SAKEIMA ($\mathrm{SK}$), for various datasets and values of θ. (c) Running time of $\mathrm{SPRISS}$ ($\mathrm{SP}$), SAKEIMA ($\mathrm{SK}$) and the exact computation ($\mathrm{E}$)—see also legend of (b). (d) Fraction of the dataset analyzed by $\mathrm{SPRISS}$ ($\mathrm{SP}$) and by SAKEIMA ($\mathrm{SK}$)

In terms of running time, $\mathrm{SPRISS}$ required at most 64% of the time required by the exact approach KMC (Fig. 1c). In addition, $\mathrm{SPRISS}$ used at most 30% of the RAM memory required by the exact approach KMC. This is due to $\mathrm{SPRISS}$ requiring to analyze at most 34% of the entire dataset (Fig. 1d). Note that the use of collections of bags of reads is crucial to achieve useful sample size, i.e. lower than the whole dataset. In fact, the sample sizes obtained from less sophisticated statistical tools, e.g. Hoeffding’s inequality combined with union bound (see Supplementary Section S1), and pseudodimension without collections of bags (see Supplementary Section S2), are much greater than the dataset size:$\approx {10}^{16}$ and $\approx {10}^{15}$, respectively, which are useless sample sizes for datasets of $\approx {10}^8$ reads. These results show that $\mathrm{SPRISS}$ obtains very accurate approximations of frequent k-mers in a fraction of the time required by exact counting approaches.

We then compared $\mathrm{SPRISS}$ with SAKEIMA. In terms of quality of approximation, $\mathrm{SPRISS}$ reports approximations with an average deviation lower than SAKEIMA’s approximations, while SAKEIMA’s approximations have a lower maximum deviation. However, the ratio between the maximum deviation of $\mathrm{SPRISS}$ and the one of SAKEIMA are always below 2. Overall, the quality of the approximation provided by $\mathrm{SPRISS}$ and SAKEIMA are, thus, comparable. In terms of running time, $\mathrm{SPRISS}$ significantly improves over SAKEIMA (Fig. 1c), and processes slightly smaller portions of the dataset compared to SAKEIMA (Fig. 1d). Summarizing, $\mathrm{SPRISS}$ is able to report most of the frequent k-mers and estimate their frequencies with small errors, by analyzing small samples of the datasets and with significant improvements on running times compared to exact approaches and to state-of-the-art sampling algorithms.

4.3 Comparing metagenomic datasets

We evaluated $\mathrm{SPRISS}$ to compare metagenomic datasets by computing an approximation to the Bray-Curtis (BC) distance between pairs of datasets of reads, and using such approximations to cluster datasets.

Let ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$ be two datasets of reads. Let ${\mathcal{F}}_1=FK({\mathcal{D}}_1,k,\theta )$ and ${\mathcal{F}}_2=FK({\mathcal{D}}_2,k,\theta )$ be the set of frequent k-mers, respectively, of ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$, where θ is a minimum frequency threshold. The BC distance between ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$ considering only frequent k-mers is defined as $BC({\mathcal{D}}_1,{\mathcal{D}}_2,{\mathcal{F}}_1,{\mathcal{F}}_2)=1-2I/U$, where $I=\sum _{K\in {\mathcal{F}}_1\cap {\mathcal{F}}_2}\min \{o_{{\mathcal{D}}_1}(K),o_{{\mathcal{D}}_2}(K)\}$ and $U=\sum _{K\in {\mathcal{F}}_1}{o}_{{\mathcal{D}}_1}(K)+\sum _{K\in {\mathcal{F}}_2}{o}_{{\mathcal{D}}_2}(K).$ Conversely, the BC similarity is defined as $1-BC({\mathcal{D}}_1,{\mathcal{D}}_2,{\mathcal{F}}_1,{\mathcal{F}}_2)$.

We considered six datasets from HMP, and estimated the BC distances among them by using $\mathrm{SPRISS}$ to approximate the sets of frequent k-mers ${\mathcal{F}}_1=FK({\mathcal{D}}_1,k,\theta )$ and ${\mathcal{F}}_2=FK({\mathcal{D}}_2,k,\theta )$ for the values of θ as in Section 4.2. We compared such estimated distances with the exact BC distances and with the estimates obtained using SAKEIMA. Both $\mathrm{SPRISS}$ and SAKEIMA provide accurate estimates of the BC distances (Fig. 2a and Supplementary Fig. S3), which can be used to assess the relative similarity of pairs of datasets. However, to obtain such approximations $\mathrm{SPRISS}$ requires at most 40% of the time required by SAKEIMA and usually 30% of the time required by the exact computation with KMC (Fig. 2b). Therefore $\mathrm{SPRISS}$ provides accurate estimates of metagenomic distances in a fraction of time required by other approaches.

Fig. 2.

(a) Comparison of the approximations of the Bray-Curtis (BC) distances using approximations of frequent k-mers provided by $\mathrm{SPRISS}$ (×) and by SAKEIMA ($•$), and the exact distances, for $\theta =2.5\times {10}^{-8}$. (b) Running time to approximate BC distances for all pairs of datasets with $\mathrm{SPRISS}$, with SAKEIMA and the exact approach. (c) Average linkage hierarchical clustering of GOS datasets using Jaccard similarity. (d) Same as (c), using estimated BC similarity from $\mathrm{SPRISS}$ with 50% of the data (see also larger Supplementary Figs S4–S6 for better readability of datasets’ labels and computed clusters)

As an example of the impact in accurately estimating distances among metagenomic datasets, we used the sampling approach of $\mathrm{SPRISS}$ to approximate all pairwise BC distances among 37 small datasets from the Sorcerer II Global Ocean Sampling Expedition (GOS) (Rusch et al., 2007), and used such distances to cluster the datasets using average linkage hierarchical clustering. The k-mer-based clustering of metagenomic datasets is often performed by using presence-based distances, such as the Jaccard distance (Ondov et al., 2016), which estimates similarities between two datasets by computing the fraction of k-mers in common between the two datasets. Abundance-based distances, such as the BC distance (Benoit et al., 2016; Danovaro et al., 2017; Dickson et al., 2017), provide more detailed measures based also on the k-mers abundance, but are often not used due to the heavy computational requirements to extract all k-mers counts. However, the sampling approach of $\mathrm{SPRISS}$ can significantly speed-up the computation of all BC distances, and, thus, the entire clustering analysis. In fact, for this experiment, the use of $\mathrm{SPRISS}$ reduces the time required to analyze the datasets (i.e. obtain k-mers frequencies, compute all pairwise distances and obtain the clustering) by 62%.

We then compared the clustering obtained using the Jaccard distance (Fig. 2c) and the clustering obtained using the estimates of the BC distances (Fig. 2d) obtained using only 50% of reads in the GOS datasets, which are assigned to groups and macro-groups according to the origin of the sample (Rusch et al., 2007). Even if the BC distance is computed using only a sample of the datasets, while the Jaccard distance is computed using the entirety of all datasets, the use of approximate BC distances leads to a better clustering in terms of correspondence of clusters to groups, and to the correct cluster separation for macro-groups. In addition, the similarities among datasets in the same group and the dissimilarities among datasets in different groups are more accentuated using the approximated BC distance. In fact, the ratio between the average BC similarity among datasets in the same group and the analogous average Jaccard is in the interval $[1.25,1.75]$ for all groups. In addition, the ratio between (i) the difference of the average BC similarity within the tropical macro-group and the average BC similarity between the tropical and temperate groups, and (ii) the analogous difference using the Jaccard similarity is $\approx 1.53$. These results tell us the approximate BC-distances, computed using only half of the reads in each dataset, increase by $\approx 50\%$ the similarity signal inside all groups defined by the original study (Rusch et al., 2007), and the dissimilarities between the two macro-groups (tropical and temperate).

To conclude, the estimates of the BC similarities obtained using the sampling scheme of $\mathrm{SPRISS}$ allows to better cluster metagenomic datasets than using the Jaccard similarity, while requiring less than 40% of the time needed by the exact computation of BC similarities, even for fairly small metagenomic datasets.

4.4 Approximation of discriminative k-mers

In this section, we assess $\mathrm{SPRISS}$ for approximating discriminative k-mers in metagenomic datasets. In particular, we consider the following definition of discriminative k-mers (Liu et al., 2017). Given two datasets ${\mathcal{D}}_1,{\mathcal{D}}_2$, and a minimum frequency threshold θ, we define the set $DK({\mathcal{D}}_1,{\mathcal{D}}_2,k,\theta ,\rho )$ of ${\mathcal{D}}_1$-discriminative k-mers as the collection of k-mers K for which the following conditions both hold: (i) $K\in FK({\mathcal{D}}_1,k,\theta )$; (ii) $f_{{\mathcal{D}}_1}(K)\ge \rho f_{{\mathcal{D}}_2}(K)$, with ρ  =  2. Note that the computation of $DK({\mathcal{D}}_1,{\mathcal{D}}_2,k,\theta ,\rho )$ requires to extract $FK({\mathcal{D}}_1,k,\theta )$ and $FK({\mathcal{D}}_2,k,\theta /\rho )$. $\mathrm{SPRISS}$ can be used to approximate the set $DK({\mathcal{D}}_1,{\mathcal{D}}_2,k,\theta ,\rho )$, by computing approximations $\overline{FK}({\mathcal{D}}_i,k,\theta )$ of the sets $FK({\mathcal{D}}_i,k,\theta )$, i = 1, 2, of frequent k-mers in ${\mathcal{D}}_1,{\mathcal{D}}_2$, and then reporting a k-mer K as ${\mathcal{D}}_1$-discriminative if the following conditions both hold: (i) $K\in \overline{FK}({\mathcal{D}}_1,k,\theta )$; (ii) $K\notin \overline{FK}({\mathcal{D}}_2,k,\theta )$, or $f_{S_{\ell }^1}(K)\ge \rho f_{S_{\ell }^2}(K)$ when $K\in \overline{FK}({\mathcal{D}}_2,k,\theta )$.

To evaluate such approach, we considered two datasets from (Liu et al., 2017), and $\theta =2˟{10}^{-7}$ and ρ = 2, which are the parameters used in (Liu et al., 2017). We used the sampling approach of $\mathrm{SPRISS}$ with $\ell =\lfloor 0.02/(\theta g_{\mathcal{D},k})\rfloor$ and $\ell =\lfloor 0.04/(\theta g_{\mathcal{D},k})\rfloor$, resulting in analyzing of 5% and 10% of all reads, to approximate the sets of discriminative ${\mathcal{D}}_1$-discriminative and of ${\mathcal{D}}_2$-discriminative k-mers. When 5% of the reads are used, the false negative rate is < 0.028, while when 10% of the reads are used, the false negative rate is < 0.018. The running times are $\approx 1130$ and $\approx 1970$ s, respectively, while the exact computation of the discriminative k-mers with KMC requires $\approx {10}^4$ s (we used 32 workers for both $\mathrm{SPRISS}$ and KMC). Similar results are obtained when analyzing pairs of HMP datasets, for various values of θ (Supplementary Fig. S7). These results show that $\mathrm{SPRISS}$ can identify discriminative k-mers with small false negative rates while providing a remarkable improvement in running time compared to the exact approach.

4.5 SNP genotyping

In this section, we evaluate $\mathrm{SPRISS}$ for approximate SNP genotyping. In particular, we assess the use of $\mathrm{SPRISS}$ in combination with previously proposed algorithms for SNP genotyping in terms of precision, sensitivity and F-measure. The genotyping algorithms we used are the standard pipeline [BWA (Li and Durbin, 2009) as aligner, and BCFtools (Li, 2011) as variant caller], and VarGeno (Sun and Medvedev, 2018). We considered hg19 as reference genome, and dbSNP (Sherry, 2001) as reference SNP database. We used the gold standard of NA12878 individual provided by the Genome In A Bottle (GIAB) consortium (Zook et al., 2014). The Illumina WGS dataset $\mathcal{D}$ of reads from NA12878 we used has a coverage of $\approx 75$x. We used the sampling scheme of $\mathrm{SPRISS}$ to create samples of 12.5%, 25%, 50% and 75% of reads of $\mathcal{D}$. The standard pipeline was run with 64 threads. When evaluating the running time, we do not include the time to obtain the sample, since once the sample is created it can be reused several times. Moreover, the time to obtain the sample is always a small fraction of the overall execution time (e.g. even for a sample containing 75% of reads of $\mathcal{D}$ the required time is < 3000 s).

The performance measures of the standard pipeline on $\mathcal{D}$ are the following: 0.961 of precision, 0.959 of sensitivity and 0.960 of F-measure. Figure 3 describes the running times and the performance measures of the standard pipeline using samples of $\mathcal{D}$ from $\mathrm{SPRISS}$. Considering a sample of just 25% of reads of $\mathcal{D}$, the sensitivity and the F-measure decrease, respectively, by 0.02 and 0.004, while the precision increases by 0.012. The increment of the precision is due to a decrement in the number of false positive calls, which is achieved by the reads sampling of $\mathrm{SPRISS}$ that filters out low coverage regions and erroneous k-mers. The speed-up of using a sample of 25% of reads of $\mathcal{D}$ instead of the entire dataset $\mathcal{D}$ is $\approx 3.9$x.

Fig. 3.

As function of the sample rate, experimental results of combining $\mathrm{SPRISS}$ with VarGeno and the standard pipeline in the SNP genotyping process: VarGeno’s precision (a), sensitivity (c) F-measure (e), running time (g) and standard pipeline’s precision (b), sensitivity (d) F-measure (f), running time (h)

VarGeno achieves on $\mathcal{D} 0.974$ of precision, 0.585 of sensitivity and 0.731 of F-measure. With a sample from $\mathrm{SPRISS}$ of just 25% of reads of $\mathcal{D}$, we obtain a decrement of the performance of VarGeno of 0.003 in precision, 0.015 in sensitivity, 0.013 in F-measure and a speed-up of $\approx 4.5$x with respect to the time required to analyze the entire dataset $\mathcal{D}$. The results for the other sample sizes are described in Figure 3.

To conclude, the sampling scheme of $\mathrm{SPRISS}$ is very useful to remarkably speed up genotyping algorithms, while achieving very small decrements in the performance measures, and even improving the precision in some cases.

5 Discussion

We presented $\mathrm{SPRISS}$, an efficient algorithm to compute rigorous approximations of frequent k-mers and their frequencies by sampling reads. $\mathrm{SPRISS}$ builds on pseudodimension, an advanced concept from statistical learning theory. Our extensive experimental evaluation shows that $\mathrm{SPRISS}$ provides high-quality approximations and can be employed to speed-up exploratory analyses in various applications, such as the analysis of metagenomic datasets, the identification of discriminative k-mers and SNP genotyping. Overall, the sampling approach used by $\mathrm{SPRISS}$ provides an efficient way to obtain a representative subset of the data that can be used to perform complex analyses more efficiently than examining the whole data, while obtaining representative results.

Funding

Part of this work was supported by the Italian Ministry of Education, University and Research (MIUR), under PRIN Project No. 20174LF3T8 AHeAD (Efficient Algorithms for HArnessing Networked Data) and the initiative ‘Departments of Excellence’ (Law 232/2016); and University of Padova under project SEED 2020 RATED-X.

Conflict of Interest: none declared.