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|
-- Copyright 2011 by Jannis Pohlmann
--
-- This file may be distributed and/or modified
--
-- 1. under the LaTeX Project Public License and/or
-- 2. under the GNU Public License
--
-- See the file doc/generic/pgf/licenses/LICENSE for more information
-- @release $Header: /cvsroot/pgf/pgf/generic/pgf/graphdrawing/lua/pgf/gd/force/SpringElectricalHu2006.lua,v 1.1 2012/11/27 17:24:25 tantau Exp $
local SpringElectricalHu2006 = {}
-- Imports
local declare = require("pgf.gd.interface.InterfaceToAlgorithms").declare
---
declare {
key = "spring electrical Hu 2006 layout",
algorithm = SpringElectricalHu2006,
preconditions = {
connected = true,
loop_free = true,
simple = true,
},
old_graph_model = true,
summary = [["
Implementation of a spring electrical graph drawing algorithm based on
a paper by Hu.
"]],
documentation = [["
\begin{itemize}
\item
Y. Hu.
\newblock Efficient, high-quality force-directed graph drawing.
\newblock \emph{The Mathematica Journal}, 2006.
\end{itemize}
There are some modifications compared to the original algorithm,
see the Diploma thesis of Pohlmann for details.
"]]
}
-- Imports
local PathLengths = require "pgf.gd.lib.PathLengths"
local Vector = require "pgf.gd.deprecated.Vector"
local QuadTree = require "pgf.gd.force.QuadTree"
local CoarseGraph = require "pgf.gd.force.CoarseGraph"
local lib = require "pgf.gd.lib"
function SpringElectricalHu2006:run()
-- Setup properties
local options = self.digraph.options
self.iterations = options['iterations']
self.cooling_factor = options['cooling factor']
self.initial_step_length = options['initial step length']
self.convergence_tolerance = options['convergence tolerance']
self.natural_spring_length = options['node distance']
self.spring_constant = options['spring constant']
self.approximate_repulsive_forces = options['approximate remote forces']
self.repulsive_force_order = options['electric force order']
self.coarsen = options['coarsen']
self.downsize_ratio = options['downsize ratio']
self.minimum_graph_size = options['minimum coarsening size']
-- Adjust types
self.downsize_ratio = math.max(0, math.min(1, self.downsize_ratio))
self.graph_size = #self.graph.nodes
self.graph_density = (2 * #self.graph.edges) / (#self.graph.nodes * (#self.graph.nodes - 1))
-- validate input parameters
assert(self.iterations >= 0, 'iterations (value: ' .. self.iterations .. ') need to be greater than 0')
assert(self.cooling_factor >= 0 and self.cooling_factor <= 1, 'the cooling factor (value: ' .. self.cooling_factor .. ') needs to be between 0 and 1')
assert(self.initial_step_length >= 0, 'the initial step length (value: ' .. self.initial_step_length .. ') needs to be greater than or equal to 0')
assert(self.convergence_tolerance >= 0, 'the convergence tolerance (value: ' .. self.convergence_tolerance .. ') needs to be greater than or equal to 0')
assert(self.natural_spring_length >= 0, 'the natural spring dimension (value: ' .. self.natural_spring_length .. ') needs to be greater than or equal to 0')
assert(self.spring_constant >= 0, 'the spring constant (value: ' .. self.spring_constant .. ') needs to be greater or equal to 0')
assert(self.downsize_ratio >= 0 and self.downsize_ratio <= 1, 'the downsize ratio (value: ' .. self.downsize_ratio .. ') needs to be between 0 and 1')
assert(self.minimum_graph_size >= 2, 'the minimum coarsening size of coarse graphs (value: ' .. self.minimum_graph_size .. ') needs to be greater than or equal to 2')
-- initialize node weights
for _,node in ipairs(self.graph.nodes) do
node.weight = node:getOption('electric charge')
end
-- initialize edge weights
for _,edge in ipairs(self.graph.edges) do
edge.weight = 1
end
-- initialize the coarse graph data structure. note that the algorithm
-- is the same regardless whether coarsening is used, except that the
-- number of coarsening steps without coarsening is 0
local coarse_graph = CoarseGraph.new(self.graph)
-- check if the multilevel approach should be used
if self.coarsen then
-- coarsen the graph repeatedly until only minimum_graph_size nodes
-- are left or until the size of the coarse graph was not reduced by
-- at least the downsize ratio configured by the user
while coarse_graph:getSize() > self.minimum_graph_size
and coarse_graph:getRatio() <= (1 - self.downsize_ratio)
do
coarse_graph:coarsen()
end
end
if self.coarsen then
-- use the natural spring length as the initial natural spring length
local spring_length = self.natural_spring_length
-- compute a random initial layout for the coarsest graph
self:computeInitialLayout(coarse_graph.graph, spring_length)
-- set the spring length to the average edge length of the initial layout
spring_length = 0
for _,edge in ipairs(coarse_graph.graph.edges) do
spring_length = spring_length + edge.nodes[1].pos:minus(edge.nodes[2].pos):norm()
end
spring_length = spring_length / #coarse_graph.graph.edges
-- additionally improve the layout with the force-based algorithm
-- if there are more than two nodes in the coarsest graph
if coarse_graph:getSize() > 2 then
self:computeForceLayout(coarse_graph.graph, spring_length, SpringElectricalHu2006.adaptive_step_update)
end
-- undo coarsening step by step, applying the force-based sub-algorithm
-- to every intermediate coarse graph as well as the original graph
while coarse_graph:getLevel() > 0 do
-- compute the diameter of the parent coarse graph
local parent_diameter = PathLengths.pseudoDiameter(coarse_graph.graph)
-- interpolate the previous coarse graph from its parent
coarse_graph:interpolate()
-- compute the diameter of the current coarse graph
local current_diameter = PathLengths.pseudoDiameter(coarse_graph.graph)
-- scale node positions by the quotient of the pseudo diameters
for _,node in ipairs(coarse_graph.graph) do
node.pos:update(function (n, value)
return value * (current_diameter / parent_diameter)
end)
end
-- compute forces in the graph
self:computeForceLayout(coarse_graph.graph, spring_length, SpringElectricalHu2006.conservative_step_update)
end
else
-- compute a random initial layout for the coarsest graph
self:computeInitialLayout(coarse_graph.graph, self.natural_spring_length)
-- set the spring length to the average edge length of the initial layout
spring_length = 0
for _,e in ipairs(coarse_graph.graph.edges) do
spring_length = spring_length + edge.nodes[1].pos:minus(edge.nodes[2].pos):norm()
end
spring_length = spring_length / #coarse_graph.graph.edges
-- improve the layout with the force-based algorithm
self:computeForceLayout(coarse_graph.graph, spring_length, SpringElectricalHu2006.adaptive_step_update)
end
end
function SpringElectricalHu2006:computeInitialLayout(graph, spring_length)
-- TODO how can supernodes and fixed nodes go hand in hand?
-- maybe fix the supernode if at least one of its subnodes is
-- fixated?
-- fixate all nodes that have a 'desired at' option. this will set the
-- node.fixed member to true and also set node.pos.x and node.pos.y
self:fixateNodes(graph)
if #graph.nodes == 2 then
if not (graph.nodes[1].fixed and graph.nodes[2].fixed) then
local fixed_index = graph.nodes[2].fixed and 2 or 1
local loose_index = graph.nodes[2].fixed and 1 or 2
if not graph.nodes[1].fixed and not graph.nodes[2].fixed then
-- both nodes can be moved, so we assume node 1 is fixed at (0,0)
graph.nodes[1].pos.x = 0
graph.nodes[1].pos.y = 0
end
-- position the loose node relative to the fixed node, with
-- the displacement (random direction) matching the spring length
local direction = Vector.new{x = math.random(1, spring_length), y = math.random(1, spring_length)}
local distance = 3 * spring_length * self.graph_density * math.sqrt(self.graph_size) / 2
local displacement = direction:normalized():timesScalar(distance)
graph.nodes[loose_index].pos = graph.nodes[fixed_index].pos:plus(displacement)
else
-- both nodes are fixed, initial layout may be far from optimal
end
else
-- use a random positioning technique
local function positioning_func(n)
local radius = 3 * spring_length * self.graph_density * math.sqrt(self.graph_size) / 2
return math.random(-radius, radius)
end
-- compute initial layout based on the random positioning technique
for _,node in ipairs(graph.nodes) do
if not node.fixed then
node.pos.x = positioning_func(1)
node.pos.y = positioning_func(2)
end
end
end
end
function SpringElectricalHu2006:computeForceLayout(graph, spring_length, step_update_func)
-- global (=repulsive) force function
function accurate_repulsive_force(distance, weight)
-- note: the weight is taken into the equation here. unlike in the original
-- algorithm different electric charges are allowed for each node in this
-- implementation
return - weight * self.spring_constant * math.pow(spring_length, self.repulsive_force_order + 1) / math.pow(distance, self.repulsive_force_order)
end
-- global (=repulsive, approximated) force function
function approximated_repulsive_force(distance, mass)
return - mass * self.spring_constant * math.pow(spring_length, self.repulsive_force_order + 1) / math.pow(distance, self.repulsive_force_order)
end
-- local (spring) force function
function attractive_force(distance)
return (distance * distance) / spring_length
end
-- define the Barnes-Hut opening criterion
function barnes_hut_criterion(cell, particle)
local distance = particle.pos:minus(cell.center_of_mass):norm()
return cell.width / distance <= 1.2
end
-- fixate all nodes that have a 'desired at' option. this will set the
-- node.fixed member to true and also set node.pos.x and node.pos.y
self:fixateNodes(graph)
-- adjust the initial step length automatically if desired by the user
local step_length = self.initial_step_length == 0 and spring_length or self.initial_step_length
-- convergence criteria etc.
local converged = false
local energy = math.huge
local iteration = 0
local progress = 0
while not converged and iteration < self.iterations do
-- remember old node positions
local old_positions = lib.map(graph.nodes, function (node)
return node.pos:copy(), node
end)
-- remember the old system energy and reset it for the current iteration
local old_energy = energy
energy = 0
-- build the quadtree for approximating repulsive forces, if desired
local quadtree = nil
if self.approximate_repulsive_forces then
quadtree = self:buildQuadtree(graph)
end
for _,v in ipairs(graph.nodes) do
if not v.fixed then
-- vector for the displacement of v
local d = Vector.new(2)
-- compute repulsive forces
if self.approximate_repulsive_forces then
-- determine the cells that have a repulsive influence on v
local cells = quadtree:findInteractionCells(v, barnes_hut_criterion)
-- compute the repulsive force between these cells and v
for _,cell in ipairs(cells) do
-- check if the cell is a leaf
if #cell.subcells == 0 then
-- compute the forces between the node and all particles in the cell
for _,particle in ipairs(cell.particles) do
local real_particles = lib.copy(particle.subparticles)
table.insert(real_particles, particle)
for _,real_particle in ipairs(real_particles) do
local delta = real_particle.pos:minus(v.pos)
-- enforce a small virtual distance if the node and the cell's
-- center of mass are located at (almost) the same position
if delta:norm() < 0.1 then
delta:update(function (n, value) return 0.1 + math.random() * 0.1 end)
end
-- compute the repulsive force vector
local repulsive_force = approximated_repulsive_force(delta:norm(), real_particle.mass)
local force = delta:normalized():timesScalar(repulsive_force)
-- move the node v accordingly
d = d:plus(force)
end
end
else
-- compute the distance between the node and the cell's center of mass
local delta = cell.center_of_mass:minus(v.pos)
-- enforce a small virtual distance if the node and the cell's
-- center of mass are located at (almost) the same position
if delta:norm() < 0.1 then
delta:update(function (n, value) return 0.1 + math.random() * 0.1 end)
end
-- compute the repulsive force vector
local repulsive_force = approximated_repulsive_force(delta:norm(), cell.mass)
local force = delta:normalized():timesScalar(repulsive_force)
-- move the node v accordingly
d = d:plus(force)
end
end
else
for _,u in ipairs(graph.nodes) do
if v ~= u then
-- compute the distance between u and v
local delta = u.pos:minus(v.pos)
-- enforce a small virtual distance if the nodes are
-- located at (almost) the same position
if delta:norm() < 0.1 then
delta:update(function (n, value) return 0.1 + math.random() * 0.1 end)
end
-- compute the repulsive force vector
local repulsive_force = accurate_repulsive_force(delta:norm(), u.weight)
local force = delta:normalized():timesScalar(repulsive_force)
-- move the node v accordingly
d = d:plus(force)
end
end
end
-- compute attractive forces between v and its neighbours
for _,edge in ipairs(v.edges) do
local u = edge:getNeighbour(v)
-- compute the distance between u and v
local delta = u.pos:minus(v.pos)
-- enforce a small virtual distance if the nodes are
-- located at (almost) the same position
if delta:norm() < 0.1 then
delta:update(function (n, value) return 0.1 + math.random() * 0.1 end)
end
-- compute the spring force vector between u and v
local attr_force = attractive_force(delta:norm())
local force = delta:normalized():timesScalar(attr_force)
-- move the node v accordingly
d = d:plus(force)
end
-- really move the node now
-- TODO note how all nodes are moved by the same amount (step_length)
-- while Walshaw multiplies the normalized force with min(step_length,
-- d:norm()). could that improve this algorithm even further?
v.pos = v.pos:plus(d:normalized():timesScalar(step_length))
-- TODO Hu doesn't mention this but the energy of a particle is
-- typically considered as the product of its mass and the square of
-- its forces. This means we should probably take the weight of
-- the node v into the equation, doesn't it?
--
-- update the energy function
energy = energy + math.pow(d:norm(), 2)
end
end
-- update the step length and progress counter
step_length, progress = step_update_func(step_length, self.cooling_factor, energy, old_energy, progress)
-- compute the maximum node movement in this iteration
local max_movement = 0
for _,x in ipairs(graph.nodes) do
local delta = x.pos:minus(old_positions[x])
max_movement = math.max(delta:norm(), max_movement)
end
-- the algorithm will converge if the maximum movement is below a
-- threshold depending on the spring length and the convergence
-- tolerance
if max_movement < spring_length * self.convergence_tolerance then
converged = true
end
-- increment the iteration counter
iteration = iteration + 1
end
end
-- Fixes nodes at their specified positions.
--
function SpringElectricalHu2006:fixateNodes(graph)
local number_of_fixed_nodes = 0
for _,node in ipairs(graph.nodes) do
-- read the 'desired at' option of the node
local coordinate = node:getOption('desired at')
if coordinate then
-- apply the coordinate
node.pos.x = coordinate.x
node.pos.y = coordinate.y
-- mark the node as fixed
node.fixed = true
number_of_fixed_nodes = number_of_fixed_nodes + 1
end
end
if number_of_fixed_nodes > 1 then
self.growth_direction = "fixed" -- do not grow, orientation is now fixed
end
end
function SpringElectricalHu2006:buildQuadtree(graph)
-- compute the minimum x and y coordinates of all nodes
local min_pos = graph.nodes[1].pos
for _,node in ipairs(graph.nodes) do
min_pos = Vector.new(2, function (n) return math.min(min_pos[n], node.pos[n]) end)
end
-- compute maximum x and y coordinates of all nodes
local max_pos = graph.nodes[1].pos
for _,node in ipairs(graph.nodes) do
max_pos = Vector.new(2, function (n) return math.max(max_pos[n], node.pos[n]) end)
end
-- make sure the maximum position is at least a tiny bit
-- larger than the minimum position
if min_pos:equals(max_pos) then
max_pos = max_pos:plus(Vector.new(2, function (n)
return 0.1 + math.random() * 0.1
end))
end
-- make sure to make the quadtree area slightly larger than required
-- in theory; for some reason Lua will otherwise think that nodes with
-- min/max x/y coordinates are outside the box... weird? yes.
min_pos = min_pos:minus({1,1})
max_pos = max_pos:plus({1,1})
-- create the quadtree
quadtree = QuadTree.new(min_pos.x, min_pos.y,
max_pos.x - min_pos.x,
max_pos.y - min_pos.y)
-- insert nodes into the quadtree
for _,node in ipairs(graph.nodes) do
local particle = QuadTree.Particle.new(node.pos, node.weight)
particle.node = node
quadtree:insert(particle)
end
return quadtree
end
function SpringElectricalHu2006.conservative_step_update(step, cooling_factor)
return cooling_factor * step, nil
end
function SpringElectricalHu2006.adaptive_step_update(step, cooling_factor, energy, old_energy, progress)
if energy < old_energy then
progress = progress + 1
if progress >= 5 then
progress = 0
step = step / cooling_factor
end
else
progress = 0
step = cooling_factor * step
end
return step, progress
end
-- done
return SpringElectricalHu2006
|