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< h1 id = "minimizer-selection" > Minimizer selection</ h1 >
< h2 id = "definition" > Definition</ h2 >
< p > A < strong > minimizer</ strong > of a kmer window is the m-mer (< span class = "arithmatex" > \(m < k\)</ span > ) that is smallest, among all < span class = "arithmatex" > \(k - m + 1\)</ span > 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.</ p >
< p > The minimizer partitions a sequence into super-kmers: maximal runs of overlapping kmers that share the same minimizer (see < a href = "../kmers_and_superkmers/" > Kmers and super-kmers</ a > ).</ p >
< h2 id = "hash-based-random-minimizer" > Hash-based ("random") minimizer</ h2 >
< p >< code > obikmer</ code > 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.</ p >
< p > Instead, a well-distributed hash function < span class = "arithmatex" > \(H\)</ span > is applied to the canonical (lexicographically minimal) form of each m-mer, and the m-mer with the smallest < span class = "arithmatex" > \(H\)</ span > value wins. Because < span class = "arithmatex" > \(H\)</ span > is a bijection with good avalanche properties, every distinct m-mer in a window has an equal chance of holding the minimum hash value, independent of its nucleotide composition.</ p >
< p > The canonical form used as input to < span class = "arithmatex" > \(H\)</ span > is still the lexicographic minimum of forward/reverse-complement — hashing is applied on top of it, not used to redefine it. Defining canonicity by hash value instead would bias the < em > distribution of hash values themselves</ em > toward small values (the minimum of two independent hashes is not uniformly distributed), reintroducing a bias one layer down.</ p >
< h3 id = "hash-function" > Hash function</ h3 >
< p > The hash function is a 64-bit mixing function (splitmix64-style finalizer) applied to the m-mer XORed with a fixed non-zero seed:</ p >
< div class = "arithmatex" > \[H(x) = \text{mix64}(x \oplus s), \quad s = \lfloor 2^{64}/\varphi \rfloor = \texttt{0x9e3779b97f4a7c15}\]</ div >
< 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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</ code ></ pre ></ div >
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< p > The XOR seed avoids the finalizer's fixed point at 0 (< span class = "arithmatex" > \(\text{mix64}(0) = 0\)</ span > ), which would otherwise make an all-A m-mer (canonical value 0) win every window comparison.</ p >
< h2 id = "partition-routing-is-independent-of-minimizer-selection" > Partition routing is independent of minimizer selection</ h2 >
< p > The hash used to select a minimizer within a window (the minimum of several hash values) and the hash used to route a super-kmer to a storage partition are computed separately:</ p >
< ul >
< li >< strong > Selection</ strong > uses < span class = "arithmatex" > \(H\)</ span > applied to every candidate m-mer in the window, keeping the minimum.</ li >
< li >< strong > Partition routing</ strong > recomputes < span class = "arithmatex" > \(H\)</ span > on the single selected minimizer only, once its position is fixed. This is a hash of one specific value, not the minimum of several, so it is uniformly distributed and safe to use directly for routing.</ li >
</ ul >
< p > See < a href = "../indexing_architecture/" > Partitioning and indexing architecture</ a > for how the routing value is turned into a partition index.</ p >
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