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<title>Polylogarithmic function</title>
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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Polylogarithmic function</span></span>
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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Polylogarithm" title="Polylogarithm">Polylogarithm</a>.</div>
<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>polylogarithmic function</b> in <span class="texhtml mvar" style="font-style:italic;">n</span> is a <a href="Polynomial" title="Polynomial">polynomial</a> in the <a href="Logarithm" title="Logarithm">logarithm</a> of <span class="texhtml mvar" style="font-style:italic;">n</span>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
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<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{k}(\log n)^{k}+a_{k-1}(\log n)^{k-1}+\cdots +a_{1}(\log n)+a_{0}.}">
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<annotation encoding="application/x-tex">{\displaystyle a_{k}(\log n)^{k}+a_{k-1}(\log n)^{k-1}+\cdots +a_{1}(\log n)+a_{0}.}</annotation>
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</math></span><img src="./d0a3f45af82e7455d44541124c95edcec15e9a45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.004ex; height:3.176ex;" alt="{\displaystyle a_{k}(\log n)^{k}+a_{k-1}(\log n)^{k-1}+\cdots +a_{1}(\log n)+a_{0}.}" loading="lazy"></span></dd></dl>
<p>The notation <span class="texhtml">log<sup><i>k</i></sup><i>n</i></span> is often used as a <a href="Shorthand" title="Shorthand">shorthand</a> for <span class="texhtml">(log <i>n</i>)<sup><i>k</i></sup></span>, analogous to <span class="texhtml">sin<sup>2</sup><i>θ</i></span> for <span class="texhtml">(sin <i>θ</i>)<sup>2</sup></span>.
</p><p>In <a href="Computer_science" title="Computer science">computer science</a>, polylogarithmic functions occur as the <a href="Big_O_notation" title="Big O notation">order</a> of <a href="Time_complexity" title="Time complexity">time</a> for some <a href="Data_structure" title="Data structure">data structure</a> operations. Additionally, the <a href="Exponential_function" title="Exponential function">exponential function</a> of a polylogarithmic function produces a function with <a href="Quasi-polynomial_growth" title="Quasi-polynomial growth">quasi-polynomial growth</a>, and algorithms with this as their <a href="Time_complexity" title="Time complexity">time complexity</a> are said to take <a href="Quasi-polynomial_time" title="Quasi-polynomial time">quasi-polynomial time</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p><p>All polylogarithmic functions of <span class="texhtml mvar" style="font-style:italic;">n</span> are <span class="texhtml">o(<i>n</i><sup><i>ε</i></sup>)</span> for every exponent <span class="texhtml"><i>ε</i> > 0</span> (for the meaning of this symbol, see <a href="Small_o_notation" class="mw-redirect" title="Small o notation">small o notation</a>), that is, a polylogarithmic function grows more slowly than any positive exponent. This observation is the basis for the <a href="Soft_O_notation" class="mw-redirect" title="Soft O notation">soft O notation</a> <span class="texhtml">Õ(<i>n</i>)</span>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>
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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFBlack2004" class="citation web cs1">Black, Paul E. (2004-12-17). <a rel="nofollow" class="external text" href="http://xlinux.nist.gov/dads/HTML/polylogarith.html">"polylogarithmic"</a>. <i>Dictionary of Algorithms and Data Structures</i>. U.S. National Institute of Standards and Technology<span class="reference-accessdate">. Retrieved <span class="nowrap">2010-01-10</span></span>.</cite></span>
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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><i><a href="Complexity_Zoo" class="mw-redirect" title="Complexity Zoo">Complexity Zoo</a></i>: <a rel="nofollow" class="external text" href="https://complexityzoo.net/Complexity_Zoo:Q#qp">Class QP: Quasipolynomial-Time</a></span>
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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFCormenLeisersonRivestStein2022" class="citation book cs1">Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2022). <a rel="nofollow" class="external text" href="https://mitpress.mit.edu/9780262046305/introduction-to-algorithms/"><i>Introduction to Algorithms</i></a> (4th ed.). Cambridge, Mass.: The MIT Press. pp. <span class="nowrap">74–</span>75. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780262046305</bdi>.</cite></span>
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