From 0b79a8ee7469a01c44672d8b0229f93feba5977f Mon Sep 17 00:00:00 2001
From: Eric Coissac <eric@coissac.eu>
Date: Sat, 12 Sep 2026 16:26:13 +0200
Subject: [PATCH] Define k-mer minimizer selection {} Introduces the definition
 of a minimizer for k-mer windows, defining the canonical form as the
 lexicographic minimum between the m-mer and its reverse complement. Selection
 is determined by a deterministic hash function that chooses the m-mer whose
 canonical form minimizes the hash value, including a warning about potential
 biases in the ordering.

---
 theory-minimizer_selection.md | 24 ++++++++++++++++++++++--
 1 file changed, 22 insertions(+), 2 deletions(-)

diff --git a/theory-minimizer_selection.md b/theory-minimizer_selection.md
index 6b83a47..e58787d 100644
--- a/theory-minimizer_selection.md
+++ b/theory-minimizer_selection.md
@@ -2,11 +2,31 @@
 
 ## Definition
 
-A **minimizer** of a kmer window is the m-mer ($m < k$) that is smallest, among all $k - m + 1$ overlapping m-mers in the window, under a chosen ordering. The minimizer is always taken in canonical form (lexicographic minimum of forward and reverse complement) so that selection is strand-independent.
+The **minimizer** of a k-mer is the canonical form of the m-mer that is smallest according to a chosen ordering among the $k-m+1$ overlapping m-mers contained in the k-mer, with $m<k$ \@Roberts2004-rz.
+
+For an m-mer $s$, its **canonical form** is defined as
+
+$$s^c = \min_{\mathrm{lex}}\left(s,\operatorname{RC}(s)\right),$$
+
+where $\operatorname{RC}(s)$ denotes the reverse complement of $s$.
+
+The ordering used to select the minimizer is independent of this canonicalization. In **OBIkmer**, canonical m-mers are ordered according to a deterministic hash function $h$:
+
+$$s_1^c <_h s_2^c \quad\Longleftrightarrow\quad h(s_1^c) < h(s_2^c).$$
+
+Thus, for a k-mer $K$, let
+
+$$M(K)=\{s_1,\ldots,s_{k-m+1}\}$$
+
+be its set of overlapping m-mers. The minimizer is
+
+$$\operatorname{minimizer}(K) = s_j^c, \qquad j=\underset{i}{\operatorname{argmin}}\;h(s_i^c).$$
+
+In other words, the hash function defines the ordering of canonical m-mers, while the minimizer itself is the canonical form of the m-mer selected by that ordering.
 
 The minimizer partitions a sequence into super-kmers: maximal runs of overlapping kmers that share the same minimizer (see [Kmers and super-kmers](theory-kmers_and_superkmers)).
 
-## Hash-based ("random") minimizer
+## Hash-based minimizer ordering
 
 `obikmer` selects minimizers by hash order rather than plain lexicographic order. Ordering m-mers lexicographically on their 2-bit encoding systematically favors AT-rich m-mers (an all-A m-mer always encodes to 0), which causes low-complexity regions to dominate as minimizers and produces unbalanced partitions ([Golan & Shur 2025](#ref-Golan2025-xf); [Kille et al. 2023](#ref-Kille2023-px); [Pan & Reinert 2024](#ref-Pan2024-hb); [Zheng et al. 2020](#ref-Zheng2020-ji); [2021](#ref-Zheng2021-cc)).