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<h1><a name="sec:disjoint-sets" id="sec:disjoint-sets"></a> Disjoint
Sets</h1>
<pre>
disjoint_sets<Rank, Parent, FindCompress>
</pre>
<p>This is class that provides disjoint sets operations with <i>union by
rank</i> and <i>path compression</i>. A disjoint-sets data structure
maintains a collection <i>S = {S<sub>1</sub>, S<sub>2</sub>, ...,
S<sub>k</sub>}</i> of disjoint sets. Each set is identified by a
<i>representative</i> which is some member of of the set. Sets are
represented by rooted trees which are encoded in the <tt>Parent</tt>
property map. Two heuristics: "union by rank" and "path compression" are
used to speed up the operations [<a href=
"./bibliography.html#tarjan83:_data_struct_network_algo">1</a>, <a href=
"./bibliography.html#clr90">2</a>].</p>
<h3>Where Defined</h3><a href=
"../../boost/pending/disjoint_sets.hpp"><tt>boost/pending/disjoint_sets.hpp</tt></a>
<h3>Template Parameters</h3>
<table border summary="">
<tr>
<td><tt>Rank</tt></td>
<td>must be a model of <a href=
"../property_map/doc/ReadWritePropertyMap.html">ReadWritePropertyMap</a>
with an integer value type and a key type equal to the set's element
type.</td>
</tr>
<tr>
<td><tt>Parent</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 set's element type.</td>
</tr>
<tr>
<td><tt>FindCompress</tt></td>
<td>should be one of the find representative and path compress function
objects.</td>
</tr>
</table>
<h3>Example</h3>
<p>A typical usage pattern for <tt>disjoint_sets</tt> can be seen in the
<a href=
"../graph/doc/kruskal_min_spanning_tree.html"><tt>kruskal_minimum_spanning_tree()</tt></a>
algorithm. In this example, we call <tt>link()</tt> instead of
<tt>union_set()</tt> because <tt>u</tt> and <tt>v</tt> were obtained from
<tt>find_set()</tt> and therefore are already the representatives for their
sets.</p>
<pre>
...
disjoint_sets<Rank, Parent, FindCompress> dsets(rank, p);
for (ui = vertices(G).first; ui != vertices(G).second; ++ui)
dsets.make_set(*ui);
...
while ( !Q.empty() ) {
e = Q.front();
Q.pop();
u = dsets.find_set(source(e));
v = dsets.find_set(target(e));
if ( u != v ) {
*out++ = e;
dsets.link(u, v);
}
}
</pre>
<h3>Members</h3>
<table border summary="">
<tr>
<th>Member</th>
<th>Description</th>
</tr>
<tr>
<td><tt>disjoint_sets(Rank r, Parent p)</tt></td>
<td>Constructor.</td>
</tr>
<tr>
<td><tt>disjoint_sets(const disjoint_sets& x)</tt></td>
<td>Copy constructor.</td>
</tr>
<tr>
<td><tt>template <class Element><br>
void make_set(Element x)</tt></td>
<td>Creates a singleton set containing Element <tt>x</tt>.</td>
</tr>
<tr>
<td><tt>template <class Element><br>
void link(Element x, Element y)</tt></td>
<td>Union the two sets <i>represented</i> by element <tt>x</tt> and
<tt>y</tt>.</td>
</tr>
<tr>
<td><tt>template <class Element><br>
void union_set(Element x, Element y)</tt></td>
<td>Union the two sets that <i>contain</i> elements <tt>x</tt> and
<tt>y</tt>. This is equivalent to
<tt>link(find_set(x),find_set(y))</tt>.</td>
</tr>
<tr>
<td><tt>template <class Element><br>
Element find_set(Element x)</tt></td>
<td>Return the representative for the set containing element
<tt>x</tt>.</td>
</tr>
<tr>
<td><tt>template <class ElementIterator><br>
std::size_t count_sets(ElementIterator first, ElementIterator
last)</tt></td>
<td>Returns the number of disjoint sets.</td>
</tr>
<tr>
<td><tt>template <class ElementIterator><br>
void compress_sets(ElementIterator first, ElementIterator
last)</tt></td>
<td>Flatten the parents tree so that the parent of every element is its
representative.</td>
</tr>
</table>
<h3>Complexity</h3>
<p>The time complexity is <i>O(m alpha(m,n))</i>, where <i>alpha</i> is the
inverse Ackermann's function, <i>m</i> is the number of disjoint-set
operations (<tt>make_set()</tt>, <tt>find_set()</tt>, and <tt>link()</tt>
and <i>n</i> is the number of elements. The <i>alpha</i> function grows
very slowly, much more slowly than the <i>log</i> function.</p>
<h3>See Also</h3><a href=
"../graph/doc/incremental_components.html"><tt>incremental_connected_components()</tt></a>
<hr>
<pre>
disjoint_sets_with_storage<ID,InverseID,FindCompress>
</pre>
<p>This class manages the storage for the rank and parent properties
internally. The storage is in arrays, which are indexed by element ID,
hence the requirement for the <tt>ID</tt> and <tt>InverseID</tt> functors.
The rank and parent properties are initialized during construction so the
each element is in a set by itself (so it is not necessary to initialize
objects of this class with the <a href=
"../graph/doc/incremental_components.html#sec:initialize-incremental-components">
<tt>initialize_incremental_components()</tt></a> function). This class is
especially useful when computing the (dynamic) connected components of an
<tt>edge_list</tt> graph which does not provide a place to store vertex
properties.</p>
<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 integers between zero 0 and N, the total number of
elements in the sets.</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 integers to elements.</td>
<td><tt>boost::identity_property_map</tt></td>
</tr>
<tr>
<td><tt>FindCompress</tt></td>
<td>should be one of the find representative and path compress function
objects.</td>
<td><tt>representative_with_full_path_compression</tt></td>
</tr>
</table>
<h3>Members</h3>
<p>This class has all of the members in <tt>disjoint_sets</tt> as well as
the following members.</p>
<pre>
disjoint_sets_with_storage(size_type n = 0,
ID id = ID(),
InverseID inv = InverseID())
</pre>Constructor.
<pre>
template <class ElementIterator>
void disjoint_sets_with_storage::
normalize_sets(ElementIterator first, ElementIterator last)
</pre>This rearranges the representatives such that the representative of
each set is the element with the smallest ID.<br>
Postcondition: <tt>v >= parent[v]</tt><br>
Precondition: the disjoint sets structure must be compressed.<br>
<hr>
<h2><a name="sec:representative-with-path-halving" id=
"sec:representative-with-path-halving"></a></h2>
<pre>
representative_with_path_halving<Parent>
</pre>
<p>This is a functor which finds the representative vertex for the same
component as the element <tt>x</tt>. While traversing up the representative
tree, the functor also applies the path halving technique to shorten the
height of the tree.</p>
<pre>
Element operator()(Parent p, Element x)
</pre>
<hr>
<h2><a name="sec:representative-with-full-path-compression" id=
"sec:representative-with-full-path-compression"></a><br></h2>
<pre>
representative_with_full_path_compression<Parent>
</pre>
<p>This is a functor which finds the representative element for the set
that element <tt>x</tt> belongs to.</p>
<pre>
Element operator()(Parent p, Element x)
</pre>
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<p>Revised
<!--webbot bot="Timestamp" s-type="EDITED" s-format="%d %B, %Y" startspan -->01
December, 2006<!--webbot bot="Timestamp" endspan i-checksum="38508" --></p>
<table summary="">
<tr valign="top">
<td nowrap><i>Copyright © 2000</i></td>
<td><i><a href="http://www.boost.org/people/jeremy_siek.htm">Jeremy Siek</a>, Univ.of
Notre Dame (<a href="mailto:jsiek@lsc.nd.edu">jsiek@lsc.nd.edu</a>)<br>
<a href="http://www.boost.org/people/liequan_lee.htm">Lie-Quan Lee</a>,
Univ.of Notre Dame (<a href=
"mailto:llee1@lsc.nd.edu">llee1@lsc.nd.edu</a>)<br>
<a href="http://www.lsc.nd.edu/~lums">Andrew Lumsdaine</a>, Univ.of
Notre Dame (<a href=
"mailto:lums@lsc.nd.edu">lums@lsc.nd.edu</a>)</i></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>
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