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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">\(m &lt; k\)</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 &gt;&gt; 30)
x ← x × 0xbf58476d1ce4e5b9
x ← x ⊕ (x &gt;&gt; 27)
x ← x × 0x94d049bb133111eb
return x ⊕ (x &gt;&gt; 31)
</code></pre></div>
<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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