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# Minimizer selection

## Definition

A minimizer of a k-mer window is the m-mer (m < k) with the smallest value under some total order ≺ among all k − m + 1 overlapping m-mers in the window. The minimizer is always taken in canonical form (lexicographic minimum of forward and reverse complement) to ensure strand-independence.

The minimizer partitions the sequence into super-kmers: maximal contiguous runs of overlapping k-mers that share the same minimizer. A single minimizer anchors each super-kmer, enabling partitioned storage and indexing.

## Lexicographic ordering and its bias

The classical definition uses lexicographic order on the canonical m-mer value. In 2-bit encoding (A=00, C=01, G=10, T=11), the canonical form is \(\min_{\text{lex}}(\text{fwd}, \text{rc})\), so AT-rich m-mers have systematically small values:

\[\text{canonical}(\text{AAAA}\cdots\text{A}) = \text{canonical}(\text{TTTT}\cdots\text{T}) = 0\]

Since small values always win the lex comparison, low-complexity AT-rich m-mers dominate as minimizers across large genomic regions. On real metagenomics data with k=31, m=11 and 256 partitions, this produces a max/min partition ratio of ≈ 2.75 — and a single pathological partition when the hash function has a fixed point at 0.

## Random minimizer

A random minimizer replaces lex order with a hash order: define \(H : \{0,1\}^{2m} \to \{0,1\}^{64}\) and select the m-mer with the minimum \(H\) value in the window.

The key property: because \(H\) is a bijection with well-distributed outputs, each distinct m-mer in the window has equal probability of holding the minimum hash value. Selection probability is no longer correlated with nucleotide composition.

## Why the canonical form remains lexicographic

An apparent alternative is to redefine the canonical form of each m-mer as the strand with the smaller hash value:

\[\text{canonical}_H(v) = \arg\min(H(\text{fwd}),\ H(\text{rc}))\]

This must be rejected. The hash of this new canonical is \(\min(H(\text{fwd}), H(\text{rc}))\) — the minimum of two i.i.d. Uniform\([0, 2^{64})\) values. Its distribution is:

\[F(x) = 1 - \left(1 - \frac{x}{2^{64}}\right)^2\]

with density \(f(x) = 2(1 - x/2^{64})\), which is approximately twice as large near 0 than near \(2^{64}\). The low-order partition bits inherit this bias: partition 0 receives roughly twice as many super-kmers as the last partition.

The lex canonical form does not have this problem: \(\text{canonical}_{\text{lex}}(v)\) is a fixed, deterministic representative of each equivalence class, and \(H(\text{canonical}_{\text{lex}})\) is uniformly distributed over \([0, 2^{64})\) independently of the min/max relationship between the two strands.

## Partition key independence

A further subtlety arises when the selection hash is used directly as the partition key. The selected minimizer is the m-mer with the minimum \(H\) value in a window of \(W = k - m + 1\) positions. The minimum of \(W\) i.i.d. Uniform\([0,2^{64})\) values has distribution:

\[F(x) = 1 - \left(1 - \frac{x}{2^{64}}\right)^W \approx \frac{Wx}{2^{64}}\]

concentrated near 0 relative to the full range. Using this minimum-hash directly as the partition key creates the same bias as lex ordering, just distributed differently.

The correct approach is to decouple selection from partition routing:

- Selection uses \(H(\text{canonical}_{\text{lex}}(m\text{-mer}))\) to pick the minimizer in the window.
- Partition routing recomputes \(H(\text{canonical}_{\text{lex}}(\text{minimizer}))\) from the stored minimizer position. This is the hash of a specific kmer value, not the minimum of a window — it is uniformly distributed over \([0, 2^{64})\).

## Seed and fixed-point elimination

The splitmix64 finalizer has a fixed point at 0:

\[\text{mix64}(0) = 0\]

Since \(\text{canonical}_{\text{lex}}(\text{AAAA}\cdots\text{A}) = 0\), using unseeded mix64 causes all-A m-mers to win every window comparison, recreating a pathological partition identical to the lex-ordering bias.

The fix is a non-zero XOR seed applied before mixing:

\[H(x) = \text{mix64}(x \oplus s), \quad s = \lfloor 2^{64}/\varphi \rfloor = \texttt{0x9e3779b97f4a7c15}\]

where \(\varphi\) is the golden ratio. This maps 0 to \(\text{mix64}(s)\), a well-distributed non-zero value. No canonical m-mer value has a systematically small \(H\).

Hash function \(H\)

    H(x):
        x ← x  ⊕  0x9e3779b97f4a7c15
        x ← x  ⊕  (x >> 30)
        x ← x  ×  0xbf58476d1ce4e5b9
        x ← x  ⊕  (x >> 27)
        x ← x  ×  0x94d049bb133111eb
        return x ⊕ (x >> 31)