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authorKarl Berry <karl@freefriends.org>2014-02-27 00:01:50 +0000
committerKarl Berry <karl@freefriends.org>2014-02-27 00:01:50 +0000
commit90ae413a94da014dbf7a6a8ab584f7c668483a5a (patch)
tree4e0528a2b2fb2004988e7fc804ea2f1354f1f5d5 /Master/texmf-dist/tex/generic/pgf/graphdrawing/lua/pgf/gd/layered/NetworkSimplex.lua
parent800cc56380d1edfe567a6a66bef13e8950eb2b0c (diff)
pgf 3.0.0
git-svn-id: svn://tug.org/texlive/trunk@33057 c570f23f-e606-0410-a88d-b1316a301751
Diffstat (limited to 'Master/texmf-dist/tex/generic/pgf/graphdrawing/lua/pgf/gd/layered/NetworkSimplex.lua')
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diff --git a/Master/texmf-dist/tex/generic/pgf/graphdrawing/lua/pgf/gd/layered/NetworkSimplex.lua b/Master/texmf-dist/tex/generic/pgf/graphdrawing/lua/pgf/gd/layered/NetworkSimplex.lua
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+-- Copyright 2011 by Jannis Pohlmann
+-- Copyright 2012 by Till Tantau
+--
+-- This file may be distributed an/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/layered/NetworkSimplex.lua,v 1.2 2013/05/23 20:01:27 tantau Exp $
+
+
+
+
+--- This file contains an implementation of the network simplex method
+--- for node ranking and x coordinate optimization in layered drawing
+--- algorithms, as proposed in
+---
+--- "A Technique for Drawing Directed Graphs"
+-- by Gansner, Koutsofios, North, Vo, 1993.
+
+
+local NetworkSimplex = {}
+NetworkSimplex.__index = NetworkSimplex
+
+-- Namespace
+local layered = require "pgf.gd.layered"
+layered.NetworkSimplex = NetworkSimplex
+
+
+-- Imports
+local DepthFirstSearch = require "pgf.gd.lib.DepthFirstSearch"
+local Ranking = require "pgf.gd.layered.Ranking"
+local Graph = require "pgf.gd.deprecated.Graph"
+local lib = require "pgf.gd.lib"
+
+
+
+-- Definitions
+
+NetworkSimplex.BALANCE_TOP_BOTTOM = 1
+NetworkSimplex.BALANCE_LEFT_RIGHT = 2
+
+
+function NetworkSimplex.new(graph, balancing)
+ local simplex = {
+ graph = graph,
+ balancing = balancing,
+ }
+ setmetatable(simplex, NetworkSimplex)
+ return simplex
+end
+
+
+
+function NetworkSimplex:run()
+
+ assert (#self.graph.nodes > 0, "graph must contain at least one node")
+
+ -- initialize the tree edge search index
+ self.search_index = 1
+
+ -- initialize internal edge parameters
+ self.cut_value = {}
+ for _,edge in ipairs(self.graph.edges) do
+ self.cut_value[edge] = 0
+ end
+
+ -- reset graph information needed for ranking
+ self.lim = {}
+ self.low = {}
+ self.parent_edge = {}
+ self.ranking = Ranking.new()
+
+ if #self.graph.nodes == 1 then
+ self.ranking:setRank(self.graph.nodes[1], 1)
+ else
+ self:rankNodes()
+ end
+end
+
+
+
+function NetworkSimplex:rankNodes()
+ -- construct feasible tree of tight edges
+ self:constructFeasibleTree()
+
+ -- iteratively replace edges with negative cut values
+ -- with non-tree edges (chosen by minimum slack)
+ local leave_edge = self:findNegativeCutEdge()
+ while leave_edge do
+ local enter_edge = self:findReplacementEdge(leave_edge)
+
+ assert(enter_edge, 'no non-tree edge to replace ' .. tostring(leave_edge) .. ' could be found')
+
+ -- exchange leave_edge and enter_edge in the tree, updating
+ -- the ranks and cut values of all nodes
+ self:exchangeTreeEdges(leave_edge, enter_edge)
+
+ -- find the next tree edge with a negative cut value, if
+ -- there are any left
+ leave_edge = self:findNegativeCutEdge()
+ end
+
+ if self.balancing == NetworkSimplex.BALANCE_TOP_BOTTOM then
+ -- normalize by setting the least rank to zero
+ self.ranking:normalizeRanks()
+
+ -- move nodes to feasible ranks with the least number of nodes
+ -- in order to avoid crowding and to improve the overall aspect
+ -- ratio of the drawing
+ self:balanceRanksTopBottom()
+ elseif self.balancing == NetworkSimplex.BALANCE_LEFT_RIGHT then
+ self:balanceRanksLeftRight()
+ end
+end
+
+
+
+function NetworkSimplex:constructFeasibleTree()
+
+ self:computeInitialRanking()
+
+ -- find a maximal tree of tight edges in the graph
+ while self:findTightTree() < #self.graph.nodes do
+
+ local min_slack_edge = nil
+
+ for _,node in ipairs(self.graph.nodes) do
+ local out_edges = node:getOutgoingEdges()
+ for _,edge in ipairs(out_edges) do
+ if not self.tree_edge[edge] and self:isIncidentToTree(edge) then
+ if not min_slack_edge or self:edgeSlack(edge) < self:edgeSlack(min_slack_edge) then
+ min_slack_edge = edge
+ end
+ end
+ end
+ end
+
+ if min_slack_edge then
+ local delta = self:edgeSlack(min_slack_edge)
+
+ if delta > 0 then
+ local head = min_slack_edge:getHead()
+ local tail = min_slack_edge:getTail()
+
+ if self.tree_node[head] then
+ delta = -delta
+ end
+
+
+ for _,node in ipairs(self.tree.nodes) do
+ local rank = self.ranking:getRank(self.orig_node[node])
+ self.ranking:setRank(self.orig_node[node], rank + delta)
+ end
+ end
+ end
+ end
+
+ self:initializeCutValues()
+end
+
+
+
+function NetworkSimplex:findNegativeCutEdge()
+ local minimum_edge = nil
+
+ for n=1,#self.tree.edges do
+ local index = self:nextSearchIndex()
+
+ local edge = self.tree.edges[index]
+
+ if self.cut_value[edge] < 0 then
+ if minimum_edge then
+ if self.cut_value[minimum_edge] > self.cut_value[edge] then
+ minimum_edge = edge
+ end
+ else
+ minimum_edge = edge
+ end
+ end
+ end
+
+ return minimum_edge
+end
+
+
+
+function NetworkSimplex:findReplacementEdge(leave_edge)
+ local tail = leave_edge:getTail()
+ local head = leave_edge:getHead()
+
+ local v = nil
+ local direction = nil
+
+ if self.lim[tail] < self.lim[head] then
+ v = tail
+ direction = 'in'
+ else
+ v = head
+ direction = 'out'
+ end
+
+ local search_root = v
+ local enter_edge = nil
+ local slack = math.huge
+
+ -- TODO Janns: Get rid of this recursion:
+
+ local function find_edge(v, direction)
+
+ if direction == 'out' then
+ local out_edges = self.orig_node[v]:getOutgoingEdges()
+ for _,edge in ipairs(out_edges) do
+ local head = edge:getHead()
+ local tree_head = self.tree_node[head]
+
+ assert(head and tree_head)
+
+ if not self.tree_edge[edge] then
+ if not self:inTailComponentOf(tree_head, search_root) then
+ if self:edgeSlack(edge) < slack or not enter_edge then
+ enter_edge = edge
+ slack = self:edgeSlack(edge)
+ end
+ end
+ else
+ if self.lim[tree_head] < self.lim[v] then
+ find_edge(tree_head, 'out')
+ end
+ end
+ end
+
+ for _,edge in ipairs(v:getIncomingEdges()) do
+ if slack <= 0 then
+ break
+ end
+
+ local tail = edge:getTail()
+
+ if self.lim[tail] < self.lim[v] then
+ find_edge(tail, 'out')
+ end
+ end
+ else
+ local in_edges = self.orig_node[v]:getIncomingEdges()
+ for _,edge in ipairs(in_edges) do
+ local tail = edge:getTail()
+ local tree_tail = self.tree_node[tail]
+
+ assert(tail and tree_tail)
+
+ if not self.tree_edge[edge] then
+ if not self:inTailComponentOf(tree_tail, search_root) then
+ if self:edgeSlack(edge) < slack or not enter_edge then
+ enter_edge = edge
+ slack = self:edgeSlack(edge)
+ end
+ end
+ else
+ if self.lim[tree_tail] < self.lim[v] then
+ find_edge(tree_tail, 'in')
+ end
+ end
+ end
+
+ for _,edge in ipairs(v:getOutgoingEdges()) do
+ if slack <= 0 then
+ break
+ end
+
+ local head = edge:getHead()
+
+ if self.lim[head] < self.lim[v] then
+ find_edge(head, 'in')
+ end
+ end
+ end
+ end
+
+ find_edge(v, direction)
+
+ return enter_edge
+end
+
+
+
+function NetworkSimplex:exchangeTreeEdges(leave_edge, enter_edge)
+
+ self:rerankBeforeReplacingEdge(leave_edge, enter_edge)
+
+ local cutval = self.cut_value[leave_edge]
+ local head = self.tree_node[enter_edge:getHead()]
+ local tail = self.tree_node[enter_edge:getTail()]
+
+ local ancestor = self:updateCutValuesUpToCommonAncestor(tail, head, cutval, true)
+ local other_ancestor = self:updateCutValuesUpToCommonAncestor(head, tail, cutval, false)
+
+ assert(ancestor == other_ancestor)
+
+ -- remove the old edge from the tree
+ self:removeEdgeFromTree(leave_edge)
+
+ -- add the new edge to the tree
+ local tree_edge = self:addEdgeToTree(enter_edge)
+
+ -- set its cut value
+ self.cut_value[tree_edge] = -cutval
+
+ -- update DFS search tree traversal information
+ self:calculateDFSRange(ancestor, self.parent_edge[ancestor], self.low[ancestor])
+end
+
+
+
+function NetworkSimplex:balanceRanksTopBottom()
+
+ -- available ranks
+ local ranks = self.ranking:getRanks()
+
+ -- node to in/out weight mappings
+ local in_weight = {}
+ local out_weight = {}
+
+ -- node to lowest/highest possible rank mapping
+ local min_rank = {}
+ local max_rank = {}
+
+ -- compute the in and out weights of each node
+ for _,node in ipairs(self.graph.nodes) do
+ -- assume there are no restrictions on how to rank the node
+ min_rank[node], max_rank[node] = ranks[1], ranks[#ranks]
+
+ for _,edge in ipairs(node:getIncomingEdges()) do
+ -- accumulate the weights of all incoming edges
+ in_weight[node] = (in_weight[node] or 0) + edge.weight
+
+ -- update the minimum allowed rank (which is the maximum of
+ -- the ranks of all parent neighbours plus the minimum level
+ -- separation caused by the connecting edges)
+ local neighbour = edge:getNeighbour(node)
+ local neighbour_rank = self.ranking:getRank(neighbour)
+ min_rank[node] = math.max(min_rank[node], neighbour_rank + edge.minimum_levels)
+ end
+
+ for _,edge in ipairs(node:getOutgoingEdges()) do
+ -- accumulate the weights of all outgoing edges
+ out_weight[node] = (out_weight[node] or 0) + edge.weight
+
+ -- update the maximum allowed rank (which is the minimum of
+ -- the ranks of all child neighbours minus the minimum level
+ -- sparation caused by the connecting edges)
+ local neighbour = edge:getNeighbour(node)
+ local neighbour_rank = self.ranking:getRank(neighbour)
+ max_rank[node] = math.min(max_rank[node], neighbour_rank - edge.minimum_levels)
+ end
+
+ -- check whether the in- and outweight is the same
+ if in_weight[node] == out_weight[node] then
+
+ -- check which of the allowed ranks has the least number of nodes
+ local min_nodes_rank = min_rank[node]
+ for n = min_rank[node] + 1, max_rank[node] do
+ if #self.ranking:getNodes(n) < #self.ranking:getNodes(min_nodes_rank) then
+ min_nodes_rank = n
+ end
+ end
+
+ -- only move the node to the rank with the least number of nodes
+ -- if it differs from the current rank of the node
+ if min_nodes_rank ~= self.ranking:getRank(node) then
+ self.ranking:setRank(node, min_nodes_rank)
+ end
+
+ end
+ end
+end
+
+
+
+function NetworkSimplex:balanceRanksLeftRight()
+ for _,edge in ipairs(self.tree.edges) do
+ if self.cut_value[edge] == 0 then
+ local other_edge = self:findReplacementEdge(edge)
+ if other_edge then
+ local delta = self:edgeSlack(other_edge)
+ if delta > 1 then
+ if self.lim[edge:getTail()] < self.lim[edge:getHead()] then
+ self:rerank(edge:getTail(), delta / 2)
+ else
+ self:rerank(edge:getHead(), -delta / 2)
+ end
+ end
+ end
+ end
+ end
+end
+
+
+
+function NetworkSimplex:computeInitialRanking()
+
+ -- queue for nodes to rank next
+ local queue = {}
+
+ -- convenience functions for managing the queue
+ local function enqueue(node) table.insert(queue, node) end
+ local function dequeue() return table.remove(queue, 1) end
+
+ -- reset the two-dimensional mapping from ranks to lists
+ -- of corresponding nodes
+ self.ranking:reset()
+
+ -- mapping of nodes to the number of unscanned incoming edges
+ local remaining_edges = {}
+
+ -- add all sinks to the queue
+ for _,node in ipairs(self.graph.nodes) do
+ local edges = node:getIncomingEdges()
+
+ remaining_edges[node] = #edges
+
+ if #edges == 0 then
+ enqueue(node)
+ end
+ end
+
+ -- run long as there are nodes to be ranked
+ while #queue > 0 do
+
+ -- fetch the next unranked node from the queue
+ local node = dequeue()
+
+ -- get a list of its incoming edges
+ local in_edges = node:getIncomingEdges()
+
+ -- determine the minimum possible rank for the node
+ local rank = 1
+ for _,edge in ipairs(in_edges) do
+ local neighbour = edge:getNeighbour(node)
+ if self.ranking:getRank(neighbour) then
+ -- the minimum possible rank is the maximum of all neighbour ranks plus
+ -- the corresponding edge lengths
+ rank = math.max(rank, self.ranking:getRank(neighbour) + edge.minimum_levels)
+ end
+ end
+
+ -- rank the node
+ self.ranking:setRank(node, rank)
+
+ -- get a list of the node's outgoing edges
+ local out_edges = node:getOutgoingEdges()
+
+ -- queue neighbours of nodes for which all incoming edges have been scanned
+ for _,edge in ipairs(out_edges) do
+ local head = edge:getHead()
+ remaining_edges[head] = remaining_edges[head] - 1
+ if remaining_edges[head] <= 0 then
+ enqueue(head)
+ end
+ end
+ end
+end
+
+
+
+function NetworkSimplex:findTightTree()
+
+ -- TODO: Jannis: Remove the recursion below:
+
+ local marked = {}
+
+ local function build_tight_tree(node)
+
+ local out_edges = node:getOutgoingEdges()
+ local in_edges = node:getIncomingEdges()
+
+ local edges = lib.copy(out_edges)
+ for _,v in ipairs(in_edges) do
+ edges[#edges + 1] = v
+ end
+
+ for _,edge in ipairs(edges) do
+ local neighbour = edge:getNeighbour(node)
+ if (not marked[neighbour]) and math.abs(self:edgeSlack(edge)) < 0.00001 then
+ self:addEdgeToTree(edge)
+
+ for _,node in ipairs(edge.nodes) do
+ marked[node] = true
+ end
+
+ if #self.tree.edges == #self.graph.nodes-1 then
+ return true
+ end
+
+ if build_tight_tree(neighbour) then
+ return true
+ end
+ end
+ end
+
+ return false
+ end
+
+ for _,node in ipairs(self.graph.nodes) do
+ self.tree = Graph.new()
+ self.tree_node = {}
+ self.orig_node = {}
+ self.tree_edge = {}
+ self.orig_edge = {}
+
+ build_tight_tree(node)
+
+ if #self.tree.edges > 0 then
+ break
+ end
+ end
+
+ return #self.tree.nodes
+end
+
+
+
+function NetworkSimplex:edgeSlack(edge)
+ -- make sure this is never called with a tree edge
+ assert(not self.orig_edge[edge])
+
+ local head_rank = self.ranking:getRank(edge:getHead())
+ local tail_rank = self.ranking:getRank(edge:getTail())
+ local length = head_rank - tail_rank
+ return length - edge.minimum_levels
+end
+
+
+
+function NetworkSimplex:isIncidentToTree(edge)
+ -- make sure this is never called with a tree edge
+ assert(not self.orig_edge[edge])
+
+ local head = edge:getHead()
+ local tail = edge:getTail()
+
+ if self.tree_node[head] and not self.tree_node[tail] then
+ return true
+ elseif self.tree_node[tail] and not self.tree_node[head] then
+ return true
+ else
+ return false
+ end
+end
+
+
+
+function NetworkSimplex:initializeCutValues()
+ self:calculateDFSRange(self.tree.nodes[1], nil, 1)
+
+ local function init(search)
+ search:push({ node = self.tree.nodes[1], parent_edge = nil })
+ end
+
+ local function visit(search, data)
+ search:setVisited(data, true)
+
+ local into = data.node:getIncomingEdges()
+ local out = data.node:getOutgoingEdges()
+
+ for i=#into,1,-1 do
+ local edge = into[i]
+ if edge ~= data.parent_edge then
+ search:push({ node = edge:getTail(), parent_edge = edge })
+ end
+ end
+
+ for i=#out,1,-1 do
+ local edge = out[i]
+ if edge ~= data.parent_edge then
+ search:push({ node = edge:getHead(), parent_edge = edge })
+ end
+ end
+ end
+
+ local function complete(search, data)
+ if data.parent_edge then
+ self:updateCutValue(data.parent_edge)
+ end
+ end
+
+ DepthFirstSearch.new(init, visit, complete):run()
+end
+
+
+
+--- DFS algorithm that calculates post-order traversal indices and parent edges.
+--
+-- This algorithm performs a depth-first search in a directed or undirected
+-- graph. For each node it calculates the node's post-order traversal index, the
+-- minimum post-order traversal index of its descendants as well as the edge by
+-- which the node was reached in the depth-first traversal.
+--
+function NetworkSimplex:calculateDFSRange(root, edge_from_parent, lowest)
+
+ -- global traversal index counter
+ local lim = lowest
+
+ -- start the traversal at the root node
+ local function init(search)
+ search:push({ node = root, parent_edge = edge_from_parent, low = lowest })
+ end
+
+ -- visit nodes in depth-first order
+ local function visit(search, data)
+ -- mark node as visited so we only visit it once
+ search:setVisited(data, true)
+
+ -- remember the parent edge
+ self.parent_edge[data.node] = data.parent_edge
+
+ -- remember the minimum traversal index for this branch of the search tree
+ self.low[data.node] = lim
+
+ -- next we push all outgoing and incoming edges in reverse order
+ -- to simulate recursive calls
+
+ local into = data.node:getIncomingEdges()
+ local out = data.node:getOutgoingEdges()
+
+ for i=#into,1,-1 do
+ local edge = into[i]
+ if edge ~= data.parent_edge then
+ search:push({ node = edge:getTail(), parent_edge = edge })
+ end
+ end
+
+ for i=#out,1,-1 do
+ local edge = out[i]
+ if edge ~= data.parent_edge then
+ search:push({ node = edge:getHead(), parent_edge = edge })
+ end
+ end
+ end
+
+ -- when completing a node, store its own traversal index
+ local function complete(search, data)
+ self.lim[data.node] = lim
+ lim = lim + 1
+ end
+
+ -- kick off the depth-first search
+ DepthFirstSearch.new(init, visit, complete):run()
+
+ local lim_lookup = {}
+ local min_lim = math.huge
+ local max_lim = -math.huge
+ for _,node in ipairs(self.tree.nodes) do
+ assert(self.lim[node])
+ assert(self.low[node])
+ assert(not lim_lookup[self.lim[node]])
+ lim_lookup[self.lim[node]] = true
+ min_lim = math.min(min_lim, self.lim[node])
+ max_lim = math.max(max_lim, self.lim[node])
+ end
+ for n = min_lim, max_lim do
+ assert(lim_lookup[n] == true)
+ end
+end
+
+
+
+function NetworkSimplex:updateCutValue(tree_edge)
+
+ local v = nil
+ if self.parent_edge[tree_edge:getTail()] == tree_edge then
+ v = tree_edge:getTail()
+ dir = 1
+ else
+ v = tree_edge:getHead()
+ dir = -1
+ end
+
+ local sum = 0
+
+ local out_edges = self.orig_node[v]:getOutgoingEdges()
+ local in_edges = self.orig_node[v]:getIncomingEdges()
+ local edges = lib.copy(out_edges)
+ for _,v in ipairs(in_edges) do
+ edges[#edges + 1] = v
+ end
+
+ for _,edge in ipairs(edges) do
+ local other = edge:getNeighbour(self.orig_node[v])
+
+ local f = 0
+ local rv = 0
+
+ if not self:inTailComponentOf(self.tree_node[other], v) then
+ f = 1
+ rv = edge.weight
+ else
+ f = 0
+
+ if self.tree_edge[edge] then
+ rv = self.cut_value[self.tree_edge[edge]]
+ else
+ rv = 0
+ end
+
+ rv = rv - edge.weight
+ end
+
+ local d = 0
+
+ if dir > 0 then
+ if edge:isHead(self.orig_node[v]) then
+ d = 1
+ else
+ d = -1
+ end
+ else
+ if edge:isTail(self.orig_node[v]) then
+ d = 1
+ else
+ d = -1
+ end
+ end
+
+ if f > 0 then
+ d = -d
+ end
+
+ if d < 0 then
+ rv = -rv
+ end
+
+ sum = sum + rv
+ end
+
+ self.cut_value[tree_edge] = sum
+end
+
+
+
+function NetworkSimplex:inTailComponentOf(node, v)
+ return (self.low[v] <= self.lim[node]) and (self.lim[node] <= self.lim[v])
+end
+
+
+
+function NetworkSimplex:nextSearchIndex()
+ local index = 1
+
+ -- avoid tree edge index out of bounds by resetting the search index
+ -- as soon as it leaves the range of edge indices in the tree
+ if self.search_index > #self.tree.edges then
+ self.search_index = 1
+ index = 1
+ else
+ index = self.search_index
+ self.search_index = self.search_index + 1
+ end
+
+ return index
+end
+
+
+
+function NetworkSimplex:rerank(node, delta)
+ local function init(search)
+ search:push({ node = node, delta = delta })
+ end
+
+ local function visit(search, data)
+ search:setVisited(data, true)
+
+ local orig_node = self.orig_node[data.node]
+ self.ranking:setRank(orig_node, self.ranking:getRank(orig_node) - data.delta)
+
+ local into = data.node:getIncomingEdges()
+ local out = data.node:getOutgoingEdges()
+
+ for i=#into,1,-1 do
+ local edge = into[i]
+ if edge ~= self.parent_edge[data.node] then
+ search:push({ node = edge:getTail(), delta = data.delta })
+ end
+ end
+
+ for i=#out,1,-1 do
+ local edge = out[i]
+ if edge ~= self.parent_edge[data.node] then
+ search:push({ node = edge:getHead(), delta = data.delta })
+ end
+ end
+ end
+
+ DepthFirstSearch.new(init, visit):run()
+end
+
+
+
+function NetworkSimplex:rerankBeforeReplacingEdge(leave_edge, enter_edge)
+ local delta = self:edgeSlack(enter_edge)
+
+ if delta > 0 then
+ local tail = leave_edge:getTail()
+
+ if #tail.edges == 1 then
+ self:rerank(tail, delta)
+ else
+ local head = leave_edge:getHead()
+
+ if #head.edges == 1 then
+ self:rerank(head, -delta)
+ else
+ if self.lim[tail] < self.lim[head] then
+ self:rerank(tail, delta)
+ else
+ self:rerank(head, -delta)
+ end
+ end
+ end
+ end
+end
+
+
+
+function NetworkSimplex:updateCutValuesUpToCommonAncestor(v, w, cutval, dir)
+
+ while not self:inTailComponentOf(w, v) do
+ local edge = self.parent_edge[v]
+
+ if edge:isTail(v) then
+ d = dir
+ else
+ d = not dir
+ end
+
+ if d then
+ self.cut_value[edge] = self.cut_value[edge] + cutval
+ else
+ self.cut_value[edge] = self.cut_value[edge] - cutval
+ end
+
+ if self.lim[edge:getTail()] > self.lim[edge:getHead()] then
+ v = edge:getTail()
+ else
+ v = edge:getHead()
+ end
+ end
+
+ return v
+end
+
+
+
+function NetworkSimplex:addEdgeToTree(edge)
+ assert(not self.tree_edge[edge])
+
+ -- create the new tree edge
+ local tree_edge = edge:copy()
+ self.orig_edge[tree_edge] = edge
+ self.tree_edge[edge] = tree_edge
+
+ -- create tree nodes if necessary
+ for _,node in ipairs(edge.nodes) do
+ local tree_node
+
+ if self.tree_node[node] then
+ tree_node = self.tree_node[node]
+ else
+ tree_node = node:copy()
+ self.orig_node[tree_node] = node
+ self.tree_node[node] = tree_node
+ end
+
+ self.tree:addNode(tree_node)
+ tree_edge:addNode(tree_node)
+ end
+
+ self.tree:addEdge(tree_edge)
+
+ return tree_edge
+end
+
+
+
+function NetworkSimplex:removeEdgeFromTree(edge)
+ self.tree:deleteEdge(edge)
+ self.tree_edge[self.orig_edge[edge]] = nil
+ self.orig_edge[edge] = nil
+end
+
+
+
+
+-- Done
+
+return NetworkSimplex \ No newline at end of file