author | Dremov Kirill (Nokia-D-MSW/Tampere) <kirill.dremov@nokia.com> |
Tue, 02 Feb 2010 02:01:42 +0200 | |
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// Copyright 2004 The Trustees of Indiana University. |
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// Distributed under the Boost Software License, Version 1.0. |
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// (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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// Authors: Douglas Gregor |
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// Andrew Lumsdaine |
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#ifndef BOOST_GRAPH_KAMADA_KAWAI_SPRING_LAYOUT_HPP |
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#define BOOST_GRAPH_KAMADA_KAWAI_SPRING_LAYOUT_HPP |
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#include <boost/graph/graph_traits.hpp> |
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#include <boost/graph/johnson_all_pairs_shortest.hpp> |
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#include <boost/type_traits/is_convertible.hpp> |
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#include <utility> |
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#include <iterator> |
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#include <vector> |
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#include <boost/limits.hpp> |
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#include <cmath> |
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namespace boost { |
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namespace detail { namespace graph { |
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/** |
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* Denotes an edge or display area side length used to scale a |
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* Kamada-Kawai drawing. |
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*/ |
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template<bool Edge, typename T> |
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struct edge_or_side |
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{ |
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explicit edge_or_side(T value) : value(value) {} |
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T value; |
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}; |
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/** |
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* Compute the edge length from an edge length. This is trivial. |
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*/ |
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template<typename Graph, typename DistanceMap, typename IndexMap, |
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typename T> |
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T compute_edge_length(const Graph&, DistanceMap, IndexMap, |
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edge_or_side<true, T> length) |
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{ return length.value; } |
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/** |
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* Compute the edge length based on the display area side |
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length. We do this by dividing the side length by the largest |
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shortest distance between any two vertices in the graph. |
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*/ |
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template<typename Graph, typename DistanceMap, typename IndexMap, |
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typename T> |
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T |
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compute_edge_length(const Graph& g, DistanceMap distance, IndexMap index, |
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edge_or_side<false, T> length) |
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{ |
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T result(0); |
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typedef typename graph_traits<Graph>::vertex_iterator vertex_iterator; |
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for (vertex_iterator ui = vertices(g).first, end = vertices(g).second; |
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ui != end; ++ui) { |
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vertex_iterator vi = ui; |
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for (++vi; vi != end; ++vi) { |
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T dij = distance[get(index, *ui)][get(index, *vi)]; |
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if (dij > result) result = dij; |
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} |
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} |
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return length.value / result; |
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} |
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/** |
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* Implementation of the Kamada-Kawai spring layout algorithm. |
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*/ |
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template<typename Graph, typename PositionMap, typename WeightMap, |
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typename EdgeOrSideLength, typename Done, |
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typename VertexIndexMap, typename DistanceMatrix, |
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typename SpringStrengthMatrix, typename PartialDerivativeMap> |
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struct kamada_kawai_spring_layout_impl |
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{ |
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typedef typename property_traits<WeightMap>::value_type weight_type; |
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typedef std::pair<weight_type, weight_type> deriv_type; |
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typedef typename graph_traits<Graph>::vertex_iterator vertex_iterator; |
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typedef typename graph_traits<Graph>::vertex_descriptor |
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vertex_descriptor; |
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kamada_kawai_spring_layout_impl( |
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const Graph& g, |
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PositionMap position, |
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WeightMap weight, |
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EdgeOrSideLength edge_or_side_length, |
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Done done, |
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weight_type spring_constant, |
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VertexIndexMap index, |
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DistanceMatrix distance, |
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SpringStrengthMatrix spring_strength, |
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PartialDerivativeMap partial_derivatives) |
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: g(g), position(position), weight(weight), |
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edge_or_side_length(edge_or_side_length), done(done), |
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spring_constant(spring_constant), index(index), distance(distance), |
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spring_strength(spring_strength), |
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partial_derivatives(partial_derivatives) {} |
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// Compute contribution of vertex i to the first partial |
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// derivatives (dE/dx_m, dE/dy_m) (for vertex m) |
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104 |
deriv_type |
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compute_partial_derivative(vertex_descriptor m, vertex_descriptor i) |
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106 |
{ |
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107 |
#ifndef BOOST_NO_STDC_NAMESPACE |
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using std::sqrt; |
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#endif // BOOST_NO_STDC_NAMESPACE |
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110 |
|
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deriv_type result(0, 0); |
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if (i != m) { |
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weight_type x_diff = position[m].x - position[i].x; |
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weight_type y_diff = position[m].y - position[i].y; |
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weight_type dist = sqrt(x_diff * x_diff + y_diff * y_diff); |
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result.first = spring_strength[get(index, m)][get(index, i)] |
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* (x_diff - distance[get(index, m)][get(index, i)]*x_diff/dist); |
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result.second = spring_strength[get(index, m)][get(index, i)] |
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* (y_diff - distance[get(index, m)][get(index, i)]*y_diff/dist); |
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} |
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121 |
|
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return result; |
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} |
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124 |
|
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// Compute partial derivatives dE/dx_m and dE/dy_m |
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deriv_type |
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compute_partial_derivatives(vertex_descriptor m) |
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128 |
{ |
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129 |
#ifndef BOOST_NO_STDC_NAMESPACE |
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using std::sqrt; |
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#endif // BOOST_NO_STDC_NAMESPACE |
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132 |
|
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deriv_type result(0, 0); |
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134 |
|
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// TBD: looks like an accumulate to me |
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std::pair<vertex_iterator, vertex_iterator> verts = vertices(g); |
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for (/* no init */; verts.first != verts.second; ++verts.first) { |
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vertex_descriptor i = *verts.first; |
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deriv_type deriv = compute_partial_derivative(m, i); |
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result.first += deriv.first; |
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result.second += deriv.second; |
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} |
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143 |
|
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return result; |
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} |
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146 |
|
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// The actual Kamada-Kawai spring layout algorithm implementation |
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bool run() |
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{ |
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150 |
#ifndef BOOST_NO_STDC_NAMESPACE |
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using std::sqrt; |
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#endif // BOOST_NO_STDC_NAMESPACE |
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153 |
|
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// Compute d_{ij} and place it in the distance matrix |
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if (!johnson_all_pairs_shortest_paths(g, distance, index, weight, |
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weight_type(0))) |
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return false; |
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158 |
|
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// Compute L based on side length (if needed), or retrieve L |
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weight_type edge_length = |
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detail::graph::compute_edge_length(g, distance, index, |
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edge_or_side_length); |
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163 |
|
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// Compute l_{ij} and k_{ij} |
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const weight_type K = spring_constant; |
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vertex_iterator ui, end = vertices(g).second; |
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for (ui = vertices(g).first; ui != end; ++ui) { |
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vertex_iterator vi = ui; |
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for (++vi; vi != end; ++vi) { |
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weight_type dij = distance[get(index, *ui)][get(index, *vi)]; |
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if (dij == (std::numeric_limits<weight_type>::max)()) |
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return false; |
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distance[get(index, *ui)][get(index, *vi)] = edge_length * dij; |
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distance[get(index, *vi)][get(index, *ui)] = edge_length * dij; |
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spring_strength[get(index, *ui)][get(index, *vi)] = K/(dij*dij); |
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spring_strength[get(index, *vi)][get(index, *ui)] = K/(dij*dij); |
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177 |
} |
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} |
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179 |
|
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// Compute Delta_i and find max |
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vertex_descriptor p = *vertices(g).first; |
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weight_type delta_p(0); |
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183 |
|
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for (ui = vertices(g).first; ui != end; ++ui) { |
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deriv_type deriv = compute_partial_derivatives(*ui); |
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put(partial_derivatives, *ui, deriv); |
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187 |
|
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weight_type delta = |
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sqrt(deriv.first*deriv.first + deriv.second*deriv.second); |
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190 |
|
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if (delta > delta_p) { |
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p = *ui; |
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delta_p = delta; |
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} |
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} |
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196 |
|
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while (!done(delta_p, p, g, true)) { |
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// The contribution p makes to the partial derivatives of |
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// each vertex. Computing this (at O(n) cost) allows us to |
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// update the delta_i values in O(n) time instead of O(n^2) |
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// time. |
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std::vector<deriv_type> p_partials(num_vertices(g)); |
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for (ui = vertices(g).first; ui != end; ++ui) { |
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vertex_descriptor i = *ui; |
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p_partials[get(index, i)] = compute_partial_derivative(i, p); |
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} |
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207 |
|
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do { |
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// Compute the 4 elements of the Jacobian |
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weight_type dE_dx_dx = 0, dE_dx_dy = 0, dE_dy_dx = 0, dE_dy_dy = 0; |
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for (ui = vertices(g).first; ui != end; ++ui) { |
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vertex_descriptor i = *ui; |
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if (i != p) { |
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weight_type x_diff = position[p].x - position[i].x; |
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weight_type y_diff = position[p].y - position[i].y; |
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weight_type dist = sqrt(x_diff * x_diff + y_diff * y_diff); |
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weight_type dist_cubed = dist * dist * dist; |
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weight_type k_mi = spring_strength[get(index,p)][get(index,i)]; |
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weight_type l_mi = distance[get(index, p)][get(index, i)]; |
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dE_dx_dx += k_mi * (1 - (l_mi * y_diff * y_diff)/dist_cubed); |
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dE_dx_dy += k_mi * l_mi * x_diff * y_diff / dist_cubed; |
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dE_dy_dx += k_mi * l_mi * x_diff * y_diff / dist_cubed; |
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dE_dy_dy += k_mi * (1 - (l_mi * x_diff * x_diff)/dist_cubed); |
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} |
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} |
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|
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// Solve for delta_x and delta_y |
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weight_type dE_dx = get(partial_derivatives, p).first; |
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weight_type dE_dy = get(partial_derivatives, p).second; |
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|
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weight_type delta_x = |
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(dE_dx_dy * dE_dy - dE_dy_dy * dE_dx) |
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/ (dE_dx_dx * dE_dy_dy - dE_dx_dy * dE_dy_dx); |
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|
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weight_type delta_y = |
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(dE_dx_dx * dE_dy - dE_dy_dx * dE_dx) |
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/ (dE_dy_dx * dE_dx_dy - dE_dx_dx * dE_dy_dy); |
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|
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239 |
|
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// Move p by (delta_x, delta_y) |
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position[p].x += delta_x; |
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position[p].y += delta_y; |
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|
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// Recompute partial derivatives and delta_p |
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deriv_type deriv = compute_partial_derivatives(p); |
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put(partial_derivatives, p, deriv); |
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|
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delta_p = |
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sqrt(deriv.first*deriv.first + deriv.second*deriv.second); |
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} while (!done(delta_p, p, g, false)); |
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251 |
|
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// Select new p by updating each partial derivative and delta |
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vertex_descriptor old_p = p; |
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for (ui = vertices(g).first; ui != end; ++ui) { |
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deriv_type old_deriv_p = p_partials[get(index, *ui)]; |
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deriv_type old_p_partial = |
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compute_partial_derivative(*ui, old_p); |
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deriv_type deriv = get(partial_derivatives, *ui); |
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|
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deriv.first += old_p_partial.first - old_deriv_p.first; |
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deriv.second += old_p_partial.second - old_deriv_p.second; |
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|
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put(partial_derivatives, *ui, deriv); |
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weight_type delta = |
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sqrt(deriv.first*deriv.first + deriv.second*deriv.second); |
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266 |
|
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if (delta > delta_p) { |
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p = *ui; |
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delta_p = delta; |
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} |
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} |
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} |
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|
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return true; |
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} |
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276 |
|
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const Graph& g; |
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PositionMap position; |
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WeightMap weight; |
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EdgeOrSideLength edge_or_side_length; |
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Done done; |
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weight_type spring_constant; |
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VertexIndexMap index; |
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DistanceMatrix distance; |
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SpringStrengthMatrix spring_strength; |
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PartialDerivativeMap partial_derivatives; |
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}; |
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} } // end namespace detail::graph |
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|
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/// States that the given quantity is an edge length. |
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template<typename T> |
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inline detail::graph::edge_or_side<true, T> |
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edge_length(T x) |
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{ return detail::graph::edge_or_side<true, T>(x); } |
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|
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/// States that the given quantity is a display area side length. |
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template<typename T> |
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inline detail::graph::edge_or_side<false, T> |
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side_length(T x) |
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{ return detail::graph::edge_or_side<false, T>(x); } |
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|
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/** |
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* \brief Determines when to terminate layout of a particular graph based |
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* on a given relative tolerance. |
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*/ |
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template<typename T = double> |
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struct layout_tolerance |
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308 |
{ |
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layout_tolerance(const T& tolerance = T(0.001)) |
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: tolerance(tolerance), last_energy((std::numeric_limits<T>::max)()), |
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last_local_energy((std::numeric_limits<T>::max)()) { } |
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312 |
|
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template<typename Graph> |
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314 |
bool |
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operator()(T delta_p, |
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typename boost::graph_traits<Graph>::vertex_descriptor p, |
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const Graph& g, |
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bool global) |
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{ |
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if (global) { |
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if (last_energy == (std::numeric_limits<T>::max)()) { |
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last_energy = delta_p; |
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return false; |
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} |
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325 |
|
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T diff = last_energy - delta_p; |
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if (diff < T(0)) diff = -diff; |
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bool done = (delta_p == T(0) || diff / last_energy < tolerance); |
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last_energy = delta_p; |
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return done; |
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} else { |
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if (last_local_energy == (std::numeric_limits<T>::max)()) { |
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last_local_energy = delta_p; |
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return delta_p == T(0); |
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} |
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336 |
|
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T diff = last_local_energy - delta_p; |
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bool done = (delta_p == T(0) || (diff / last_local_energy) < tolerance); |
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last_local_energy = delta_p; |
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return done; |
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} |
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} |
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343 |
|
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private: |
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T tolerance; |
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T last_energy; |
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T last_local_energy; |
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}; |
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349 |
|
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/** \brief Kamada-Kawai spring layout for undirected graphs. |
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351 |
* |
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352 |
* This algorithm performs graph layout (in two dimensions) for |
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* connected, undirected graphs. It operates by relating the layout |
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* of graphs to a dynamic spring system and minimizing the energy |
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355 |
* within that system. The strength of a spring between two vertices |
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* is inversely proportional to the square of the shortest distance |
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* (in graph terms) between those two vertices. Essentially, |
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* vertices that are closer in the graph-theoretic sense (i.e., by |
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* following edges) will have stronger springs and will therefore be |
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* placed closer together. |
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361 |
* |
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* Prior to invoking this algorithm, it is recommended that the |
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* vertices be placed along the vertices of a regular n-sided |
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* polygon. |
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* |
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* \param g (IN) must be a model of Vertex List Graph, Edge List |
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* Graph, and Incidence Graph and must be undirected. |
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368 |
* |
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* \param position (OUT) must be a model of Lvalue Property Map, |
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* where the value type is a class containing fields @c x and @c y |
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* that will be set to the @c x and @c y coordinates of each vertex. |
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* |
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* \param weight (IN) must be a model of Readable Property Map, |
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* which provides the weight of each edge in the graph @p g. |
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* |
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* \param edge_or_side_length (IN) provides either the unit length |
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* @c e of an edge in the layout or the length of a side @c s of the |
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* display area, and must be either @c boost::edge_length(e) or @c |
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* boost::side_length(s), respectively. |
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* |
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* \param done (IN) is a 4-argument function object that is passed |
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* the current value of delta_p (i.e., the energy of vertex @p p), |
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* the vertex @p p, the graph @p g, and a boolean flag indicating |
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384 |
* whether @p delta_p is the maximum energy in the system (when @c |
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* true) or the energy of the vertex being moved. Defaults to @c |
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* layout_tolerance instantiated over the value type of the weight |
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* map. |
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388 |
* |
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389 |
* \param spring_constant (IN) is the constant multiplied by each |
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390 |
* spring's strength. Larger values create systems with more energy |
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* that can take longer to stabilize; smaller values create systems |
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392 |
* with less energy that stabilize quickly but do not necessarily |
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* result in pleasing layouts. The default value is 1. |
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394 |
* |
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395 |
* \param index (IN) is a mapping from vertices to index values |
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* between 0 and @c num_vertices(g). The default is @c |
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* get(vertex_index,g). |
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398 |
* |
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399 |
* \param distance (UTIL/OUT) will be used to store the distance |
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400 |
* from every vertex to every other vertex, which is computed in the |
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401 |
* first stages of the algorithm. This value's type must be a model |
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402 |
* of BasicMatrix with value type equal to the value type of the |
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403 |
* weight map. The default is a a vector of vectors. |
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404 |
* |
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405 |
* \param spring_strength (UTIL/OUT) will be used to store the |
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406 |
* strength of the spring between every pair of vertices. This |
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407 |
* value's type must be a model of BasicMatrix with value type equal |
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408 |
* to the value type of the weight map. The default is a a vector of |
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409 |
* vectors. |
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410 |
* |
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411 |
* \param partial_derivatives (UTIL) will be used to store the |
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412 |
* partial derivates of each vertex with respect to the @c x and @c |
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413 |
* y coordinates. This must be a Read/Write Property Map whose value |
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414 |
* type is a pair with both types equivalent to the value type of |
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415 |
* the weight map. The default is an iterator property map. |
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416 |
* |
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417 |
* \returns @c true if layout was successful or @c false if a |
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418 |
* negative weight cycle was detected. |
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419 |
*/ |
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420 |
template<typename Graph, typename PositionMap, typename WeightMap, |
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421 |
typename T, bool EdgeOrSideLength, typename Done, |
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422 |
typename VertexIndexMap, typename DistanceMatrix, |
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423 |
typename SpringStrengthMatrix, typename PartialDerivativeMap> |
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424 |
bool |
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425 |
kamada_kawai_spring_layout( |
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426 |
const Graph& g, |
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427 |
PositionMap position, |
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428 |
WeightMap weight, |
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429 |
detail::graph::edge_or_side<EdgeOrSideLength, T> edge_or_side_length, |
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430 |
Done done, |
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431 |
typename property_traits<WeightMap>::value_type spring_constant, |
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432 |
VertexIndexMap index, |
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433 |
DistanceMatrix distance, |
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434 |
SpringStrengthMatrix spring_strength, |
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435 |
PartialDerivativeMap partial_derivatives) |
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436 |
{ |
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437 |
BOOST_STATIC_ASSERT((is_convertible< |
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438 |
typename graph_traits<Graph>::directed_category*, |
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439 |
undirected_tag* |
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440 |
>::value)); |
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441 |
|
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442 |
detail::graph::kamada_kawai_spring_layout_impl< |
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443 |
Graph, PositionMap, WeightMap, |
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444 |
detail::graph::edge_or_side<EdgeOrSideLength, T>, Done, VertexIndexMap, |
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445 |
DistanceMatrix, SpringStrengthMatrix, PartialDerivativeMap> |
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446 |
alg(g, position, weight, edge_or_side_length, done, spring_constant, |
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447 |
index, distance, spring_strength, partial_derivatives); |
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448 |
return alg.run(); |
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449 |
} |
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450 |
|
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451 |
/** |
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452 |
* \overload |
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453 |
*/ |
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454 |
template<typename Graph, typename PositionMap, typename WeightMap, |
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455 |
typename T, bool EdgeOrSideLength, typename Done, |
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456 |
typename VertexIndexMap> |
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457 |
bool |
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458 |
kamada_kawai_spring_layout( |
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459 |
const Graph& g, |
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460 |
PositionMap position, |
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461 |
WeightMap weight, |
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462 |
detail::graph::edge_or_side<EdgeOrSideLength, T> edge_or_side_length, |
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463 |
Done done, |
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464 |
typename property_traits<WeightMap>::value_type spring_constant, |
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465 |
VertexIndexMap index) |
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466 |
{ |
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typedef typename property_traits<WeightMap>::value_type weight_type; |
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468 |
|
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typename graph_traits<Graph>::vertices_size_type n = num_vertices(g); |
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typedef std::vector<weight_type> weight_vec; |
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471 |
|
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472 |
std::vector<weight_vec> distance(n, weight_vec(n)); |
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473 |
std::vector<weight_vec> spring_strength(n, weight_vec(n)); |
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474 |
std::vector<std::pair<weight_type, weight_type> > partial_derivatives(n); |
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475 |
|
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476 |
return |
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477 |
kamada_kawai_spring_layout( |
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478 |
g, position, weight, edge_or_side_length, done, spring_constant, index, |
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479 |
distance.begin(), |
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480 |
spring_strength.begin(), |
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481 |
make_iterator_property_map(partial_derivatives.begin(), index, |
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482 |
std::pair<weight_type, weight_type>())); |
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483 |
} |
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|
484 |
|
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485 |
/** |
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486 |
* \overload |
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487 |
*/ |
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|
488 |
template<typename Graph, typename PositionMap, typename WeightMap, |
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489 |
typename T, bool EdgeOrSideLength, typename Done> |
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490 |
bool |
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491 |
kamada_kawai_spring_layout( |
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492 |
const Graph& g, |
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493 |
PositionMap position, |
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494 |
WeightMap weight, |
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|
495 |
detail::graph::edge_or_side<EdgeOrSideLength, T> edge_or_side_length, |
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496 |
Done done, |
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497 |
typename property_traits<WeightMap>::value_type spring_constant) |
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|
498 |
{ |
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Revision: 201002
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|
499 |
return kamada_kawai_spring_layout(g, position, weight, edge_or_side_length, |
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500 |
done, spring_constant, |
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Revision: 201002
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501 |
get(vertex_index, g)); |
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502 |
} |
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|
503 |
|
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504 |
/** |
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Revision: 201002
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|
505 |
* \overload |
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|
506 |
*/ |
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Revision: 201002
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|
507 |
template<typename Graph, typename PositionMap, typename WeightMap, |
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|
508 |
typename T, bool EdgeOrSideLength, typename Done> |
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|
509 |
bool |
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Revision: 201002
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510 |
kamada_kawai_spring_layout( |
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Revision: 201002
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511 |
const Graph& g, |
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Revision: 201002
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512 |
PositionMap position, |
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Revision: 201002
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513 |
WeightMap weight, |
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detail::graph::edge_or_side<EdgeOrSideLength, T> edge_or_side_length, |
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Done done) |
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{ |
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typedef typename property_traits<WeightMap>::value_type weight_type; |
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return kamada_kawai_spring_layout(g, position, weight, edge_or_side_length, |
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done, weight_type(1)); |
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} |
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|
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/** |
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* \overload |
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*/ |
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template<typename Graph, typename PositionMap, typename WeightMap, |
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typename T, bool EdgeOrSideLength> |
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bool |
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kamada_kawai_spring_layout( |
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const Graph& g, |
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PositionMap position, |
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WeightMap weight, |
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detail::graph::edge_or_side<EdgeOrSideLength, T> edge_or_side_length) |
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{ |
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typedef typename property_traits<WeightMap>::value_type weight_type; |
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return kamada_kawai_spring_layout(g, position, weight, edge_or_side_length, |
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layout_tolerance<weight_type>(), |
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weight_type(1.0), |
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get(vertex_index, g)); |
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} |
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} // end namespace boost |
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|
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#endif // BOOST_GRAPH_KAMADA_KAWAI_SPRING_LAYOUT_HPP |