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|
local E = {}
require("pgf.gd.planar").Embedding = E
-- includes
local LinkedList = require("pgf.gd.planar.LinkedList")
E.vertexmetatable = {
__tostring = function(v)
if v.name then
return v.name
elseif v.inputvertex then
return v.inputvertex.name
else
return tostring(v)
end
end
}
E.halfedgemetatable = {
__tostring = function(e)
return tostring(e.twin.target)
.. " -> "
.. tostring(e.target)
end
}
-- create class properties
E.__index = E
function E.new()
local t = {
vertices = {},
}
setmetatable(t, E)
return t
end
function E:add_vertex(name, inputvertex, virtual)
virtual = virtual or nil
local vertex = {
adjmat = {},
name = name,
inputvertex = inputvertex,
virtual = virtual,
}
setmetatable(vertex, E.vertexmetatable)
table.insert(self.vertices, vertex)
return vertex
end
function E:add_edge(v1, v2, after1, after2, virtual)
assert(v1.link == nil or v1 == after1.twin.target)
assert(v2.link == nil or v2 == after2.twin.target)
assert(v1.adjmat[v2] == nil)
assert(v2.adjmat[v1] == nil)
virtual = virtual or nil
local halfedge1 = {
target = v2,
virtual = virtual,
links = {},
}
local halfedge2 = {
target = v1,
virtual = virtual,
links = {},
}
halfedge1.twin = halfedge2
halfedge2.twin = halfedge1
setmetatable(halfedge1, E.halfedgemetatable)
setmetatable(halfedge2, E.halfedgemetatable)
if v1.link == nil then
v1.link = halfedge1
halfedge1.links[0] = halfedge1
halfedge1.links[1] = halfedge1
else
halfedge1.links[0] = after1.links[0]
after1.links[0].links[1] = halfedge1
halfedge1.links[1] = after1
after1.links[0] = halfedge1
end
if v2.link == nil then
v2.link = halfedge2
halfedge2.links[0] = halfedge2
halfedge2.links[1] = halfedge2
else
halfedge2.links[0] = after2.links[0]
after2.links[0].links[1] = halfedge2
halfedge2.links[1] = after2
after2.links[0] = halfedge2
end
v1.adjmat[v2] = halfedge1
v2.adjmat[v1] = halfedge2
return halfedge1, halfedge2
end
function E:remove_virtual()
local virtuals = {}
for i, v in ipairs(self.vertices) do
if v.virtual then
table.insert(virtuals, i)
else
local start = v.link
local current = start
repeat
current = current.links[0]
if current.virtual then
current.links[0].links[1] = current.links[1]
current.links[1].links[0] = current.links[0]
v.adjmat[current.target] = nil
current.target.adjmat[v] = nil
end
until current == start
end
end
for i = #virtuals, 1, -1 do
self.vertices[virtuals[i]] = self.vertices[#self.vertices]
table.remove(self.vertices)
end
end
-- for the use in for-loops
-- iterates over the adjacency list of a vertex
-- given a half edge to start and a direction (0 or 1, default 0)
function E.adjacency_iterator(halfedge, direction)
direction = direction or 0
local function next_edge(startedge, prevedge)
if prevedge == nil then
return startedge
else
local nextedge = prevedge.links[direction]
if nextedge ~= startedge then
return nextedge
else
return nil
end
end
end
return next_edge, halfedge, nil
end
function E.face_iterator(halfedge, direction)
direction = direction or 0
local function next_edge(startedge, prevedge)
if prevedge == nil then
return startedge
else
local nextedge = prevedge.twin.links[1 - direction]
if nextedge ~= startedge then
return nextedge
else
return nil
end
end
end
return next_edge, halfedge, nil
end
function E:triangulate()
local visited = {}
for _, vertex in ipairs(self.vertices) do
for start in E.adjacency_iterator(vertex.link) do
if not visited[start] then
local prev = start
local beforestart = start.links[0].twin
local current = start.twin.links[1]
local next = current.twin.links[1]
visited[start] = true
visited[current] = true
visited[next] = true
while next ~= beforestart do
local halfedge1, halfedge2
if vertex ~= current.target
and not vertex.adjmat[current.target] then
halfedge1, halfedge2 = self:add_edge(
vertex, current.target,
prev, next,
true
)
prev = halfedge1
current = next
next = next.twin.links[1]
elseif not prev.target.adjmat[next.target] then
halfedge1, halfedge2 = self:add_edge(
prev.target, next.target,
current, next.twin.links[1],
true
)
current = halfedge1
next = halfedge2.links[1]
else
local helper = next.twin.links[1]
halfedge1, halfedge2 = self:add_edge(
current.target, helper.target,
next, helper.twin.links[1],
true
)
next = halfedge1
end
visited[next] = true
visited[halfedge1] = true
visited[halfedge2] = true
end
end
end
end
end
function E:canonical_order(v1, v2, vn)
local n = #self.vertices
local order = { v1 }
local marks = { [v1] = "ordered", [v2] = 0 }
local visited = {}
local vk = v1
local candidates = LinkedList.new()
local listelements = {}
for k = 1, n-2 do
for halfedge in E.adjacency_iterator(vk.link) do
local vertex = halfedge.target
if vertex ~= vn then
local twin = halfedge.twin
visited[twin] = true
if marks[vertex] == nil then
marks[vertex] = "visited"
elseif marks[vertex] ~= "ordered" then
local neighbor1 = visited[twin.links[0]]
local neighbor2 = visited[twin.links[1]]
if marks[vertex] == "visited" then
if neighbor1 or neighbor2 then
marks[vertex] = 1
listelements[vertex] = candidates:addback(vertex)
else
marks[vertex] = 2
end
else
if neighbor1 == neighbor2 then
if neighbor1 and neighbor2 then
marks[vertex] = marks[vertex] - 1
else
marks[vertex] = marks[vertex] + 1
end
if marks[vertex] == 1 then
listelements[vertex]
= candidates:addback(vertex)
elseif listelements[vertex] then
candidates:remove(listelements[vertex])
listelements[vertex] = nil
end
end
end
end
end
end
vk = candidates:popfirst()
order[k+1] = vk
marks[vk] = "ordered"
end
order[n] = vn
return order
end
function E:get_biggest_face()
local number = 0
local edge
local visited = {}
for _, vertex in ipairs(self.vertices) do
for start in E.adjacency_iterator(vertex.link) do
local count = 0
if not visited[start] then
visited[start] = true
local current = start
repeat
count = count + 1
current = current.twin.links[1]
until current == start
if count > number then
number = count
edge = start
end
end
end
end
return edge, number
end
function E:surround_by_triangle(faceedge, facesize)
local divisor = 3
if facesize > 3 then
divisor = 4
end
local basenodes = math.floor(facesize / divisor)
local extranodes = facesize % divisor
local attachnodes = { basenodes, basenodes, basenodes }
if facesize > 3 then
attachnodes[2] = basenodes * 2
end
for i = 1,extranodes do
attachnodes[i] = attachnodes[i] + 1
end
local v = {
self:add_vertex("$v_1$", nil, true),
self:add_vertex("$v_n$", nil, true),
self:add_vertex("$v_2$", nil, true)
}
for i = 1,3 do
local currentv = v[i]
local nextv = v[i % 3 + 1]
self:add_edge(currentv, nextv, currentv.link, nextv.link, true)
end
local current = faceedge
local next = current.twin.links[1]
for i = 1,3 do
local vertex = v[i]
local otheredge = vertex.adjmat[v[i % 3 + 1]]
local previnserted = otheredge.links[1]
for count = 1, attachnodes[i] do
if not vertex.adjmat[current.target] then
previnserted, _ = self:add_edge(vertex, current.target,
previnserted, next,
true
)
end
current = next
next = next.twin.links[1]
end
if not vertex.adjmat[current.target] then
previnserted, _ = self:add_edge(
vertex, current.target,
previnserted, next,
true
)
current = previnserted
end
end
return v[1], v[3], v[2]
end
function E:improve()
local pairdata = {}
local inpair = {}
for i, v1 in ipairs(self.vertices) do
for j = i + 1, #self.vertices do
local v2 = self.vertices[j]
local pd = self:find_pair_components(v1, v2)
if pd then
inpair[v1] = true
inpair[v2] = true
table.insert(pairdata, pd)
end
end
if not inpair[v1] then
local pd = self:find_pair_components(v1, nil)
if pd then
inpair[v1] = true
table.insert(pairdata, pd)
end
end
end
local changed
local runs = 1
local edgepositions = {}
repeat
changed = false
for i, pd in ipairs(pairdata) do
self:improve_separation_pair(pd)
end
-- check for changes
for i, v in ipairs(self.vertices) do
local start = v.link
local current = start
local counter = 1
repeat
if counter ~= edgepositions[current] then
changed = true
edgepositions[current] = counter
end
counter = counter + 1
current = current.links[0]
until current == start
end
runs = runs + 1
until changed == false or runs > 100
end
function E:find_pair_components(v1, v2)
local visited = {}
local companchors = {}
local edgecomps = {}
local compvertices = {}
local islinear = {}
local edgeindices = {}
local pair = { v1, v2 }
local start = v1.link
local current = start
local edgeindex = 1
-- start searches from v1
repeat
edgeindices[current] = edgeindex
edgeindex = edgeindex + 1
if not edgecomps[current] then
local compindex = #companchors + 1
local ca, il
edgecomps[current] = compindex
compvertices[compindex] = {}
local target = current.target
if target == v2 then
edgecomps[current.twin] = compindex
ca = 3
il = true
else
ca, il = self:component_dfs(
target,
pair,
visited,
edgecomps,
compvertices[compindex],
compindex
)
end
companchors[compindex] = ca
islinear[compindex] = il
end
current = current.links[0]
until current == start
if v2 then
start = v2.link
current = start
local lastincomp = true
local edgeindex = 1
-- now find the remaining blocks at v2
repeat
edgeindices[current] = edgeindex
edgeindex = edgeindex + 1
if not edgecomps[current] then
local compindex = #companchors + 1
edgecomps[current] = compindex
compvertices[compindex] = {}
self:component_dfs(
current.target,
pair,
visited,
edgecomps,
compvertices[compindex],
compindex
)
companchors[compindex] = 2
end
current = current.links[0]
until current == start
end
-- init compedges, tricomps, twocomps
local tricomps = {}
local twocomps = {{}, {}}
for i, anchors in ipairs(companchors) do
if anchors == 3 then
table.insert(tricomps, i)
else
table.insert(twocomps[anchors], i)
end
end
local flipimmune = #tricomps == 2
and (islinear[tricomps[1]] or islinear[tricomps[2]])
if (#tricomps < 2 or flipimmune)
and (v2 ~= nil or #twocomps[1] < 2) then
return nil
end
-- order tri comps cyclic
local function sorter(a, b)
return #compvertices[a] < #compvertices[b]
end
table.sort(tricomps, sorter)
-- determine order of comps
local numtricomps = #tricomps
local comporder = { {}, {} }
local bottom = math.ceil(numtricomps / 2)
local top = bottom + 1
for i, comp in ipairs(tricomps) do
if i % 2 == 1 then
comporder[1][bottom] = comp
comporder[2][numtricomps - bottom + 1] = comp
bottom = bottom - 1
else
comporder[1][top] = comp
comporder[2][numtricomps - top + 1] = comp
top = top + 1
end
end
local pairdata = {
pair = pair,
companchors = companchors,
edgecomps = edgecomps,
edgeindices = edgeindices,
compvertices = compvertices,
tricomps = tricomps,
twocomps = twocomps,
comporder = comporder,
}
return pairdata
end
function E:component_dfs(v, pair, visited, edgecomps, compvertices, compindex)
visited[v] = true
local start = v.link
local current = start
local companchors = 1
local numedges = 0
local islinear = true
table.insert(compvertices, v)
repeat
numedges = numedges + 1
local target = current.target
if target == pair[1] or target == pair[2] then
edgecomps[current.twin] = compindex
if target == pair[2] then
companchors = 3
end
elseif not visited[target] then
local ca, il = self:component_dfs(
target,
pair,
visited,
edgecomps,
compvertices,
compindex
)
if ca == 3 then
companchors = 3
end
islinear = islinear and il
end
current = current.links[0]
until current == start
return companchors, islinear and numedges == 2
end
function E:improve_separation_pair(pairdata)
local pair = pairdata.pair
local companchors = pairdata.companchors
local edgecomps = pairdata.edgecomps
local edgeindices = pairdata.edgeindices
local compvertices = pairdata.compvertices
local tricomps = pairdata.tricomps
local twocomps = pairdata.twocomps
local comporder = pairdata.comporder
local v1 = pair[1]
local v2 = pair[2]
local compedges = {}
for i = 1, #companchors do
compedges[i] = {{}, {}}
end
local numtricomps = #tricomps
local numtwocomps = { #twocomps[1], #twocomps[2] }
-- find compedges
for i = 1, #pair do
-- first find an edge that is the first of a triconnected component
local start2
if v2 then
start = pair[i].link
current = start
local last
repeat
local comp = edgecomps[current]
if companchors[comp] == 3 then
if last == nil then
last = comp
elseif last ~= comp then
start2 = current
break
end
end
current = current.links[0]
until current == start
else
start2 = pair[i].link
end
-- now list the edges by components
current = start2
repeat
table.insert(compedges[edgecomps[current]][i], current)
current = current.links[0]
until current == start2
end
-- count edges on each side of tri comps
local edgecount = {}
for _, comp in ipairs(tricomps) do
edgecount[comp] = {}
for i = 1, #pair do
local count = 1
local current = compedges[comp][i][1]
local other = pair[3 - i]
while current.target ~= other do
count = count + 1
current = current.twin.links[0]
end
edgecount[comp][i] = count
end
end
-- determine which comps have to be flipped
local flips = {}
local numflips = 0
local allflipped = true
for i, comp in ipairs(comporder[1]) do
local side1, side2
if i > numtricomps / 2 then
side1 = edgecount[comp][1]
side2 = edgecount[comp][2]
else
side1 = edgecount[comp][2]
side2 = edgecount[comp][1]
end
if side1 > side2 then
numflips = numflips + 1
flips[comp] = true
elseif side1 < side2 then
allflipped = false
end
end
if allflipped then
for i, comp in ipairs(tricomps) do
flips[comp] = false
end
else
for i, comp in ipairs(tricomps) do
if flips[comp] then
for _, v in ipairs(compvertices[comp]) do
local start = v.link
local current = start
repeat
current.links[0], current.links[1]
= current.links[1], current.links[0]
current = current.links[1]
until current == start
end
end
end
end
-- order edges cyclic per component (one cycle for all tri comps)
for i = 1, #pair do
if v2 then
local co
if allflipped then
co = comporder[3 - i]
else
co = comporder[i]
end
local id = co[numtricomps]
lastedges = compedges[id][i]
if flips[id] then
lastedge = lastedges[1]
else
lastedge = lastedges[#lastedges]
end
-- tri comps
for _, id in ipairs(co) do
local edges = compedges[id][i]
local from
local to
local step
if flips[id] then
from = #edges
to = 1
step = -1
else
from = 1
to = #edges
step = 1
end
for k = from, to, step do
local edge = edges[k]
lastedge.links[0] = edge
edge.links[1] = lastedge
lastedge = edge
end
end
end
-- two comps
for _, id in ipairs(twocomps[i]) do
lastedges = compedges[id][i]
lastedge = lastedges[#lastedges]
for _, edge in ipairs(compedges[id][i]) do
lastedge.links[0] = edge
edge.links[1] = lastedge
lastedge = edge
end
end
end
-- now merge the cycles
for i = 1, #pair do
local outeredges = {}
-- find the biggest face of the tri comps
if v2 then
local biggestedge
local biggestsize
local biggestindex
local start = compedges[tricomps[1]][i][1]
local current = start
repeat
local size = self:get_face_size(current)
if not biggestedge or size > biggestsize
or (size == biggestsize
and edgeindices[current] > biggestindex) then
biggestedge = current
biggestsize = size
biggestindex = edgeindices[current]
end
current = current.links[0]
until current == start
outeredges[1] = biggestedge
end
-- now for every two comp
for _, id in ipairs(twocomps[i]) do
local biggestedge
local biggestsize
local biggestindex
local start = compedges[id][i][1]
local current = start
repeat
local size = self:get_face_size(current)
if not biggestedge or size > biggestsize
or (size == biggestsize
and edgeindices[current] > biggestindex) then
biggestedge = current
biggestsize = size
biggestindex = edgeindices[current]
end
current = current.links[0]
until current == start
table.insert(outeredges, biggestedge)
end
-- now merge all comps at the outer edges
local lastedge = outeredges[#outeredges].links[0]
for _, edge in ipairs(outeredges) do
local nextlastedge = edge.links[0]
lastedge.links[1] = edge
edge.links[0] = lastedge
lastedge = nextlastedge
end
end
end
function E:get_face_size(halfedge)
local size = 0
local current = halfedge
repeat
size = size + 1
current = current.twin.links[1]
until current == halfedge
return size
end
return E
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