Minimizers on CuriousCoding

Understanding GreedyMini

Sun, 27 Apr 2025 00:00:00 +0200

In this post, we will look at the minimizer schemes generated by the greedy minimizer (Golan et al. 2025).

Low Density Minimizers

Tue, 08 Apr 2025 00:00:00 +0200

Table of Contents

1 Theory of Sampling Schemes
- 1.1 Introduction
- 1.2 Overview
- 1.3 Theory of sampling schemes
- 1.4 Notation
- 1.5 Types of sampling schemes
- 1.6 Computing the density
- 1.7 The density of random minimizers
- 1.8 Universal hitting sets
- 1.9 Asymptotic results
- 1.10 Variants
2 Lower Bounds on Sampling Scheme Density
3 Practical Sampling Schemes
- 3.1 Variants of lexicographic minimizers
  - Evaluation
- 3.2 UHS-inspired schemes
- 3.3 Syncmer-based schemes
- 3.4 Open-closed minimizer
  - Evaluation
- 3.5 Mod-minimizer
  - Theoretical density
  - Evaluation
- 3.6 Discussion
4 Towards Optimal Selection Schemes
- 4.1 Bidirectional anchors
- 4.2 Sus-anchors
  - Evaluation
- 4.3 Discussion

This is Part 2 of my thesis (Groot Koerkamp 2025), containing chapters 6 to 9 on Low Density Minimizers. Please cite the thesis instead of this post.

Near-optimal sampling schemes

Thu, 27 Feb 2025 00:00:00 +0100

1 Warming up: A cute prolblem Link to heading

Given a string, choose one character.
- CABAACBD
Given a rotation, choose one character.
- ACBDCABA
Can we always choose the same character?
Yes: e.g. the smallest rotation (bd-anchor):
- CABAACBD
- ACBDCABA

1.1 This talk: what if one character is hidden? Link to heading

Given a string (length $w$), choose one character.
- CABAACBDX
Given a rotation (of the hidden $w+1$ string), choose one character.
- ACBDXCABA
Can we always choose the same character?
Maybe?
- CABAACBD
- ACBDXCAB 🤔

1.2 The answer is no! Link to heading

CABAACBDX rotations:

Minimizer papers

Mon, 17 Feb 2025 00:00:00 +0100

Table of Contents

1 Overview
2 Introduction - Previous reviews
3 Theory of sampling schemes
- 3.1 Questions
- 3.2 Types of schemes
- 3.3 Parameter regimes
- 3.4 Different perspectives
- 3.5 UHS vs minimizer scheme
- 3.6 (Asymptotic) bounds
- 3.7 Lower bounds
4 Minimizer schemes
- 4.1 Orders
- 4.2 UHS-based and search-based schemes
- 4.3 Pure schemes
- 4.4 Other variants
  - Selection schemes
  - Canonical minimizers
- 4.5 Non-overlapping string sets

This post is simply a list of brief comments on many papers related to minimizers, and forms the basis of /posts/minimizers/.

Comments on Brisk

Fri, 29 Nov 2024 00:00:00 +0100

Table of Contents

Overview
Detailed comments

These are some (biased) comments on Brisk, a dynamic k-mer dictionary (Smith et al. 2024).

Comments on GreedyMini

Mon, 04 Nov 2024 00:00:00 +0100

Table of Contents

Overview
Detailed comments
Comments on “Expected density of random minimizers”

These are some (biased) comments on “Greedymini: Generating Low-Density Dna Minimizers” (Golan et al. 2024), which introduces the GreedyMini minimizer scheme. (Meanwhile, this has been published as Golan et al. (2025).)

At the bottom, there are also some comment on Golan and Shur (2025).

Comments on 'When Less is More' minimizer review

Tue, 15 Oct 2024 00:00:00 +0200

Table of Contents

The importance of ordering
Asymptotically optimal minimizers

These are some (biased) comments on “When Less Is More: Sketching with Minimizers in Genomics” (Ndiaye et al. 2024).

The importance of ordering Link to heading

the interest lies in constructing a minimizer with a density within a constant factor, i.e., $O(1/w)$ for any $k$. With lexicographic ordering, minimizers can achieve such density, but with large $k$ values ($\geq \log_{|Σ|}(w)-c$ for a constant $c$), which might not be desirable (Zheng, Kingsford, and Marçais 2020). However, random ordering can result in a lower density than that of the lexicographic ordering. Thus, random ordering (implemented with pseudo-random hash functions) is usually used in practice.

Practical minimizers

Thu, 12 Sep 2024 00:00:00 +0200

Table of Contents

1 Sampling schemes
- 1.1 Definitions
- 1.2 Miniception
- 1.3 Mod-minimizer
- 1.4 Forward scheme lower bound
- 1.5 Open syncmer minimizer
- 1.6 Open-closed minimizer
- 1.7 New: General mod-minimizer
- 1.8 Variant: Open-closed minimizer using offsets
2 Selection schemes
- 2.1 Definition
- 2.2 Bd-anchors
- 2.3 New: Smallest unique substring anchors
- 2.4 New: Anti lexicographic sorting
3 More sampling schemes
- 3.1 Anti-lex sus-anchors
- 3.2 Threshold anchors
- 3.3 The $t$-gap disappears for large alphabets
4 Computing the density of forward schemes
- 4.1 WIP: Anti lexicographic sus-anchor density
5 Open questions
6 Ideas
7 Optimal schemes for $k \in \{w, w+1\}$

Most of the content here has now been absorbed into my thesis chapter on minimizers.

SimdMinimizers: Computing random minimizers, fast

Fri, 12 Jul 2024 00:00:00 +0200

Table of Contents

1 Introduction
- 1.1 Results
2 Random minimizers
3 Algorithms
4 Analysing what we have so far
5 Rolling our own hash
6 SIMD sliding window
- 6.1 Results
  - Human genome results
7 Extending into something useful
- 7.1 Collecting minimizer positions
- 7.2 Deduplicating the minimizer positions
- 7.3 Super-k-mers
- 7.4 Canonical k-mers
- 7.5 AntiLex hash
8 Conclusion
- 8.1 Future work

SimdMinimizers has been published as a paper: DOI, PDF:

A near-tight lower bound on minimizer density

Tue, 25 Jun 2024 00:00:00 +0200

Table of Contents

Succinct background
- Definitions
- Lower bounds
A new lower bound
Discussion
Post scriptum
Acknowledgement

The results of this post are now published in Bioinformatics: DOI, PDF:

Kille, Bryce, Ragnar Groot Koerkamp, Drake McAdams, Alan Liu, and Todd J Treangen. 2024. “A near-tight Lower Bound on the Density of Forward Sampling Schemes.” Edited by Yann Ponty. Bioinformatics, December. https://doi.org/10.1093/bioinformatics/btae736.

This content has also been absorbed into my thesis chapter on minimizers.

Mod-minimizers and other minimizers

Thu, 18 Jan 2024 00:00:00 +0100

\[ \newcommand{\d}{\mathrm{d}} \newcommand{\L}{\mathcal{L}} \]

This post introduces some background for minimizers and some experiments for a new minimizer variant. That new variant is now called the mod-minimizer and published at WABI24 (DOI, PDF) (Groot Koerkamp and Pibiri 2024). The paper also includes a review of existing methods, including pseudocode for most of the methods covered below.

Notes on bidirectional anchors

Mon, 15 Jan 2024 00:00:00 +0100

Table of Contents

Paper overview
Remarks on the paper
Thoughts

\[ \newcommand{\A}{\mathcal{A}_\ell} \newcommand{\T}{\mathcal{T}_\ell} \]

These are some notes on Bidirectional String Anchors (Loukides, Pissis, and Sweering 2023), also called bd-anchors.

Resources:

Loukides and Pissis (2021): preceding conference paper with subset of content.
Loukides, Pissis, and Sweering (2023): The paper discussed here.
Ayad, Loukides, and Pissis (2023): follow-up/second paper containing
- a faster average-case $O(n)$ construction algorithm;
- a more memory efficient construction algorithms for the index.
https://github.com/solonas13/bd-anchors: code for first paper
https://github.com/lorrainea/BDA-index: code for follow-up paper

The remainder of this post is split into an overview of the paper, Remarks on the paper, and further Thoughts.

Perfect NtHash for Robust Minimizers

Sun, 31 Dec 2023 00:00:00 +0100

Table of Contents

NtHash
Minimizers
- Robust minimizers
Is NtHash injective on kmers?
- Searching for a collision
- Proving perfection
Alternatives
SmHasher results

NtHash Link to heading

NtHash (Mohamadi et al. 2016) is a rolling hash suitable for hashing any kind of text, but made for DNA originally. For a string of length $k$ it is a $64$ bit value computed as: