Document

**Published in:** LIPIcs, Volume 312, 24th International Workshop on Algorithms in Bioinformatics (WABI 2024)

Motivation. Given a string S, a minimizer scheme is an algorithm defined by a triple (k,w,𝒪) that samples a subset of k-mers (k-long substrings) from a string S. Specifically, it samples the smallest k-mer according to the order 𝒪 from each window of w consecutive k-mers in S. Because consecutive windows can sample the same k-mer, the set of the sampled k-mers is typically much smaller than S. More generally, we consider substring sampling algorithms that respect a window guarantee: at least one k-mer must be sampled from every window of w consecutive k-mers. As a sampled k-mer is uniquely identified by its absolute position in S, we can define the density of a sampling algorithm as the fraction of distinct sampled positions. Good methods have low density which, by respecting the window guarantee, is lower bounded by 1/w. It is however difficult to design a sequence-agnostic algorithm with provably optimal density. In practice, the order 𝒪 is usually implemented using a pseudo-random hash function to obtain the so-called random minimizer. This scheme is simple to implement, very fast to compute even in streaming fashion, and easy to analyze. However, its density is almost a factor of 2 away from the lower bound for large windows.
Methods. In this work we introduce mod-sampling, a two-step sampling algorithm to obtain new minimizer schemes. Given a (small) parameter t, the mod-sampling algorithm finds the position p of the smallest t-mer in a window. It then samples the k-mer at position pod w. The lr-minimizer uses t = k-w and the mod-minimizer uses t≡ k (mod w).
Results. These new schemes have provably lower density than random minimizers and other schemes when k is large compared to w, while being as fast to compute. Importantly, the mod-minimizer achieves optimal density when k → ∞. Although the mod-minimizer is not the first method to achieve optimal density for large k, its proof of optimality is simpler than previous work.
We provide pseudocode for a number of other methods and compare to them. In practice, the mod-minimizer has considerably lower density than the random minimizer and other state-of-the-art methods, like closed syncmers and miniception, when k > w. We plugged the mod-minimizer into SSHash, a k-mer dictionary based on minimizers. For default parameters (w,k) = (11,21), space usage decreases by 15% when indexing the whole human genome (GRCh38), while maintaining its fast query time.

Ragnar Groot Koerkamp and Giulio Ermanno Pibiri. The mod-minimizer: A Simple and Efficient Sampling Algorithm for Long k-mers. In 24th International Workshop on Algorithms in Bioinformatics (WABI 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 312, pp. 11:1-11:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{grootkoerkamp_et_al:LIPIcs.WABI.2024.11, author = {Groot Koerkamp, Ragnar and Pibiri, Giulio Ermanno}, title = {{The mod-minimizer: A Simple and Efficient Sampling Algorithm for Long k-mers}}, booktitle = {24th International Workshop on Algorithms in Bioinformatics (WABI 2024)}, pages = {11:1--11:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-340-9}, ISSN = {1868-8969}, year = {2024}, volume = {312}, editor = {Pissis, Solon P. and Sung, Wing-Kin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2024.11}, URN = {urn:nbn:de:0030-drops-206552}, doi = {10.4230/LIPIcs.WABI.2024.11}, annote = {Keywords: Minimizers, Randomized algorithms, Sketching, Hashing} }

Document

**Published in:** LIPIcs, Volume 273, 23rd International Workshop on Algorithms in Bioinformatics (WABI 2023)

The problem of sequence identification or matching - determining the subset of reference sequences from a given collection that are likely to contain a short, queried nucleotide sequence - is relevant for many important tasks in Computational Biology, such as metagenomics and pan-genome analysis. Due to the complex nature of such analyses and the large scale of the reference collections a resource-efficient solution to this problem is of utmost importance. This poses the threefold challenge of representing the reference collection with a data structure that is efficient to query, has light memory usage, and scales well to large collections.
To solve this problem, we describe how recent advancements in associative, order-preserving, k-mer dictionaries can be combined with a compressed inverted index to implement a fast and compact colored de Bruijn graph data structure. This index takes full advantage of the fact that unitigs in the colored de Bruijn graph are monochromatic (all k-mers in a unitig have the same set of references of origin, or "color"), leveraging the order-preserving property of its dictionary. In fact, k-mers are kept in unitig order by the dictionary, thereby allowing for the encoding of the map from k-mers to their inverted lists in as little as 1+o(1) bits per unitig. Hence, one inverted list per unitig is stored in the index with almost no space/time overhead. By combining this property with simple but effective compression methods for inverted lists, the index achieves very small space.
We implement these methods in a tool called Fulgor. Compared to Themisto, the prior state of the art, Fulgor indexes a heterogeneous collection of 30,691 bacterial genomes in 3.8× less space, a collection of 150,000 Salmonella enterica genomes in approximately 2× less space, is at least twice as fast for color queries, and is 2-6 × faster to construct.

Jason Fan, Noor Pratap Singh, Jamshed Khan, Giulio Ermanno Pibiri, and Rob Patro. Fulgor: A Fast and Compact {k-mer} Index for Large-Scale Matching and Color Queries. In 23rd International Workshop on Algorithms in Bioinformatics (WABI 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 273, pp. 18:1-18:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{fan_et_al:LIPIcs.WABI.2023.18, author = {Fan, Jason and Singh, Noor Pratap and Khan, Jamshed and Pibiri, Giulio Ermanno and Patro, Rob}, title = {{Fulgor: A Fast and Compact \{k-mer\} Index for Large-Scale Matching and Color Queries}}, booktitle = {23rd International Workshop on Algorithms in Bioinformatics (WABI 2023)}, pages = {18:1--18:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-294-5}, ISSN = {1868-8969}, year = {2023}, volume = {273}, editor = {Belazzougui, Djamal and Ouangraoua, A\"{i}da}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2023.18}, URN = {urn:nbn:de:0030-drops-186446}, doi = {10.4230/LIPIcs.WABI.2023.18}, annote = {Keywords: k-mers, Colored de Bruijn Graph, Compression, Read-mapping} }

Document

**Published in:** LIPIcs, Volume 242, 22nd International Workshop on Algorithms in Bioinformatics (WABI 2022)

We consider the problem of representing a set of k-mers and their abundance counts, or weights, in compressed space so that assessing membership and retrieving the weight of a k-mer is efficient. The representation is called a weighted dictionary of k-mers and finds application in numerous tasks in Bioinformatics that usually count k-mers as a pre-processing step. In fact, k-mer counting tools produce very large outputs that may result in a severe bottleneck for subsequent processing.
In this work we extend the recently introduced SSHash dictionary (Pibiri, Bioinformatics 2022) to also store compactly the weights of the k-mers. From a technical perspective, we exploit the order of the k-mers represented in SSHash to encode runs of weights, hence allowing (several times) better compression than the empirical entropy of the weights. We also study the problem of reducing the number of runs in the weights to improve compression even further and illustrate a lower bound for this problem. We propose an efficient, greedy, algorithm to reduce the number of runs and show empirically that it performs well, i.e., very similarly to the lower bound. Lastly, we corroborate our findings with experiments on real-world datasets and comparison with competitive alternatives. Up to date, SSHash is the only k-mer dictionary that is exact, weighted, associative, fast, and small.

Giulio Ermanno Pibiri. On Weighted k-mer Dictionaries. In 22nd International Workshop on Algorithms in Bioinformatics (WABI 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 242, pp. 9:1-9:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{pibiri:LIPIcs.WABI.2022.9, author = {Pibiri, Giulio Ermanno}, title = {{On Weighted k-mer Dictionaries}}, booktitle = {22nd International Workshop on Algorithms in Bioinformatics (WABI 2022)}, pages = {9:1--9:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-243-3}, ISSN = {1868-8969}, year = {2022}, volume = {242}, editor = {Boucher, Christina and Rahmann, Sven}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2022.9}, URN = {urn:nbn:de:0030-drops-170439}, doi = {10.4230/LIPIcs.WABI.2022.9}, annote = {Keywords: K-Mers, Weights, Compression, Hashing} }

Document

**Published in:** LIPIcs, Volume 78, 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)

We show that it is possible to store a dynamic ordered set S of n integers drawn from a bounded universe of size u in space close to the information-theoretic lower bound and preserve, at the same time, the asymptotic time optimality of the operations. Our results leverage on the Elias-Fano representation of monotone integer sequences, which can be shown to be less than half a bit per element away from the information-theoretic minimum.
In particular, considering a RAM model with memory word size Theta(log u) bits, when integers are drawn from a polynomial universe of size u = n^gamma for any gamma = Theta(1), we add o(n) bits to the static Elias-Fano representation in order to:
1. support static predecessor/successor queries in O(min{1+log(u/n), loglog n});
2. make S grow in an append-only fashion by spending O(1) per inserted element;
3. describe a dynamic data structure supporting random access in O(log n / loglog n) worst-case, insertions/deletions in O(log n / loglog n) amortized and predecessor/successor queries in O(min{1+log(u/n), loglog n}) worst-case time. These time bounds are optimal.

Giulio Ermanno Pibiri and Rossano Venturini. Dynamic Elias-Fano Representation. In 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 78, pp. 30:1-30:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

Copy BibTex To Clipboard

@InProceedings{pibiri_et_al:LIPIcs.CPM.2017.30, author = {Pibiri, Giulio Ermanno and Venturini, Rossano}, title = {{Dynamic Elias-Fano Representation}}, booktitle = {28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)}, pages = {30:1--30:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-039-2}, ISSN = {1868-8969}, year = {2017}, volume = {78}, editor = {K\"{a}rkk\"{a}inen, Juha and Radoszewski, Jakub and Rytter, Wojciech}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2017.30}, URN = {urn:nbn:de:0030-drops-73244}, doi = {10.4230/LIPIcs.CPM.2017.30}, annote = {Keywords: succinct data structures, integer sets, predecessor problem, Elias-Fano} }