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<div class="ChapSects"><a href="chap33_mj.html#X7988ECB7803BA915">33 <span class="Heading"> Finite metric spaces and their filtered complexes </span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap33_mj.html#X7CFDEEC07F15CF82">33.1 <span class="Heading">  </span></a>
</span>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33_mj.html#X7F8113757F7DD2F4">33.1-1 CayleyMetric</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33_mj.html#X79DA33CB7D46CAB4">33.1-2 HammingMetric</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33_mj.html#X7BD62D75829F8701">33.1-3 KendallMetric</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33_mj.html#X789AE7CE8445A67C">33.1-4 EuclideanSquaredMetric</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33_mj.html#X79497F698557427E">33.1-5 EuclideanApproximatedMetric</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33_mj.html#X8763D1167EF519A1">33.1-6 ManhattanMetric</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33_mj.html#X7C86B58A7CEA5513">33.1-7 VectorsToSymmetricMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33_mj.html#X7895AABC7904E9CA">33.1-8 SymmetricMatDisplay</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33_mj.html#X79CA51F27C07435C">33.1-9 SymmetricMatrixToFilteredGraph</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33_mj.html#X8720F77B7ED74747">33.1-10 PermGroupToFilteredGraph</a></span>
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<h3>33 <span class="Heading"> Finite metric spaces and their filtered complexes </span></h3>

<p><a id="X7CFDEEC07F15CF82" name="X7CFDEEC07F15CF82"></a></p>

<h4>33.1 <span class="Heading">  </span></h4>

<p><a id="X7F8113757F7DD2F4" name="X7F8113757F7DD2F4"></a></p>

<h5>33.1-1 CayleyMetric</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CayleyMetric</code>( <var class="Arg">g</var>, <var class="Arg">h</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CayleyMetric</code>( <var class="Arg">g</var>, <var class="Arg">h</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs two permutations <span class="SimpleMath">\(g,h\)</span> and optionally the degree <span class="SimpleMath">\(N\)</span> of a symmetric group containing them. It returns the minimum number of transpositions needed to express <span class="SimpleMath">\(g*h^-1\)</span> as a product of transpositions.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutMetrics.html">1</a></span> </p>

<p><a id="X79DA33CB7D46CAB4" name="X79DA33CB7D46CAB4"></a></p>

<h5>33.1-2 HammingMetric</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HammingMetric</code>( <var class="Arg">g</var>, <var class="Arg">h</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HammingMetric</code>( <var class="Arg">g</var>, <var class="Arg">h</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs two permutations <span class="SimpleMath">\(g,h\)</span> and optionally the degree <span class="SimpleMath">\(N\)</span> of a symmetric group containing them. It returns the number of integers moved by the permutation <span class="SimpleMath">\(g*h^-1\)</span>.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X7BD62D75829F8701" name="X7BD62D75829F8701"></a></p>

<h5>33.1-3 KendallMetric</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ KendallMetric</code>( <var class="Arg">g</var>, <var class="Arg">h</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ KendallMetric</code>( <var class="Arg">g</var>, <var class="Arg">h</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs two permutations <span class="SimpleMath">\(g,h\)</span> and optionally the degree <span class="SimpleMath">\(N\)</span> of a symmetric group containing them. It returns the minimum number of adjacent transpositions needed to express <span class="SimpleMath">\(g^-1*h\)</span> as a product of adjacent transpositions. An adjacent transposition has the form <span class="SimpleMath">\((i,i+1)\)</span>.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X789AE7CE8445A67C" name="X789AE7CE8445A67C"></a></p>

<h5>33.1-4 EuclideanSquaredMetric</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EuclideanSquaredMetric</code>( <var class="Arg">v</var>, <var class="Arg">w</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs two vectors <span class="SimpleMath">\(v,w\)</span> of equal length and returns the sum of the squares of the components of <span class="SimpleMath">\(v-w\)</span>. In other words, it returns the square of the Euclidean distance between <span class="SimpleMath">\(v\)</span> and <span class="SimpleMath">\(w\)</span>.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X79497F698557427E" name="X79497F698557427E"></a></p>

<h5>33.1-5 EuclideanApproximatedMetric</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EuclideanApproximatedMetric</code>( <var class="Arg">v</var>, <var class="Arg">w</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs two vectors <span class="SimpleMath">\(v,w\)</span> of equal length and returns a rational approximation to the square root of the sum of the squares of the components of <span class="SimpleMath">\(v-w\)</span>. In other words, it returns an approximation to the Euclidean distance between <span class="SimpleMath">\(v\)</span> and <span class="SimpleMath">\(w\)</span>.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap5.html">1</a></span> , <span class="URL"><a href="../tutorial/chap10.html">2</a></span> </p>

<p><a id="X8763D1167EF519A1" name="X8763D1167EF519A1"></a></p>

<h5>33.1-6 ManhattanMetric</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ManhattanMetric</code>( <var class="Arg">v</var>, <var class="Arg">w</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs two vectors <span class="SimpleMath">\(v,w\)</span> of equal length and returns the sum of the absolute values of the components of <span class="SimpleMath">\(v-w\)</span>. This is often referred to as the taxi-cab distance between <span class="SimpleMath">\(v\)</span> and <span class="SimpleMath">\(w\)</span>.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutMetrics.html">1</a></span> </p>

<p><a id="X7C86B58A7CEA5513" name="X7C86B58A7CEA5513"></a></p>

<h5>33.1-7 VectorsToSymmetricMatrix</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ VectorsToSymmetricMatrix</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ VectorsToSymmetricMatrix</code>( <var class="Arg">L</var>, <var class="Arg">D</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a list <span class="SimpleMath">\(L\)</span> of vectors and optionally a metric <span class="SimpleMath">\(D\)</span>. The default is <span class="SimpleMath">\(D=ManhattanMetric\)</span>. It returns the symmetric matrix whose i-j-entry is <span class="SimpleMath">\(S[i][j]=D(L[i],L[j])\)</span>.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap5.html">1</a></span> , <span class="URL"><a href="../tutorial/chap10.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutMetrics.html">3</a></span> </p>

<p><a id="X7895AABC7904E9CA" name="X7895AABC7904E9CA"></a></p>

<h5>33.1-8 SymmetricMatDisplay</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SymmetricMatDisplay</code>( <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SymmetricMatDisplay</code>( <var class="Arg">L</var>, <var class="Arg">V</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an <span class="SimpleMath">\(n \times n\)</span> symmetric matrix <span class="SimpleMath">\(S\)</span> of non-negative integers and an integer <span class="SimpleMath">\(t\)</span> in <span class="SimpleMath">\([0 .. 100]\)</span>. Optionally it inputs a list <span class="SimpleMath">\(V=[V_1, ... , V_k]\)</span> of disjoint subsets of <span class="SimpleMath">\([1 .. n]\)</span>. It displays the graph with vertex set <span class="SimpleMath">\([1 .. n]\)</span> and with an edge between <span class="SimpleMath">\(i\)</span> and <span class="SimpleMath">\(j\)</span> if <span class="SimpleMath">\(S[i][j] < t\)</span>. If the optional list <span class="SimpleMath">\(V\)</span> is input then the vertices in <span class="SimpleMath">\(V_i\)</span> will be given a common colour distinct from other vertices.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutMetrics.html">1</a></span> </p>

<p><a id="X79CA51F27C07435C" name="X79CA51F27C07435C"></a></p>

<h5>33.1-9 SymmetricMatrixToFilteredGraph</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SymmetricMatrixToFilteredGraph</code>( <var class="Arg">S</var>, <var class="Arg">t</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an integer symmetric matrix <span class="SimpleMath">\(S\)</span>, a positive integer <span class="SimpleMath">\(t\)</span> and a positive integer <span class="SimpleMath">\(m\)</span>. The function returns a filtered graph of filtration length <span class="SimpleMath">\(t\)</span>. The <span class="SimpleMath">\(k\)</span>-th term of the filtration is a graph with one vertex for each row of <span class="SimpleMath">\(S\)</span>. There is an edge in this graph between the <span class="SimpleMath">\(i\)</span>-th and <span class="SimpleMath">\(j\)</span>-th vertices if the entry <span class="SimpleMath">\(S[i][j]\)</span> is less than or equal to <span class="SimpleMath">\(k*m/t\)</span>.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap5.html">1</a></span> , <span class="URL"><a href="../tutorial/chap10.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPersistent.html">3</a></span> </p>

<p><a id="X8720F77B7ED74747" name="X8720F77B7ED74747"></a></p>

<h5>33.1-10 PermGroupToFilteredGraph</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PermGroupToFilteredGraph</code>( <var class="Arg">S</var>, <var class="Arg">D</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a permutation group <span class="SimpleMath">\(G\)</span> and a metric <span class="SimpleMath">\(D\)</span> defined on permutations. The function returns a filtered graph. The <span class="SimpleMath">\(k\)</span>-th term of the filtration is a graph with one vertex for each element of the group <span class="SimpleMath">\(G\)</span>. There is an edge in this graph between vertices <span class="SimpleMath">\(g\)</span> and <span class="SimpleMath">\(h\)</span> if <span class="SimpleMath">\(D(g,h)\)</span> is less than some integer threshold <span class="SimpleMath">\(t_k\)</span>. The thresholds <span class="SimpleMath">\(t_1 < t_2 < ... < t_N\)</span> are chosen to form as long a sequence as possible subject to each term of the filtration being a distinct graph.</p>

<p><strong class="button">Examples:</strong></p>


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