packages feed

graph-matchings (empty) → 0.1.0.0

raw patch · 4 files changed

+734/−0 lines, 4 filesdep +basedep +containersdep +fglsetup-changed

Dependencies added: base, containers, fgl

Files

+ Data/Graph/Inductive/Query/Matchings.hs view
@@ -0,0 +1,160 @@+{-|+Module      : Matchings+Description : Provides some functions on matchings in graphs+Licence     : LGPL-2.1+Maintainer  : Manuel Eberl <last name + m _at_ in.tum.de>+Stability   : experimental++This module provides some algorithms related to matchings in graphs, most notably an implementation of +Edmond's blossom algorithm, which computes maximum matchings for graphs. ++Definitions:++  * A /matching/ is a subset of the edges of a graph such that no two edges in it are incident to the same node. +  +  * A /maximal matching/ is a matching that cannot be made any larger, i.e. no additional+edge can be added to it without violating the property of node-disjoint edges.++  * A /maximum matching/ is a matching such that no other matching contains more edges.++In this list, the given notions are strictly increasing in strength. In particular, note that every maximum matching +is also maximal, but not every maximal matching is a maximum one.++Our implementation of Edmond's blossom algorithm is an adaptation of the Java implementation in JGraphT: +<https://github.com/jgrapht/jgrapht/blob/master/jgrapht-core/src/main/java/org/jgrapht/alg/EdmondsBlossomShrinking.java>+-}++module Data.Graph.Inductive.Query.Matchings+    (isMatching, isMaximalMatching, isMaximumMatching, maximalMatchings, maximumMatching) where++import Control.Monad+import Data.Graph.Inductive.Graph+import Data.Graph.Inductive.Basic+import Data.Maybe (fromJust)+import Data.Set (Set)+import qualified Data.Set as S+import Data.Map (Map)+import qualified Data.Map as M+import Data.List (nub, (\\))++type Matching = Map Node Node++-- | Inserts all the elements of a list into a set.+insertList :: Ord a => [a] -> Set a -> Set a+insertList xs s = S.union s (S.fromList xs)++-- | Updates a map such that all keys from the list point to the given value.+updateList :: Ord k => [k] -> v -> Map k v -> Map k v+updateList ks v m = M.union (M.fromList [(k,v) | k <- ks]) m++-- | A lookup in the List monad instead of Maybe.+lookup' :: Ord k => k -> Map k v -> [v]+lookup' k m = maybe [] (\x -> [x]) (M.lookup k m)++-- | Determines whether a given set of edges is a matching.+isMatching :: Graph gr => gr a b -> [Edge] -> Bool+isMatching g m = null (m \\ edges g) && distinct (concatMap (\(a,b) -> [a,b]) m)+  where distinct xs = nub xs == xs++-- | Determines whether a given set of edges is a maximal matching, i.e. a matching that cannot be +--   extended by adding another edge.+isMaximalMatching :: Graph gr => gr a b -> [Edge] -> Bool+isMaximalMatching g m = isMatching g m && and [a `elem` ns || b `elem` ns | (a,b) <- edges g]+  where ns = nub $ concatMap (\(a,b) -> [a,b]) m++-- | Computes all maximal matchings in a graph.+maximalMatchings :: Graph gr => gr a b -> [[Edge]]+maximalMatchings g = f [([], edges g)] []+  where  -- First parameter:  list of matchings with the set of edges that could still be added to it+         -- Second parameter: accumulator of matchings that have already been processed in this pass, but can still be +         --                   enlarged in the next pass.+         -- When we have a matching that cannot be enlarged with an additional edge, we return it; otherwise, +         -- we put in all the ways in which it can be enlarged for the next pass.+         f :: [([Edge], [Edge])] -> [([Edge], [Edge])] -> [[Edge]]+         f [] []           = []+         f [] ms'          = f ms' []+         f ((m,[]):ms) ms' = if isMaximalMatching g m then m : f ms ms' else f ms ms'+         f ((m,es):ms) ms' = f ms (m' ++ ms')+           where m' = do ((a,b), es') <- suffixes es+                         return ((a,b):m, [(c,d) | (c,d) <- es', a /= c, b /= c, a /= d, b /= d])+         -- Chooses an element of the list and returns that element and all its successors.+         suffixes :: [c] -> [(c, [c])]+         suffixes []       = []+         suffixes (x:xs)   = (x,xs) : suffixes xs++-- | Determines whether the given set of edges is a maximum matching.+isMaximumMatching :: Graph gr => gr a b -> [Edge] -> Bool+isMaximumMatching g m = isMatching g m && length m == length (maximumMatching g)+++-- | Computes a maximum matching of the given graph using Edmond's blossom algorithm.+maximumMatching :: Graph gr => gr a b -> [Edge]+maximumMatching g = [if (v,w) `S.member` edgeSet then (v,w) else (w,v) +                         | v <- nodes g, w <- lookup' v matching, v <= w]+  where edgeSet = S.fromList (edges g)+        matching = foldl (matchNode g) M.empty (nodes g)++matchNode :: Graph gr => gr a b -> Matching -> Node -> Matching+matchNode g matching root+  | root `M.member` matching = matching+  | otherwise = constructMatching $ +                    matchNode' ([root], S.singleton root, M.empty, M.fromList [(i,i) | i <- nodes g])+  where constructMatching :: (Matching, Maybe Node) -> Matching+        constructMatching (p, v) = +          let constructMatching' matching Nothing  = matching+              constructMatching' matching (Just v) = +                  let w = fromJust (M.lookup v p) +                  in  constructMatching' (M.insert w v (M.insert v w matching)) (M.lookup w matching)+          in  constructMatching' matching v++        matchNode' :: ([Node], Set Node, Map Node Node, Map Node Node) -> (Map Node Node, Maybe Node)+        matchNode' (q, used, p, base) =+          case q of+            []   -> (p, Nothing)+            v:q' -> either id matchNode' (foldM (go v) (q',used,p,base) (neighbors g v))+        +        go :: Node -> ([Node], Set Node, Map Node Node, Map Node Node) -> Node -> +                  Either (Map Node Node, Maybe Node) ([Node], Set Node, Map Node Node, Map Node Node)+        go v (q, used, p, base) w+          | M.lookup v base     == M.lookup w base = Right (q, used, p, base)+          | M.lookup v matching == Just w          = Right (q, used, p, base)+          | w == root || maybe False (`M.member` p) (M.lookup w matching) =+              let curBase       = lca base p v w+                  (blossom, p') = markPath base w curBase v (markPath base v curBase w (S.empty, p))+                  xs            = [x | x <- nodes g, y <- lookup' x base, y `S.member` blossom]+                  xs'           = [x | x <- xs, x `S.notMember` used]+              in  Right (xs' ++ q, insertList xs' used, p', updateList xs curBase base)+          | w `M.notMember` p =+              let p' = M.insert w v p+              in  case M.lookup w matching of+                    Nothing -> Left (p', Just w)+                    Just w' -> Right (w':q, S.insert w' used, p', base)+          | otherwise = Right (q, used, p, base)++        markPath :: Map Node Node -> Node -> Node -> Node -> (Set Node, Map Node Node) -> +                        (Set Node, Map Node Node)+        markPath base v b c (blossom, p)+          | M.lookup v base == Just b = (blossom, p)+          | otherwise =+              let w  = fromJust $ M.lookup v matching+                  bv = fromJust $ M.lookup v base+                  bw = fromJust $ M.lookup w base+                  v' = fromJust $ M.lookup w p+              in  markPath base v' b w (S.insert bw (S.insert bv blossom), M.insert v c p)+        +        lca :: Map Node Node -> Map Node Node -> Node -> Node -> Node+        lca base p a b = g b+          where f seen x =+                  let x' = fromJust $ M.lookup x base+                      seen' = S.insert x' seen+                  in  case M.lookup x' matching of+                        Nothing -> seen'+                        Just y  -> f seen' (fromJust (M.lookup y p))+                seen = f S.empty a+                g x =+                  let x' = fromJust $ M.lookup x base+                  in  if x' `S.member` seen then+                        x'+                      else+                        g (fromJust (M.lookup (fromJust (M.lookup x' matching)) p))+
+ LICENSE view
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See the GNU+    Lesser General Public License for more details.++    You should have received a copy of the GNU Lesser General Public+    License along with this library; if not, write to the Free Software+    Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301+    USA++Also add information on how to contact you by electronic and paper mail.++You should also get your employer (if you work as a programmer) or your+school, if any, to sign a "copyright disclaimer" for the library, if+necessary.  Here is a sample; alter the names:++  Yoyodyne, Inc., hereby disclaims all copyright interest in the+  library `Frob' (a library for tweaking knobs) written by James Random+  Hacker.++  {signature of Ty Coon}, 1 April 1990+  Ty Coon, President of Vice++That's all there is to it!+
+ Setup.hs view
@@ -0,0 +1,2 @@+import Distribution.Simple+main = defaultMain
+ graph-matchings.cabal view
@@ -0,0 +1,67 @@+-- Initial graph-matchings.cabal generated by cabal init.  For further +-- documentation, see http://haskell.org/cabal/users-guide/++-- The name of the package.+name:                graph-matchings++-- The package version.  See the Haskell package versioning policy (PVP) +-- for standards guiding when and how versions should be incremented.+-- http://www.haskell.org/haskellwiki/Package_versioning_policy+-- PVP summary:      +-+------- breaking API changes+--                   | | +----- non-breaking API additions+--                   | | | +--- code changes with no API change+version:             0.1.0.0++-- A short (one-line) description of the package.+synopsis:            An implementation of algorithms for matchings in graphs++-- A longer description of the package.+description:         This package provides algorithms on matchings in graphs; in particular, determining whether a given set of edges is a (maximal/maximum) matching and finding maximal/maximum matchings.++-- The license under which the package is released.+license:             LGPL-2.1++-- The file containing the license text.+license-file:        LICENSE++-- The package author(s).+author:              Manuel Eberl++-- An email address to which users can send suggestions, bug reports, and +-- patches.+maintainer:          last name + m _at_ in.tum.de++-- A copyright notice.+-- copyright:           ++category:            Data Structures, Graphs++build-type:          Simple++-- Extra files to be distributed with the package, such as examples or a +-- README.+-- extra-source-files:  ++-- Constraint on the version of Cabal needed to build this package.+cabal-version:       >=1.10+++library+  -- Modules exported by the library.+  exposed-modules:     Data.Graph.Inductive.Query.Matchings+  +  -- Modules included in this library but not exported.+  -- other-modules:    Test+  +  -- LANGUAGE extensions used by modules in this package.+  other-extensions:    ScopedTypeVariables+  +  -- Other library packages from which modules are imported.+  build-depends:       base >=3 && <5, containers >= 0.5 && < 0.6, fgl >= 5.4 && < 5.6+  +  -- Directories containing source files.+  -- hs-source-dirs:      +  +  -- Base language which the package is written in.+  default-language:    Haskell2010+