<!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 Disjoint Sets</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: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/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> <p><br></p> <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 -->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> </body> </html>