1288 lines
23 KiB
HTML
1288 lines
23 KiB
HTML
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<link rel="next" href="../indexing_architecture/">
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<meta name="generator" content="mkdocs-1.6.1, mkdocs-material-9.7.6">
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<title>Minimizer selection - obikmer — User Guide</title>
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<style>:root{--md-text-font:"Roboto";--md-code-font:"Roboto Mono"}</style>
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<div data-md-component="skip">
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<a href="#minimizer-selection" class="md-skip">
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Skip to content
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obikmer — User Guide
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<div class="md-header__topic" data-md-component="header-topic">
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<span class="md-ellipsis">
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Minimizer selection
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</span>
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</div>
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</div>
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</div>
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</nav>
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<div class="md-container" data-md-component="container">
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<main class="md-main" data-md-component="main">
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obikmer — User Guide
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<ul class="md-nav__list" data-md-scrollfix>
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Home
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<span class="md-ellipsis">
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Installation
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</span>
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</a>
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</li>
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<li class="md-nav__item md-nav__item--active md-nav__item--nested">
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<label class="md-nav__link" for="__nav_3" id="__nav_3_label" tabindex="0">
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<span class="md-ellipsis">
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Theory
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</span>
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="true">
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<label class="md-nav__title" for="__nav_3">
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<span class="md-nav__icon md-icon"></span>
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Theory
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../kmers_and_superkmers/" class="md-nav__link">
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<span class="md-ellipsis">
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Kmers and super-kmers
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../encoding/" class="md-nav__link">
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<span class="md-ellipsis">
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DNA encoding
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../entropy_filter/" class="md-nav__link">
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<span class="md-ellipsis">
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Low-complexity kmer filter
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</span>
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</a>
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</li>
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<li class="md-nav__item md-nav__item--active">
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<input class="md-nav__toggle md-toggle" type="checkbox" id="__toc">
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<label class="md-nav__link md-nav__link--active" for="__toc">
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<span class="md-ellipsis">
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Minimizer selection
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</span>
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<span class="md-nav__icon md-icon"></span>
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</label>
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<a href="./" class="md-nav__link md-nav__link--active">
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<span class="md-ellipsis">
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Minimizer selection
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</span>
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</a>
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<nav class="md-nav md-nav--secondary" aria-label="Table of contents">
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<label class="md-nav__title" for="__toc">
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<span class="md-nav__icon md-icon"></span>
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Table of contents
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</label>
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="#definition" class="md-nav__link">
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<span class="md-ellipsis">
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Definition
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||
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#hash-based-random-minimizer" class="md-nav__link">
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||
<span class="md-ellipsis">
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||
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||
Hash-based ("random") minimizer
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||
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||
</span>
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||
</a>
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||
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||
<nav class="md-nav" aria-label="Hash-based ("random") minimizer">
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||
<ul class="md-nav__list">
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||
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<li class="md-nav__item">
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||
<a href="#hash-function" class="md-nav__link">
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||
<span class="md-ellipsis">
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||
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Hash function
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||
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||
</span>
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||
</a>
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||
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||
</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item">
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<a href="#partition-routing-is-independent-of-minimizer-selection" class="md-nav__link">
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<span class="md-ellipsis">
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Partition routing is independent of minimizer selection
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</span>
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item">
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||
<a href="../indexing_architecture/" class="md-nav__link">
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||
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||
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<span class="md-ellipsis">
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Partitioning and indexing architecture
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||
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||
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</span>
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
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<label class="md-nav__link" for="__nav_4" id="__nav_4_label" tabindex="0">
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||
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<span class="md-ellipsis">
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Usage
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||
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</span>
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||
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<span class="md-nav__icon md-icon"></span>
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||
</label>
|
||
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||
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
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||
<label class="md-nav__title" for="__nav_4">
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<span class="md-nav__icon md-icon"></span>
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Usage
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||
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||
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</label>
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||
<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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||
<a href="../../usage/superkmer/" class="md-nav__link">
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<span class="md-ellipsis">
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superkmer
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||
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</span>
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||
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</a>
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</li>
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||
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<li class="md-nav__item">
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||
<a href="../../usage/index_command/" class="md-nav__link">
|
||
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||
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||
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||
<span class="md-ellipsis">
|
||
|
||
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||
index
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||
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||
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||
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||
</span>
|
||
|
||
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||
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||
</a>
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||
</li>
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||
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||
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||
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||
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||
|
||
|
||
|
||
|
||
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||
|
||
<li class="md-nav__item">
|
||
<a href="../../usage/merge/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
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||
merge
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
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||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../usage/filter/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
filter
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../usage/select/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
select
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../usage/query/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
query
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
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||
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|
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|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../usage/dump/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
dump
|
||
|
||
|
||
|
||
</span>
|
||
|
||
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||
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||
</a>
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||
</li>
|
||
|
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|
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|
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|
||
|
||
|
||
|
||
|
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|
||
<li class="md-nav__item">
|
||
<a href="../../usage/annotate/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
annotate
|
||
|
||
|
||
|
||
</span>
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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 × 0xbf58476d1ce4e5b9
|
||
x ← x ⊕ (x >> 27)
|
||
x ← x × 0x94d049bb133111eb
|
||
return x ⊕ (x >> 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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