# 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 $H$.

!!! abstract "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)
    ```
