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Link/Cut-Trees implementation #186

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Link-Cut-Tree implementation and unit tests
yi-ji Sep 28, 2019
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fix stdc++ compatibility issues; parameterize type for id_to_vertex; …
yi-ji Oct 6, 2019
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add missing headers
yi-ji Oct 8, 2019
95ee00c
Parameterize the index map type, some code style & doc fixes
yi-ji Jun 20, 2020
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Add same root assertion for `link` and `lowest_common_ancestor`; Move…
yi-ji Jun 23, 2020
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remove `inline` function modifiers
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5 changes: 5 additions & 0 deletions doc/bibliography.html
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Expand Up @@ -453,6 +453,11 @@ <H2>Bibliography</H2>
<em>Data Structures for Weighted Matching and Nearest Common Ancestors with Linking</em><br>
Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 434-443, 1990.

<p></p><dt><a name="tarjan83">77</a>
<dd>Robert Endre Tarjan<br>
<em>Linking and Cutting Trees</em><br>
Data Structures and Network Algorithms, ISBN: 978-0-89871-187-5, pp. 59-70, 1983.

</dl>

<br>
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209 changes: 209 additions & 0 deletions doc/link_cut_trees.html
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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">

<html>
<head>
<meta http-equiv="Content-Language" content="en-us">
<meta http-equiv="Content-Type" content="text/html; charset=us-ascii">

<title>Boost Link/Cut Trees</title>
</head>

<body bgcolor="#FFFFFF" link="#0000EE" text="#000000" vlink="#551A8B" alink=
"#FF0000">
<img src="../../../boost.png" alt="C++ Boost" width="277" height=
"86"><br clear="none">

<h1><a name="sec:link-cut-trees" id="sec:link-cut-trees"></a> Link/Cut Trees</h1>
<pre>link_cut_trees&lt;ElementParentMap, ElementChildMap&gt;</pre>

<p>A link/cut-trees data structure
maintains a forest of element nodes subject to dynamic linking and cutting operations.
Rooted trees are encoded in both the <tt>ElementParentMap</tt> and <tt>ElementChildMap</tt>
property maps. Splay trees are used internally to represent every link/cut-tree [<a href="bibliography.html#tarjan83">77</a>]..</p>

<h3>Where Defined</h3><a href=
"../../../boost/graph/link_cut_trees.hpp"><tt>boost/graph/link_cut_trees.hpp</tt></a>

<h3>Template Parameters</h3>

<table border summary="">
<tr>
<td><tt>ElementParentMap</tt></td>

<td>Must be a model of <a href=
"../../property_map/doc/ReadWritePropertyMap.html">ReadWritePropertyMap</a>
and the key and value type the same as the trees' element type.</td>
</tr>

<tr>
<td><tt>ElementChildMap</tt></td>

<td>Default the same as <tt>ElementParentMap</tt>. Also must be a model of <a href=
"../../property_map/doc/ReadWritePropertyMap.html">ReadWritePropertyMap</a>
and the key and value type the same as the trees' element type.</td>
</tr>
</table>

<h3>Example</h3>

<p>A typical usage pattern for <tt>link_cut_trees</tt> can be seen in the
dynamic connectivity problem for acyclic graphs. Given two nodes <tt>x</tt> and <tt>y</tt>,
they are connected if and only if <tt>find_root(x) == find_root(y)</tt>.</p>

<pre>
...
link_cut_trees&lt;ElementParentMap, ElementChildMap&gt; lct(parent_map, left_child_map, right_child_map);

for (ui = vertices(G).first; ui != vertices(G).second; ++ui)
lct.make_tree(*ui);
...
while ( !Q.empty() ) {
e = Q.front();
Q.pop();
u = lct.find_root(source(e));
v = lct.find_root(target(e));
if ( u != v ) {
*out++ = e;
lct.link(u, v);
}
}
...
for (ui = vertices(G).first; ui != vertices(G).second; ++ui)
lct.cut(*ui);
</pre>

<h3>Members</h3>

<table border summary="">
<tr>
<th>Member</th>

<th>Description</th>
</tr>

<tr>
<td><tt>link_cut_trees(ElementParentMap p, ElementChildMap l, ElementChildMap r)</tt></td>

<td>Constructor.</td>
</tr>

<tr>
<td><tt>link_cut_trees(const link_cut_trees&amp; c)</tt></td>

<td>Copy constructor.</td>
</tr>

<tr>
<td><tt>template &lt;class Element&gt;<br>
void make_tree(Element x)</tt></td>

<td>Create a singleton tree containing element <tt>x</tt>.</td>
</tr>

<tr>
<td><tt>template &lt;class Element&gt;<br>
Element find_root(Element x)</tt></td>

<td>Return the root of the tree containing element <tt>x</tt>.</td>
</tr>

<tr>
<td><tt>template &lt;class Element&gt;<br>
void link(Element x, Element y)</tt></td>

<td>Make the tree rooted at element <tt>x</tt> a subtree of Element <tt>y</tt>. Element <tt>x</tt> must be a tree root.</td>
</tr>

<tr>
<td><tt>template &lt;class Element&gt;<br>
void cut(Element x)</tt></td>

<td>Remove the edge connecting <tt>x</tt> to its parent and make <tt>x</tt> a tree root.</td>
</tr>

<tr>
<td><tt>template &lt;class Element&gt;<br>
Element lowest_common_ancestor(Element x, Element y)</tt></td>

<td>Return the lowest (i.e. nearest) common ancestor of elements <tt>x</tt> and <tt>y</tt>. Elements <tt>x</tt> and <tt>y</tt> must have the same root.</td>
</tr>

</table>

<h3>Complexity</h3>

<p>The amortized time complexity is <i>O(log n)</i> for <tt>link</tt>, <tt>cut</tt>, <tt>find_root</tt> and <tt>lowest_common_ancestor</tt> operations,
where <i>n</i> is the number of tree nodes. Operation <tt>make_tree</tt> is of constant time complexity.</p>

<hr>
<pre>link_cut_trees_with_storage&lt;ID, InverseID&gt;</pre>

<p>This class manages the storage for the parent and children properties
internally. The storage is in <tt>boost::unordered_map</tt>, which is indexed by element ID,
hence the requirement for the <tt>ID</tt> and <tt>InverseID</tt> functors.

<h3>Template Parameters</h3>

<table border summary="">
<tr>
<th>Parameter</th>

<th>Description</th>

<th>Default</th>
</tr>

<tr>
<td><tt>ID</tt></td>

<td>must be a model of <a href=
"../../property_map/doc/ReadablePropertyMap.html">ReadablePropertyMap</a> that
maps elements to values of any type that can be hashed to <tt>std::size_t</tt>.</td>

<td><tt>boost::identity_property_map</tt></td>
</tr>

<tr>
<td><tt>InverseID</tt></td>

<td>must be a model of <a href=
"../../property_map/doc/ReadablePropertyMap.html">ReadablePropertyMap</a> that
maps values of <tt>property_traits&lt;ID&gt;::value_type</tt> to elements.</td>

<td><tt>boost::unordered_map</tt></td>
</tr>

</table>

<h3>Members</h3>

<p>This class has all of the members in <tt>link_cut_trees</tt> as well as
the following constructor.</p>
<pre>link_cut_trees_with_storage(ID id = ID(), InverseID inverse_id = InverseID())</pre>

<hr>

<p><a href="http://validator.w3.org/check?uri=referer"><img border="0" src=
"../../doc/images/valid-html401.png" alt="Valid HTML 4.01 Transitional"
height="31" width="88"></a></p>

<p>Revised
<!--webbot bot="Timestamp" s-type="EDITED" s-format="%d %B, %Y" startspan -->
June, 2020<!--webbot bot="Timestamp" endspan i-checksum="38508" --></p>

<table summary="">
<tr valign="top">
<td nowrap><i>Copyright &copy; 2019</i></td>

<td>
<i><a>Yi Ji</a>, Peking University (<a href="mailto:[email protected]">[email protected]</a>)<br>
</td>
</tr>
</table>

<p><i>Distributed under the Boost Software License, Version 1.0. (See
accompanying file <a href="../../LICENSE_1_0.txt">LICENSE_1_0.txt</a> or
copy at <a href=
"http://www.boost.org/LICENSE_1_0.txt">http://www.boost.org/LICENSE_1_0.txt</a>)</i></p>
</body>
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