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< h1 id = "minimizer-selection" > Minimizer selection</ h1 >
< h2 id = "definition" > Definition</ h2 >
< p > A < strong > minimizer</ strong > of a k-mer window is the m-mer (m < k) with the smallest value under some total order ≺ among all k − < strong > canonical form</ strong > (lexicographic minimum of forward and reverse complement) to ensure strand-independence.</ p >
< p > The minimizer partitions the sequence into < strong > super-kmers</ strong > : maximal contiguous runs of overlapping k-mers that share the same minimizer. A single minimizer anchors each super-kmer, enabling partitioned storage and indexing.</ p >
< h2 id = "lexicographic-ordering-and-its-bias" > Lexicographic ordering and its bias</ h2 >
< p > 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 < span class = "arithmatex" > \(\min_{\text{lex}}(\text{fwd}, \text{rc})\)</ span > , so AT-rich m-mers have systematically small values:</ p >
< div class = "arithmatex" > \[\text{canonical}(\text{AAAA}\cdots\text{A}) = \text{canonical}(\text{TTTT}\cdots\text{T}) = 0\]</ div >
< p > 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.</ p >
< h2 id = "random-minimizer" > Random minimizer</ h2 >
< p > A < strong > random minimizer</ strong > replaces lex order with a hash order: define < span class = "arithmatex" > \(H : \{0,1\}^{2m} \to \{0,1\}^{64}\)</ span > and select the m-mer with the < strong > minimum < span class = "arithmatex" > \(H\)</ span > value</ strong > in the window.</ p >
< p > The key property: because < span class = "arithmatex" > \(H\)</ span > 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.</ p >
< h2 id = "why-the-canonical-form-remains-lexicographic" > Why the canonical form remains lexicographic</ h2 >
< p > An apparent alternative is to redefine the canonical form of each m-mer as the strand with the smaller hash value:</ p >
< div class = "arithmatex" > \[\text{canonical}_H(v) = \arg\min(H(\text{fwd}),\ H(\text{rc}))\]</ div >
< p > This must be rejected. The hash of this new canonical is < span class = "arithmatex" > \(\min(H(\text{fwd}), H(\text{rc}))\)</ span > — the minimum of two i.i.d. Uniform< span class = "arithmatex" > \([0, 2^{64})\)</ span > values. Its distribution is:</ p >
< div class = "arithmatex" > \[F(x) = 1 - \left(1 - \frac{x}{2^{64}}\right)^2\]</ div >
< p > with density < span class = "arithmatex" > \(f(x) = 2(1 - x/2^{64})\)</ span > , which is approximately < strong > twice as large near 0 than near < span class = "arithmatex" > \(2^{64}\)</ span ></ strong > . The low-order partition bits inherit this bias: partition 0 receives roughly twice as many super-kmers as the last partition.</ p >
< p > The lex canonical form does not have this problem: < span class = "arithmatex" > \(\text{canonical}_{\text{lex}}(v)\)</ span > is a fixed, deterministic representative of each equivalence class, and < span class = "arithmatex" > \(H(\text{canonical}_{\text{lex}})\)</ span > is uniformly distributed over < span class = "arithmatex" > \([0, 2^{64})\)</ span > independently of the min/max relationship between the two strands.</ p >
< h2 id = "partition-key-independence" > Partition key independence</ h2 >
< p > A further subtlety arises when the selection hash is used directly as the partition key. The selected minimizer is the m-mer with the < strong > minimum</ strong > < span class = "arithmatex" > \(H\)</ span > value in a window of < span class = "arithmatex" > \(W = k - m + 1\)</ span > positions. The minimum of < span class = "arithmatex" > \(W\)</ span > i.i.d. Uniform< span class = "arithmatex" > \([0,2^{64})\)</ span > values has distribution:</ p >
< div class = "arithmatex" > \[F(x) = 1 - \left(1 - \frac{x}{2^{64}}\right)^W \approx \frac{Wx}{2^{64}}\]</ div >
< p > 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.</ p >
< p > The correct approach is to decouple selection from partition routing:</ p >
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< ul >
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< li >< strong > Selection</ strong > uses < span class = "arithmatex" > \(H(\text{canonical}_{\text{lex}}(m\text{-mer}))\)</ span > to pick the minimizer in the window.</ li >
< li >< strong > Partition routing</ strong > recomputes < span class = "arithmatex" > \(H(\text{canonical}_{\text{lex}}(\text{minimizer}))\)</ span > 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 < span class = "arithmatex" > \([0, 2^{64})\)</ span > .</ li >
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</ ul >
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< h2 id = "seed-and-fixed-point-elimination" > Seed and fixed-point elimination</ h2 >
< p > The splitmix64 finalizer has a fixed point at 0:</ p >
< div class = "arithmatex" > \[\text{mix64}(0) = 0\]</ div >
< p > Since < span class = "arithmatex" > \(\text{canonical}_{\text{lex}}(\text{AAAA}\cdots\text{A}) = 0\)</ span > , using unseeded mix64 causes all-A m-mers to win every window comparison, recreating a pathological partition identical to the lex-ordering bias.</ p >
< p > The fix is a non-zero XOR seed applied before mixing:</ p >
< div class = "arithmatex" > \[H(x) = \text{mix64}(x \oplus s), \quad s = \lfloor 2^{64}/\varphi \rfloor = \texttt{0x9e3779b97f4a7c15}\]</ div >
< p > where < span class = "arithmatex" > \(\varphi\)</ span > is the golden ratio. This maps 0 to < span class = "arithmatex" > \(\text{mix64}(s)\)</ span > , a well-distributed non-zero value. No canonical m-mer value has a systematically small < span class = "arithmatex" > \(H\)</ span > .</ p >
< div class = "admonition abstract" >
< p class = "admonition-title" > Hash function < span class = "arithmatex" > \(H\)</ span ></ p >
< div class = "highlight" >< pre >< span ></ span >< code > H(x):
x ← x ⊕ 0x9e3779b97f4a7c15
x ← x ⊕ (x >> 30)
x ← x ×
x ← x ⊕ (x >> 27)
x ← x ×
return x ⊕ (x >> 31)
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