Skip to content
New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Jump to bottom

Link/Cut-Trees implementation #186

Open
wants to merge 7 commits into
base: develop
Choose a base branch
from
Open

Link/Cut-Trees implementation #186

Show file tree
Hide file tree
Changes from all commits
Commits
Show all changes
7 commits
Select commit Hold shift + click to select a range
ea7bbca
Link-Cut-Tree implementation and unit tests
yi-ji Sep 28, 2019
abce5b5
fix stdc++ compatibility issues; parameterize type for id_to_vertex; …
yi-ji Oct 6, 2019
3aa8cd3
add missing headers
yi-ji Oct 8, 2019
95ee00c
Parameterize the index map type, some code style & doc fixes
yi-ji Jun 20, 2020
2c95853
Add same root assertion for `link` and `lowest_common_ancestor`; Move…
yi-ji Jun 23, 2020
372d53f
remove `inline` function modifiers
Mar 24, 2021
749c9a4
Empty commit to trigger a build
yi-ji Jun 12, 2021
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
5 changes: 5 additions & 0 deletions doc/bibliography.html
View file
Open in desktop
Original file line number Diff line number Diff line change
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.

yi-ji marked this conversation as resolved.
Show resolved Hide resolved
</dl>

<br>
Expand Down
209 changes: 209 additions & 0 deletions doc/link_cut_trees.html
View file
Open in desktop
Original file line number Diff line number Diff line change
@@ -0,0 +1,209 @@
<!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>
</html>
Loading