## root / src / Ganeti / HTools / Graph.hs @ 6d3d13ab

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{-| Algorithms on Graphs. |
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This module contains a few graph algorithms and the transoformations |

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needed for them to be used on nodes. |

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For more information about Graph Coloring see: |

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<http://en.wikipedia.org/wiki/Graph_coloring> |

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<http://en.wikipedia.org/wiki/Greedy_coloring> |

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LF-coloring is described in: |

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Welsh, D. J. A.; Powell, M. B. (1967), "An upper bound for the chromatic number |

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of a graph and its application to timetabling problems", The Computer Journal |

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10 (1): 85-86, doi:10.1093/comjnl/10.1.85 |

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<http://comjnl.oxfordjournals.org/content/10/1/85> |

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DSatur is described in: |

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Brelaz, D. (1979), "New methods to color the vertices of a graph", |

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Communications of the ACM 22 (4): 251-256, doi:10.1145/359094.359101 |

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<http://dx.doi.org/10.1145%2F359094.359101> |

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Also interesting: |

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Klotz, W. (2002). Graph coloring algorithms. Mathematics Report, Technical |

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University Clausthal, 1-9. |

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<http://www.math.tu-clausthal.de/Arbeitsgruppen/Diskrete-Optimierung |

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/publications/2002/gca.pdf> |

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-} |

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{- |

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Copyright (C) 2012, Google Inc. |

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This program is free software; you can redistribute it and/or modify |

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it under the terms of the GNU General Public License as published by |

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the Free Software Foundation; either version 2 of the License, or |

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(at your option) any later version. |

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This program is distributed in the hope that it will be useful, but |

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WITHOUT ANY WARRANTY; without even the implied warranty of |

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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |

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General Public License for more details. |

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You should have received a copy of the GNU General Public License |

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along with this program; if not, write to the Free Software |

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Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA |

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02110-1301, USA. |

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-} |

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module Ganeti.HTools.Graph |

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( -- * Types |

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Color |

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, VertColorMap |

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, ColorVertMap |

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-- * Creation |

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, emptyVertColorMap |

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-- * Coloring |

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, colorInOrder |

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, colorLF |

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, colorDsatur |

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, colorDcolor |

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, isColorable |

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-- * Color map transformations |

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, colorVertMap |

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-- * Vertex characteristics |

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, verticesByDegreeDesc |

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, verticesByDegreeAsc |

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, neighbors |

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, hasLoop |

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, isUndirected |

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) where |

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import Data.Maybe |

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import Data.Ord |

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import Data.List |

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import qualified Data.IntMap as IntMap |

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import qualified Data.IntSet as IntSet |

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import qualified Data.Graph as Graph |

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import qualified Data.Array as Array |

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-- * Type declarations |

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-- | Node colors. |

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type Color = Int |

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-- | Saturation: number of colored neighbors. |

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type Satur = Int |

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-- | Vertex to Color association. |

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type VertColorMap = IntMap.IntMap Color |

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-- | Color to Vertex association. |

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type ColorVertMap = IntMap.IntMap [Int] |

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-- * Vertices characteristics |

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-- | (vertex, degree) tuples on a graph. |

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verticesDegree :: Graph.Graph -> [(Graph.Vertex, Int)] |

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verticesDegree g = Array.assocs $ Graph.outdegree g |

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-- | vertices of a graph, sorted by ascending degree. |

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verticesByDegreeDesc :: Graph.Graph -> [Graph.Vertex] |

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verticesByDegreeDesc g = |

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map fst . sortBy (flip (comparing snd)) $ verticesDegree g |

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-- | vertices of a graph, sorted by descending degree. |

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verticesByDegreeAsc :: Graph.Graph -> [Graph.Vertex] |

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verticesByDegreeAsc g = map fst . sortBy (comparing snd) $ verticesDegree g |

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-- | Get the neighbors of a vertex. |

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neighbors :: Graph.Graph -> Graph.Vertex -> [Graph.Vertex] |

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neighbors g v = g Array.! v |

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-- | Check whether a graph has no loops. |

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-- (vertices connected to themselves) |

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hasLoop :: Graph.Graph -> Bool |

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hasLoop g = any vLoops $ Graph.vertices g |

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where vLoops v = v `elem` neighbors g v |

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-- | Check whether a graph is undirected |

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isUndirected :: Graph.Graph -> Bool |

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isUndirected g = |

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(sort . Graph.edges) g == (sort . Graph.edges . Graph.transposeG) g |

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-- * Coloring |

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-- | Empty color map. |

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emptyVertColorMap :: VertColorMap |

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emptyVertColorMap = IntMap.empty |

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-- | Check whether a graph is colorable. |

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isColorable :: Graph.Graph -> Bool |

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isColorable g = isUndirected g && not (hasLoop g) |

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-- | Get the colors of a list of vertices. |

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-- Any uncolored vertices are ignored. |

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verticesColors :: VertColorMap -> [Graph.Vertex] -> [Color] |

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verticesColors cMap = mapMaybe (`IntMap.lookup` cMap) |

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-- | Get the set of colors of a list of vertices. |

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-- Any uncolored vertices are ignored. |

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verticesColorSet :: VertColorMap -> [Graph.Vertex] -> IntSet.IntSet |

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verticesColorSet cMap = IntSet.fromList . verticesColors cMap |

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-- | Get the colors of the neighbors of a vertex. |

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neighColors :: Graph.Graph -> VertColorMap -> Graph.Vertex -> [Color] |

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neighColors g cMap v = verticesColors cMap $ neighbors g v |

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-- | Color one node. |

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colorNode :: Graph.Graph -> VertColorMap -> Graph.Vertex -> Color |

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-- use of "head" is A-ok as the source is an infinite list |

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colorNode g cMap v = head $ filter notNeighColor [0..] |

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where notNeighColor = (`notElem` neighColors g cMap v) |

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-- | Color a node returning the updated color map. |

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colorNodeInMap :: Graph.Graph -> Graph.Vertex -> VertColorMap -> VertColorMap |

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colorNodeInMap g v cMap = IntMap.insert v newcolor cMap |

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where newcolor = colorNode g cMap v |

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-- | Color greedily all nodes in the given order. |

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colorInOrder :: Graph.Graph -> [Graph.Vertex] -> VertColorMap |

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colorInOrder g = foldr (colorNodeInMap g) emptyVertColorMap |

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-- | Color greedily all nodes, larger first. |

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colorLF :: Graph.Graph -> VertColorMap |

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colorLF g = colorInOrder g $ verticesByDegreeAsc g |

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-- | (vertex, (saturation, degree)) for a vertex. |

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vertexSaturation :: Graph.Graph |

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-> VertColorMap |

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-> Graph.Vertex |

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-> (Graph.Vertex, (Satur, Int)) |

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vertexSaturation g cMap v = |

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(v, (IntSet.size (verticesColorSet cMap neigh), length neigh)) |

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where neigh = neighbors g v |

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-- | (vertex, (colordegree, degree)) for a vertex. |

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vertexColorDegree :: Graph.Graph |

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-> VertColorMap |

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-> Graph.Vertex |

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-> (Graph.Vertex, (Int, Int)) |

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vertexColorDegree g cMap v = |

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(v, (length (verticesColors cMap neigh), length neigh)) |

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where neigh = neighbors g v |

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-- | Color all nodes in a dynamic order. |

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-- We have a list of vertices still uncolored, and at each round we |

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-- choose&delete one vertex among the remaining ones. A helper function |

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-- is used to induce an order so that the next vertex can be chosen. |

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colorDynamicOrder :: Ord a |

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=> (Graph.Graph |

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-> VertColorMap |

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-> Graph.Vertex |

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-> (Graph.Vertex, a)) -- ^ Helper to induce the choice |

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-> Graph.Graph -- ^ Target graph |

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-> VertColorMap -- ^ Accumulating vertex color map |

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-> [Graph.Vertex] -- ^ List of remaining vertices |

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-> VertColorMap -- ^ Output vertex color map |

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colorDynamicOrder _ _ cMap [] = cMap |

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colorDynamicOrder ordind g cMap l = colorDynamicOrder ordind g newmap newlist |

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where newmap = colorNodeInMap g choosen cMap |

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choosen = fst . maximumBy (comparing snd) $ ordlist |

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ordlist = map (ordind g cMap) l |

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newlist = delete choosen l |

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-- | Color greedily all nodes, highest number of colored neighbors, then |

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-- highest degree. This is slower than "colorLF" as we must dynamically |

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-- recalculate which node to color next among all remaining ones but |

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-- produces better results. |

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colorDcolor :: Graph.Graph -> VertColorMap |

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colorDcolor g = |

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colorDynamicOrder vertexColorDegree g emptyVertColorMap $ Graph.vertices g |

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-- | Color greedily all nodes, highest saturation, then highest degree. |

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-- This is slower than "colorLF" as we must dynamically recalculate |

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-- which node to color next among all remaining ones but produces better |

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-- results. |

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colorDsatur :: Graph.Graph -> VertColorMap |

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colorDsatur g = |

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colorDynamicOrder vertexSaturation g emptyVertColorMap $ Graph.vertices g |

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-- | ColorVertMap from VertColorMap. |

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colorVertMap :: VertColorMap -> ColorVertMap |

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colorVertMap = IntMap.foldWithKey |

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(flip (IntMap.insertWith ((:) . head)) . replicate 1) |

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IntMap.empty |