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 +160/−0
- LICENSE +505/−0
- Setup.hs +2/−0
- graph-matchings.cabal +67/−0
+ 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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+ 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+