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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/permutation.html" class="active"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../../background/upa.html"><strong aria-hidden
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<h1><a class="header" href="#permutation-argument" id="permutation-argument">Permutation argument</a></h1>
<p>Given that gates in halo2 circuits operate &quot;locally&quot; (on cells in the current row or
defined relative rows), it is common to need to copy a value from some arbitrary cell into
the current row for use in a gate. This is performed with an equality constraint, which
enforces that the source and destination cells contain the same value.</p>
<p>We implement these equality constraints by constructing a permutation that represents the
constraints, and then using a permutation argument within the proof to enforce them.</p>
<h2><a class="header" href="#notation" id="notation">Notation</a></h2>
<p>A permutation is a one-to-one and onto mapping of a set onto itself. A permutation can be
factored uniquely into a composition of cycles (up to ordering of cycles, and rotation of
each cycle).</p>
<p>We sometimes use <a href="https://en.wikipedia.org/wiki/Permutation#Cycle_notation">cycle notation</a>
to write permutations. Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mspace"> </span><span class="mord mathnormal">b</span><span class="mspace"> </span><span class="mord mathnormal">c</span><span class="mclose">)</span></span></span></span> denote a cycle where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">a</span></span></span></span> maps to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">b</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">b</span></span></span></span> maps to
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">c</span></span></span></span>, and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">c</span></span></span></span> maps to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">a</span></span></span></span> (with the obvious generalisation to arbitrary-sized cycles).
Writing two or more cycles next to each other denotes a composition of the corresponding
permutations. For example, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mspace"> </span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mspace"> </span><span class="mopen">(</span><span class="mord mathnormal">c</span><span class="mspace"> </span><span class="mord mathnormal">d</span><span class="mclose">)</span></span></span></span> denotes the permutation that maps <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">a</span></span></span></span> to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">b</span></span></span></span>,
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">b</span></span></span></span> to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">a</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">c</span></span></span></span> to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">d</span></span></span></span>, and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">d</span></span></span></span> to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">c</span></span></span></span>.</p>
<h2><a class="header" href="#constructing-the-permutation" id="constructing-the-permutation">Constructing the permutation</a></h2>
<h3><a class="header" href="#goal" id="goal">Goal</a></h3>
<p>We want to construct a permutation in which each subset of variables that are in a
equality-constraint set form a cycle. For example, suppose that we have a circuit that
defines the following equality constraints:</p>
<ul>
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.46375em;vertical-align:0em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">b</span></span></span></span></li>
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.46375em;vertical-align:0em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">c</span></span></span></span></li>
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">e</span></span></span></span></li>
</ul>
<p>From this we have the equality-constraint sets <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">c</span><span class="mclose">}</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">d</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">e</span><span class="mclose">}</span></span></span></span>. We want to
construct the permutation:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mspace"> </span><span class="mord mathnormal">b</span><span class="mspace"> </span><span class="mord mathnormal">c</span><span class="mclose">)</span><span class="mspace"> </span><span class="mopen">(</span><span class="mord mathnormal">d</span><span class="mspace"> </span><span class="mord mathnormal">e</span><span class="mclose">)</span></span></span></span></span></p>
<p>which defines the mapping of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">c</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">d</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">e</span><span class="mclose">]</span></span></span></span> to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">c</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">e</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">d</span><span class="mclose">]</span></span></span></span>.</p>
<h3><a class="header" href="#algorithm" id="algorithm">Algorithm</a></h3>
<p>We need to keep track of the set of cycles, which is a
<a href="https://en.wikipedia.org/wiki/Disjoint-set_data_structure">set of disjoint sets</a>.
Efficient data structures for this problem are known; for the sake of simplicity we choose
one that is not asymptotically optimal but is easy to implement.</p>
<p>We represent the current state as:</p>
<ul>
<li>an array <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.87381em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathsf">m</span><span class="mord mathsf">a</span><span class="mord mathsf">p</span><span class="mord mathsf">p</span><span class="mord mathsf">i</span><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.01389em;">g</span></span></span></span></span> for the permutation itself;</li>
<li>an auxiliary array <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.44444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathsf">a</span><span class="mord mathsf">u</span><span class="mord mathsf">x</span></span></span></span></span> that keeps track of a distinguished element of each
cycle;</li>
<li>another array <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.67937em;vertical-align:0em;"></span><span class="mord"><span class="mord mathsf">s</span><span class="mord mathsf">i</span><span class="mord mathsf">z</span><span class="mord mathsf">e</span><span class="mord mathsf">s</span></span></span></span></span> that keeps track of the size of each cycle.</li>
</ul>
<p>We have the invariant that for each element <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">x</span></span></span></span> in a given cycle <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">a</span><span class="mord mathsf">u</span><span class="mord mathsf">x</span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>
points to the same element <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>. This allows us to quickly decide whether two given
elements <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">x</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> are in the same cycle, by checking whether
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">a</span><span class="mord mathsf">u</span><span class="mord mathsf">x</span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">a</span><span class="mord mathsf">u</span><span class="mord mathsf">x</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span>. Also, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">s</span><span class="mord mathsf">i</span><span class="mord mathsf">z</span><span class="mord mathsf">e</span><span class="mord mathsf">s</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">a</span><span class="mord mathsf">u</span><span class="mord mathsf">x</span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mclose">)</span></span></span></span> gives the
size of the cycle containing <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">x</span></span></span></span>. (This is guaranteed only for
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">s</span><span class="mord mathsf">i</span><span class="mord mathsf">z</span><span class="mord mathsf">e</span><span class="mord mathsf">s</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">a</span><span class="mord mathsf">u</span><span class="mord mathsf">x</span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mclose">)</span><span class="mclose">)</span></span></span></span>, not for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">s</span><span class="mord mathsf">i</span><span class="mord mathsf">z</span><span class="mord mathsf">e</span><span class="mord mathsf">s</span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>.)</p>
<p>The algorithm starts with a representation of the identity permutation:
for all <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">x</span></span></span></span>, we set <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">m</span><span class="mord mathsf">a</span><span class="mord mathsf">p</span><span class="mord mathsf">p</span><span class="mord mathsf">i</span><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.01389em;">g</span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">x</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">a</span><span class="mord mathsf">u</span><span class="mord mathsf">x</span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">x</span></span></span></span>, and
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">s</span><span class="mord mathsf">i</span><span class="mord mathsf">z</span><span class="mord mathsf">e</span><span class="mord mathsf">s</span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span>.</p>
<p>To add an equality constraint <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathit">l</span><span class="mord mathit">e</span><span class="mord mathit">f</span><span class="mord mathit">t</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathit">r</span><span class="mord mathit">i</span><span class="mord mathit">g</span><span class="mord mathit">h</span><span class="mord mathit">t</span></span></span></span></span>:</p>
<ol>
<li>Check whether <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathit">l</span><span class="mord mathit">e</span><span class="mord mathit">f</span><span class="mord mathit">t</span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathit">r</span><span class="mord mathit">i</span><span class="mord mathit">g</span><span class="mord mathit">h</span><span class="mord mathit">t</span></span></span></span></span> are already in the same cycle, i.e.
whether <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">a</span><span class="mord mathsf">u</span><span class="mord mathsf">x</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathit">l</span><span class="mord mathit">e</span><span class="mord mathit">f</span><span class="mord mathit">t</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">a</span><span class="mord mathsf">u</span><span class="mord mathsf">x</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathit">r</span><span class="mord mathit">i</span><span class="mord mathit">g</span><span class="mord mathit">h</span><span class="mord mathit">t</span></span><span class="mclose">)</span></span></span></span>. If so, there is
nothing to do.</li>
<li>Otherwise, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathit">l</span><span class="mord mathit">e</span><span class="mord mathit">f</span><span class="mord mathit">t</span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathit">r</span><span class="mord mathit">i</span><span class="mord mathit">g</span><span class="mord mathit">h</span><span class="mord mathit">t</span></span></span></span></span> belong to different cycles. Make
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathit">l</span><span class="mord mathit">e</span><span class="mord mathit">f</span><span class="mord mathit">t</span></span></span></span></span> the larger cycle and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathit">r</span><span class="mord mathit">i</span><span class="mord mathit">g</span><span class="mord mathit">h</span><span class="mord mathit">t</span></span></span></span></span> the smaller one, by swapping them
iff <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">s</span><span class="mord mathsf">i</span><span class="mord mathsf">z</span><span class="mord mathsf">e</span><span class="mord mathsf">s</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">a</span><span class="mord mathsf">u</span><span class="mord mathsf">x</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathit">l</span><span class="mord mathit">e</span><span class="mord mathit">f</span><span class="mord mathit">t</span></span><span class="mclose">)</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">s</span><span class="mord mathsf">i</span><span class="mord mathsf">z</span><span class="mord mathsf">e</span><span class="mord mathsf">s</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">a</span><span class="mord mathsf">u</span><span class="mord mathsf">x</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathit">r</span><span class="mord mathit">i</span><span class="mord mathit">g</span><span class="mord mathit">h</span><span class="mord mathit">t</span></span><span class="mclose">)</span><span class="mclose">)</span></span></span></span>.</li>
<li>Following the mapping around the right (smaller) cycle, for each element <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">x</span></span></span></span> set
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">a</span><span class="mord mathsf">u</span><span class="mord mathsf">x</span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">a</span><span class="mord mathsf">u</span><span class="mord mathsf">x</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathit">l</span><span class="mord mathit">e</span><span class="mord mathit">f</span><span class="mord mathit">t</span></span><span class="mclose">)</span></span></span></span>.</li>
<li>Splice the smaller cycle into the larger one by swapping <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">m</span><span class="mord mathsf">a</span><span class="mord mathsf">p</span><span class="mord mathsf">p</span><span class="mord mathsf">i</span><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.01389em;">g</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathit">l</span><span class="mord mathit">e</span><span class="mord mathit">f</span><span class="mord mathit">t</span></span><span class="mclose">)</span></span></span></span>
with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">m</span><span class="mord mathsf">a</span><span class="mord mathsf">p</span><span class="mord mathsf">p</span><span class="mord mathsf">i</span><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.01389em;">g</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathit">r</span><span class="mord mathit">i</span><span class="mord mathit">g</span><span class="mord mathit">h</span><span class="mord mathit">t</span></span><span class="mclose">)</span></span></span></span>.</li>
</ol>
<p>For example, given two disjoint cycles <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace"> </span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace"> </span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace"> </span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mclose">)</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace"> </span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace"> </span><span class="mord mathnormal">G</span><span class="mspace"> </span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span></span>:</p>
<pre><code class="language-plaintext">A +---&gt; B
^ +
| |
+ v
D &lt;---+ C E +---&gt; F
^ +
| |
+ v
H &lt;---+ G
</code></pre>
<p>After adding constraint <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span></span></span></span> the above algorithm produces the cycle:</p>
<pre><code class="language-plaintext">A +---&gt; B +-------------+
^ |
| |
+ v
D &lt;---+ C &lt;---+ E F
^ +
| |
+ v
H &lt;---+ G
</code></pre>
<h3><a class="header" href="#broken-alternatives" id="broken-alternatives">Broken alternatives</a></h3>
<p>If we did not check whether <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathit">l</span><span class="mord mathit">e</span><span class="mord mathit">f</span><span class="mord mathit">t</span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathit">r</span><span class="mord mathit">i</span><span class="mord mathit">g</span><span class="mord mathit">h</span><span class="mord mathit">t</span></span></span></span></span> were already in the same
cycle, then we could end up undoing an equality constraint. For example, if we have the
following constraints:</p>
<ul>
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.46375em;vertical-align:0em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">b</span></span></span></span></li>
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">c</span></span></span></span></li>
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.46375em;vertical-align:0em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">d</span></span></span></span></li>
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">d</span></span></span></span></li>
</ul>
<p>and we tried to implement adding an equality constraint just using step 4 of the above
algorithm, then we would end up constructing the cycle <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mspace"> </span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mspace"> </span><span class="mopen">(</span><span class="mord mathnormal">c</span><span class="mspace"> </span><span class="mord mathnormal">d</span><span class="mclose">)</span></span></span></span>, rather than the
correct <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mspace"> </span><span class="mord mathnormal">b</span><span class="mspace"> </span><span class="mord mathnormal">c</span><span class="mspace"> </span><span class="mord mathnormal">d</span><span class="mclose">)</span></span></span></span>.</p>
<h2><a class="header" href="#argument-specification" id="argument-specification">Argument specification</a></h2>
<p>TODO: Document what we do with the permutation once we have it.</p>
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