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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> PLONKish Arithmetization</a></li><li class="chapter-item expanded "><a href="../../concepts/chips.html"><strong aria-hidden="true">1.3.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../../concepts/gadgets.html"><strong aria-hidden="true">1.4.</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/dev-tools.html"><strong aria-hidden="true">2.1.</strong> Developer tools</a></li><li class="chapter-item expanded "><a href="../../user/simple-example.html"><strong aria-hidden="true">2.2.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../../user/lookup-tables.html"><strong aria-hidden="true">2.3.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../../user/gadgets.html"><strong aria-hidden="true">2.4.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../../user/tips-and-tricks.html"><strong aria-hidden="true">2.5.</strong> Tips and tricks</a></li><li class="chapter-item expanded "><a href="../../user/wasm-port.html"><strong aria-hidden="true">2.6.</strong> WASM Guide</a></li></ol></li><li class="chapter-item expanded "><a href="../../dev.html"><strong aria-hidden="true">3.</strong> Developer Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../dev/features.html"><strong aria-hidden="true">3.1.</strong> Feature development</a></li></ol></li><li class="chapter-item expanded "><a href="../../design.html"><strong aria-hidden="true">4.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/proving-system.html"><strong aria-hidden="true">4.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/proving-system/lookup.html"><strong aria-hidden="true">4.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/permutation.html"><strong aria-hidden="true">4.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">4.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/vanishing.html"><strong aria-hidden="true">4.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">4.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/inner-product.html"><strong aria-hidden="true">4.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/comparison.html"><strong aria-hidden="true">4.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../../design/protocol.html"><strong aria-hidden="true">4.2.</strong> Protocol Description</a></li><li class="chapter-item expanded "><a href="../../design/implementation.html"><strong aria-hidden="true">4.3.</strong> Implementation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/implementation/proofs.html"><strong aria-hidden="true">4.3.1.</strong> Proofs</a></li><li class="chapter-item expanded "><a href="../../design/implementation/fields.html"><strong aria-hidden="true">4.3.2.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../../design/implementation/selector-combining.html" class="active"><strong aria-hidden="true">4.3.3.</strong> Selector combining</a></li></ol></li><li class="chapter-item expanded "><a href="../../design/gadgets.html"><strong aria-hidden="true">4.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/gadgets/ecc.html"><strong aria-hidden="true">4.4.1.</strong> Elliptic curve cryptography</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/gadgets/ecc/witnessing-points.html"><strong aria-hidden="true">4.4.1.1.</strong> Witnessing points</a></li><li class="chapter-item expanded "><a href="../../design/gadgets/ecc/addition.html"><strong aria-hidden="true">4.4.1.2.</strong> Incomplete and complete addition</a></li><li class="chapter-item expanded "><a href="../../design/gadgets/ecc/fixed-base-scalar-mul.html"><strong aria-hidden="true">4.4.1.3.</strong> Fixed-base scalar multiplication</a></li><li class="chapter-item expanded "><a href="../../design/gadgets/ecc/var-base-scalar-mul.html"><strong aria-hidden="true">4.4.1.4.</strong> Variable-base scalar multiplication</a></li></ol></li><li class="chapter-item expanded "><a href="../../design/gadgets/sinsemilla.html"><strong aria-hidden="true">4.4.2.</strong> Sinsemilla</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/gadgets/sinsemilla/merkle-crh.html"><strong aria-hidden="true">4.4.2.1.</strong> MerkleCRH</a></li></ol></li><li class="chapter-item expanded "><a href="../../design/gadgets/decomposition.html"><strong aria-hidden="true">4.4.3.</strong> Decomposition</a></li><li class="chapter-item expanded "><a href="../../design/gadgets/sha256.html"><strong aria-hidden="true">4.4.4.</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">4.4.4.1.</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">5.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../background/fields.html"><strong aria-hidden="true">5.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../../background/polynomials.html"><strong aria-hidden="true">5.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../../background/groups.html"><strong aria-hidden="true">5.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../../background/curves.html"><strong aria-hidden="true">5.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../../background/pc-ipa.html"><strong aria-hidden="true">5.5.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../../background/recursion.html"><strong aria-hidden="true">5.6.</strong> Recursion</a></li></ol></li></ol>
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<h1 id="selector-combining"><a class="header" href="#selector-combining">Selector combining</a></h1>
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<p>Heavy use of custom gates can lead to a circuit defining many binary selectors, which
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would increase proof size and verification time.</p>
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<p>This section describes an optimization, applied automatically by halo2, that combines
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binary selector columns into fewer fixed columns.</p>
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<p>The basic idea is that if we have <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℓ</span></span></span></span> binary selectors labelled <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">ℓ</span></span></span></span> that
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are enabled on disjoint sets of rows, then under some additional conditions we can combine
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them into a single fixed column, say <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span></span></span></span>, such that:
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<span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">0</span><span class="mpunct">,</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if the selector labelled </span></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord text"><span class="mord"> is </span></span><span class="mord">1</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if all these selectors are </span></span><span class="mord">0.</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
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<p>However, the devil is in the detail.</p>
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<p>The halo2 API allows defining some selectors to be "simple selectors", subject to the
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following condition:</p>
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<blockquote>
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<p>Every polynomial constraint involving a simple selector <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">s</span></span></span></span> must be of the form
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<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mpunct">,</span></span></span></span> where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> is a polynomial involving <em>no</em> simple selectors.</p>
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</blockquote>
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<p>Suppose that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">s</span></span></span></span> has label <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> in some set of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℓ</span></span></span></span> simple selectors that are combined
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into <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span></span></span></span> as above. Then this condition ensures that replacing <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">s</span></span></span></span> by
|
||
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1858em;vertical-align:-0.4358em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∏</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1864em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mrel mtight">≤</span><span class="mord mathnormal mtight">h</span><span class="mrel mtight">≤</span><span class="mord mtight">ℓ</span><span class="mpunct mtight">,</span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mathnormal mtight">h</span><span class="mrel mtight"><span class="mrel mtight"><span class="mord vbox mtight"><span class="thinbox mtight"><span class="rlap mtight"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord mtight"><span class="mrel mtight"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel mtight">=</span></span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4358em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord mathnormal">h</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mclose">)</span></span></span></span> will not change the meaning of
|
||
any constraints.</p>
|
||
<blockquote>
|
||
<p>It would be possible to relax this condition by ensuring that every use of a binary
|
||
selector is substituted by a precise interpolation of its value from the corresponding
|
||
combined selector. However,</p>
|
||
<ul>
|
||
<li>the restriction simplifies the implementation, developer tooling, and human
|
||
understanding and debugging of the resulting constraint system;</li>
|
||
<li>the scope to apply the optimization is not impeded very much by this restriction for
|
||
typical circuits.</li>
|
||
</ul>
|
||
</blockquote>
|
||
<p>Note that replacing <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">s</span></span></span></span> by <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1858em;vertical-align:-0.4358em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∏</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1864em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mrel mtight">≤</span><span class="mord mathnormal mtight">h</span><span class="mrel mtight">≤</span><span class="mord mtight">ℓ</span><span class="mpunct mtight">,</span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mathnormal mtight">h</span><span class="mrel mtight"><span class="mrel mtight"><span class="mord vbox mtight"><span class="thinbox mtight"><span class="rlap mtight"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord mtight"><span class="mrel mtight"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel mtight">=</span></span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4358em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord mathnormal">h</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mclose">)</span></span></span></span> will
|
||
increase the degree of constraints selected by <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">s</span></span></span></span> by <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord">ℓ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>, and so we must choose the
|
||
selectors that are combined in such a way that the maximum degree bound is not exceeded.</p>
|
||
<h2 id="identifying-selectors-that-can-be-combined"><a class="header" href="#identifying-selectors-that-can-be-combined">Identifying selectors that can be combined</a></h2>
|
||
<p>We need a partition of the overall set of selectors <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> into subsets
|
||
(called "combinations"), such that no two selectors in a combination are enabled on the
|
||
same row.</p>
|
||
<p>Labels must be unique within a combination, but they are not unique across combinations.
|
||
Do not confuse a selector's index with its label.</p>
|
||
<p>Suppose that we are given <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">max_degree</span></span></span></span></span>, the degree bound of the circuit.</p>
|
||
<p>We use the following algorithm:</p>
|
||
<ol>
|
||
<li>Leave nonsimple selectors unoptimized, i.e. map each of them to a separate fixed
|
||
column.</li>
|
||
<li>Check (or ensure by construction) that all polynomial constraints involving each simple
|
||
selector <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> are of the form <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5945em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9012em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9012em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> do not involve
|
||
any simple selectors. For each <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, record the maximum degree of any <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9012em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> as
|
||
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9531em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathsf mtight">max</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span>.</li>
|
||
<li>Compute a binary "exclusion matrix" <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> is <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> whenever <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span>
|
||
and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> are enabled on the same row; and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> otherwise.
|
||
<blockquote>
|
||
<p>Since <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> is symmetric and is zero on the diagonal, we can represent it by either its
|
||
upper or lower triangular entries. The rest of the algorithm is guaranteed only to
|
||
access only the entries <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>.</p>
|
||
</blockquote>
|
||
</li>
|
||
<li>Initialize a boolean array <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9028em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">added</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0..</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> to all <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">false</span></span></span></span></span>.
|
||
<blockquote>
|
||
<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">added</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> will record whether <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> has been included in any combination.</p>
|
||
</blockquote>
|
||
</li>
|
||
<li>Iterate over the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> that have not yet been added to any combination:
|
||
<ul>
|
||
<li>a. Add <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> to a fresh combination <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span></span></span></span>, and set <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">added</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5714em;"></span><span class="mord"><span class="mord mathsf">true</span></span></span></span></span>.</li>
|
||
<li>b. Let mut <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9531em;vertical-align:-0.2587em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathsf mtight">max</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>.
|
||
<blockquote>
|
||
<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span></span></span></span> is used to keep track of the largest degree, <em>excluding</em> the selector
|
||
expression, of any gate involved in the combination <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span></span></span></span> so far.</p>
|
||
</blockquote>
|
||
</li>
|
||
<li>c. Iterate over all the selectors <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> for <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> that can potentially join <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span></span></span></span>,
|
||
i.e. for which <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">added</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> is false:
|
||
<ul>
|
||
<li>i. (Optimization) If <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">len</span></span><span class="mopen">(</span><span class="mord mathnormal">c</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">max_degree</span></span></span></span></span>, break to the
|
||
outer loop, since no more selectors can be added to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span></span></span></span>.</li>
|
||
<li>ii. Let <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathsf mtight" style="margin-right:0.01389em;">new</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1448em;vertical-align:-0.3948em;"></span><span class="mord"><span class="mord mathsf">max</span></span><span class="mopen">(</span><span class="mord mathnormal">d</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathsf mtight">max</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span>.</li>
|
||
<li>iii. If <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> is <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5714em;"></span><span class="mord"><span class="mord mathsf">true</span></span></span></span></span> for any <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal">i</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span></span></span></span>, or if
|
||
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathsf mtight" style="margin-right:0.01389em;">new</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">len</span></span><span class="mopen">(</span><span class="mord mathnormal">c</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">max_degree</span></span></span></span></span>, break to the outer
|
||
loop.
|
||
<blockquote>
|
||
<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathsf mtight" style="margin-right:0.01389em;">new</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">len</span></span><span class="mopen">(</span><span class="mord mathnormal">c</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> is the maximum degree, <em>including</em> the
|
||
selector expression, of any constraint that would result from adding <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> to the
|
||
combination <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span></span></span></span>.</p>
|
||
</blockquote>
|
||
</li>
|
||
<li>iv. Set <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathsf mtight" style="margin-right:0.01389em;">new</span></span></span></span></span></span></span></span></span></span></span></span>.</li>
|
||
<li>v. Add <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span></span></span></span> and set <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">added</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5714em;"></span><span class="mord"><span class="mord mathsf">true</span></span></span></span></span>.</li>
|
||
</ul>
|
||
</li>
|
||
<li>d. Allocate a fixed column <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, initialized to all-zeroes.</li>
|
||
<li>e. For each selector <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.791em;vertical-align:-0.0391em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span></span></span></span>:
|
||
<ul>
|
||
<li>i. Label <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> with a distinct index <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathsf">len</span></span><span class="mopen">(</span><span class="mord mathnormal">c</span><span class="mclose">)</span></span></span></span>.</li>
|
||
<li>ii. Record that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> should be substituted with
|
||
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.2247em;vertical-align:-0.4747em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∏</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2253em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mrel mtight">≤</span><span class="mord mathnormal mtight">h</span><span class="mrel mtight">≤</span><span class="mord mtight"><span class="mord mathsf mtight">len</span></span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">c</span><span class="mclose mtight">)</span><span class="mpunct mtight">,</span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mathnormal mtight">h</span><span class="mrel mtight"><span class="mrel mtight"><span class="mord vbox mtight"><span class="thinbox mtight"><span class="rlap mtight"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord mtight"><span class="mrel mtight"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel mtight">=</span></span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4747em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">h</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> in all gate
|
||
constraints.</li>
|
||
<li>iii. For each row <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> is enabled at <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span>, assign the value <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> to
|
||
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> at row <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span>.</li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
</ol>
|
||
<p>The above algorithm is implemented in
|
||
<a href="https://github.com/zcash/halo2/blob/main/halo2_proofs/src/plonk/circuit/compress_selectors.rs">halo2_proofs/src/plonk/circuit/compress_selectors.rs</a>.
|
||
This is used by the <code>compress_selectors</code> function of
|
||
<a href="https://github.com/zcash/halo2/blob/main/halo2_proofs/src/plonk/circuit.rs">halo2_proofs/src/plonk/circuit.rs</a>
|
||
which does the actual substitutions.</p>
|
||
<h2 id="writing-circuits-to-take-best-advantage-of-selector-combining"><a class="header" href="#writing-circuits-to-take-best-advantage-of-selector-combining">Writing circuits to take best advantage of selector combining</a></h2>
|
||
<p>For this optimization it is beneficial for a circuit to use simple selectors as far as
|
||
possible, rather than fixed columns. It is usually not beneficial to do manual combining
|
||
of selectors, because the resulting fixed columns cannot take part in the automatic
|
||
combining. That means that to get comparable results you would need to do a global
|
||
optimization manually, which would interfere with writing composable gadgets.</p>
|
||
<p>Whether two selectors are enabled on the same row (and so are inhibited from being
|
||
combined) depends on how regions are laid out by the floor planner. The currently
|
||
implemented floor planners do not attempt to take this into account. We suggest not
|
||
worrying about it too much — the gains that can be obtained by cajoling a floor planner to
|
||
shuffle around regions in order to improve combining are likely to be relatively small.</p>
|
||
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