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doc/lengauer_tarjan_dominator.htm
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doc/lengauer_tarjan_dominator.htm
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<HTML>
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<!--
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-- Copyright (c) JongSoo Park 2005
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--
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-- Permission to use, copy, modify, distribute and sell this software
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-- and its documentation for any purpose is hereby granted without fee,
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-- provided that the above copyright notice appears in all copies and
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-- that both that copyright notice and this permission notice appear
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-- in supporting documentation. Jeremy Siek makes no
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-- representations about the suitability of this software for any
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-- purpose. It is provided "as is" without express or implied warranty.
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-->
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<Head>
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<Title>Boost Graph Library: Lengauer-Tarjan Dominator Tree Algorithm</Title>
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<BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b"
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ALINK="#ff0000">
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<IMG SRC="../../../boost.png"
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ALT="C++ Boost" width="277" height="86">
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<BR Clear>
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<H1><A NAME="sec:lengauer-tarjan"></A>
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<TT>lengauer_tarjan_dominator_tree</TT>
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</H1>
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<P>
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<PRE>
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<i>// The simplest version:
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// Data structures for depth first search is created internally,
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// and depth first search runs internally.</i>
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template <class Graph, class DomTreePredMap>
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void
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lengauer_tarjan_dominator_tree
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(const Graph& g,
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const typename graph_traits<Graph>::vertex_descriptor& entry,
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DomTreePredMap domTreePredMap)
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<i>// The version providing data structures for depth first search:
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// After calling this function,
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// user can reuse the depth first search related information
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// filled in property maps.</i>
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template <class Graph, class IndexMap, class TimeMap, class PredMap,
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class VertexVector, class DomTreePredMap>
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void
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lengauer_tarjan_dominator_tree
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(const Graph& g,
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const typename graph_traits<Graph>::vertex_descriptor& entry,
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const IndexMap& indexMap,
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TimeMap dfnumMap, PredMap parentMap, VertexVector& verticesByDFNum,
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DomTreePredMap domTreePredMap)
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<i>// The version without depth first search:
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// The caller should provide depth first search related information
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// evaluated before.</i>
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template <class Graph, class IndexMap, class TimeMap, class PredMap,
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class VertexVector, class DomTreePredMap>
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void
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lengauer_tarjan_dominator_tree_without_dfs(
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(const Graph& g,
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const typename graph_traits<Graph>::vertex_descriptor& entry,
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const IndexMap& indexMap,
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TimeMap dfnumMap, PredMap parentMap, VertexVector& verticesByDFNum,
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DomTreePredMap domTreePredMap)
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</PRE>
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<P> This algorithm [<A
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HREF="bibliography.html#lengauer-tarjan79">65</A>,<A
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HREF="bibliography.html#muchnick97">66</A>,<A
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HREF="bibliography.html#appel98">67</A>] builds the dominator tree for
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directed graph. There are three options for dealing the depth first
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search related values. The simplest version creates data structures
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and run depth first search internally. However, chances are that a
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user wants to reuse the depth first search data, so we have two
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versions. </P>
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<P> A vertex <i>u</i> dominates a vertex <i>v</i>, if every path of
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directed graph from the entry to <i>v</i> must go through <i>u</i>. In
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the left graph of <a href="#fig:dominator-tree-example">Figure 1</a>,
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vertex 1 dominates vertex 2, 3, 4, 5, 6 and 7, because we have to pass
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vertex 1 to reach those vertex. Note that vertex 4 dominates vertex 6,
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even though vertex 4 is a successor of vertex 6. Dominator
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relationship is useful in many applications especially for compiler
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optimization. We can define the immediate dominator for each vertex
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such that <i>idom(n) dominates n</i> but does not dominate any other
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dominator of <i>n</i>. For example, vertex 1, 3 and 4 are dominators
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of vertex 6, but vertex 4 is the immediate dominator, because vertex 1
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and 3 dominates vertex 4. If we make every idom of each vertex as its
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parent, we can build the dominator tree like the right part of <a
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href="#fig:dominator-tree-example">Figure 1</a> </P>
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<P></P>
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<DIV ALIGN="CENTER"><A NAME="fig:dominator-tree-example">
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1:</STRONG>
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An example graph and its dominator tree.</CAPTION>
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<TR>
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<TD><IMG SRC="./figs/dominator-tree1.gif"></TD>
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<TD> </TD>
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<TD><IMG SRC="./figs/dominator-tree2.gif"></TD>
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</TR>
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</TABLE>
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</DIV><P></P>
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<P> An easy way to build dominator tree is to use iterative bit vector
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algorithm, but it may be slow in the worst case. We implemented
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Lengauer-Tarjan algorithm whose time complexity is
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<i>O((V+E)log(V+E))</i>. </P>
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<P> Lengauer-Tarjan algorithm utilizes two techniques. The first one
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is, as an intermediate step, finding semidominator which is relatively
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easier to evaluate than immediate dominator, and the second one is the
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path compression. For the detail of the algorithm, see [<A
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HREF="bibliography.html#lengauer-tarjan79">65</A>]. </P>
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<h3>Where Defined</h3>
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<a href="../../../boost/graph/dominator_tree.hpp"><tt>boost/graph/dominator_tree.hpp</tt></a>
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<h3>Parameters</h3>
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IN: <tt>const Graph& g</tt>
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<blockquote>
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The graph object on which the algorithm will be applied.
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The type <tt>Graph</tt> must be a model of
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<a href="./VertexListGraph.html">Vertex List Graph</a>
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and <a href="./BidirectionalGraph.html">Bidirectional Graph</a>.<br>
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</blockquote>
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IN: <tt>vertex_descriptor entry</tt>
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<blockquote>
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The entry vertex. The dominator tree will be rooted at this vertex.
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</blockquote>
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IN: <tt>IndexMap indexMap</tt>
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<blockquote>
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This maps each vertex to an integer in the range <tt>[0, num_vertices(g))</tt>.
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The type
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<tt>VertexIndexMap</tt> must be a model of
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<a href="../../property_map/ReadablePropertyMap.html">Readable Property Map</a>. The value type of the map must be an
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integer type. The vertex descriptor type of the graph needs to be
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usable as the key type of the map.
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</blockquote>
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IN/OUT: <tt>TimeMap dfnumMap</tt>
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<blockquote>
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The sequence number of depth first search. The type <tt>TimeMap</tt> must be a model of <a href="../../property_map/ReadWritePropertyMap.html">Read/Write Property Map</a>. The vertex descriptor type of the graph needs to be usable as the key type of the <tt>TimeMap</tt>.
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</blockquote>
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IN/OUT: <tt>PredMap parentMap</tt>
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<blockquote>
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The predecessor map records the parent of the depth first search tree. The <tt>PredMap</tt> type must be a <a href="../../property_map/ReadWritePropertyMap.html">Read/Write Property Map</a> whose key and value types are the same as the vertex descriptor type of the graph.
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</blockquote>
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IN/OUT: <tt>VertexVector verticesByDFNum</tt>
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<blockquote>
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The vector containing vertices in depth first search order. If we access this vector sequentially, it's equivalent to access vertices by depth first search order.
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</blockquote>
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OUT: <tt>DomTreePredMap domTreePredMap</tt>
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<blockquote>
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The dominator tree where parents are each children's immediate dominator.
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</blockquote>
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<H3>Complexity</H3>
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<P>
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The time complexity is <i>O((V+E)log(V+E))</i>.
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<H3>Example</H3>
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<P>
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See <a href="../test/dominator_tree_test.cpp">
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<TT>test/dominator_tree_test.cpp</TT></a> for an example of using Dijkstra's
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algorithm.
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<br>
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<HR>
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<TABLE>
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<TR valign="top">
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<TD>Copyright © 2005</TD><TD>
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JongSoo Park, Stanford University
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</TD></TR></TABLE>
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</BODY>
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</HTML>
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133
doc/sorted_erdos_renyi_gen.html
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doc/sorted_erdos_renyi_gen.html
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<HTML>
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<!--
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-- Copyright (c) 2004, 2005 The Trustees of Indiana University
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--
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-- Use, modification and distribution is subject to the Boost Software
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-- License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at
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-- http://www.boost.org/LICENSE_1_0.txt)
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--
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-- Authors: Jeremiah Willcock
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-- Douglas Gregor
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-- Andrew Lumsdaine
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-->
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<Head>
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<Title>Boost Graph Library: Erdös-Renyi Generator</Title>
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<script language="JavaScript" type="text/JavaScript">
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<!--
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function address(host, user) {
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var atchar = '@';
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var thingy = user+atchar+host;
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thingy = '<a hre' + 'f=' + "mai" + "lto:" + thingy + '>' + user+atchar+host + '</a>';
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document.write(thingy);
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}
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//-->
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</script>
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</head>
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<BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b"
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ALINK="#ff0000">
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<IMG SRC="../../../boost.png"
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ALT="C++ Boost" width="277" height="86">
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<tt>sorted_erdos_renyi_iterator</tt>
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<br>
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<PRE>
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template<typename RandomGenerator, typename Graph>
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class sorted_erdos_renyi_iterator
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{
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public:
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typedef std::input_iterator_tag iterator_category;
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typedef std::pair<vertices_size_type, vertices_size_type> value_type;
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typedef const value_type& reference;
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typedef const value_type* pointer;
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typedef void difference_type;
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sorted_erdos_renyi_iterator();
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sorted_erdos_renyi_iterator(RandomGenerator& gen, vertices_size_type n,
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double probability = 0.0, bool allow_self_loops = false);
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// Iterator operations
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reference operator*() const;
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pointer operator->() const;
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sorted_erdos_renyi_iterator& operator++();
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sorted_erdos_renyi_iterator operator++(int);
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bool operator==(const sorted_erdos_renyi_iterator& other) const;
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bool operator!=(const sorted_erdos_renyi_iterator& other) const;
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};
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</PRE>
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<p> This class template implements a generator for Erdös-Renyi
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graphs, suitable for initializing an <a
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href="adjacency_list.html"><tt>adjacency_list</tt></a> or other graph
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structure with iterator-based initialization. An Erdös-Renyi
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graph <em>G = (n, p)</em> is a graph with <em>n</em> vertices
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that. The probability of having an edge <em>(u, v)</em> in <em>G</em>
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is <em>p</em> for any vertices <em>u</em> and <em>v</em>. Typically,
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there are no self-loops, but the generator can optionally introduce
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self-loops with probability <em>p</em>.</p>
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<p>Erdös-Renyi graphs typically exhibit very little
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structure. For this reason, they are rarely useful in modeling
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real-world problems. However, they are often used when determining
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the theoretical complexity of complex graph algorithms.</p>
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<h3>Where Defined</h3>
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<a href="../../../boost/graph/erdos_renyi_generator.hpp"><tt>boost/graph/erdos_renyi_generator.hpp</tt></a>
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<h3>Constructors</h3>
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<a name="default-constructor"/>
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<pre>sorted_erdos_renyi_iterator();</pre>
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<blockquote>
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Constructs a past-the-end iterator.
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</blockquote>
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<pre>
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sorted_erdos_renyi_iterator(RandomGenerator& gen, vertices_size_type n,
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double probability = 0.0, bool allow_self_loops = false);
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</pre>
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<blockquote>
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Constructs an Erdös-Renyi generator iterator that creates a
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graph with <tt>n</tt> vertices and a given <tt>probability</tt> of the
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total number of edges that a simple graph may have.
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<tt>probability</tt>. Random vertices and edges are selected using the
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random number generator <tt>gen</tt>. Self-loops are permitted only when
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<tt>allow_self_loops</tt> is <tt>true</tt>.
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</blockquote>
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<H3>Example</H3>
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<pre>
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#include <boost/graph/adjacency_list.hpp>
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#include <boost/graph/erdos_renyi_generator.hpp>
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#include <boost/random/linear_congruential.hpp>
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typedef boost::adjacency_list<> Graph;
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typedef boost::sorted_erdos_renyi_iterator<boost::minstd_rand, Graph> ERGen;
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int main()
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{
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boost::minstd_rand gen;
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// Create graph with 100 nodes and edges with probability 0.05
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Graph g(ERGen(gen, 100, 0.05), ERGen(), 100);
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return 0;
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}
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</pre>
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<br>
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<HR>
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<TABLE>
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<TR valign=top>
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<TD nowrap>Copyright © 2005</TD><TD>
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Jeremiah Willcock, Indiana University (<script language="Javascript">address("cs.indiana.edu", "jewillco")</script>)<br>
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<A HREF="../../../people/doug_gregor.html">Doug Gregor</A>, Indiana University (<script language="Javascript">address("cs.indiana.edu", "dgregor")</script>)<br>
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<A HREF=http://www.osl.iu.edu/~lums>Andrew Lumsdaine</A>,
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Indiana University (<script language="Javascript">address("osl.iu.edu", "lums")</script>)
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</TD></TR></TABLE>
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</BODY>
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</HTML>
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