# Code for Manipulating Graphs in Haskell

Here’s a bunch of code I wrote, most of it about a year ago, for doing things with the graphs from the Data.Graph module in Haskell. The choice of functions, from among the many generally useful functions acting on graphs, comes from a specific project. The actually functionality is pretty generic, though. So I’m just throwing this out there. If someone else wanted to package it and throw it on Hackage (likely with a different module name), they would be welcome to do so.

{- Module containing utility functions for operating on Data.Graph. -} module GraphUtil where import Data.Graph import Data.List import Data.Tree import Data.Array import Data.Maybe import Data.Function (on) import qualified Data.Map as M import Data.Map (Map) {- Computes all possible (directed) graphs over n vertices. Allows loops and antiparallel edges, but does not allow parallel edges. -} allGraphs n = map (buildG (1,n)) (subsequences [ (a,b) | a <- [1..n], b <- [1..n] ]) {- Finds all cycles in a graph. A cycle is given as a finite list of the vertices in order of occurrence, where each vertex only appears once. The point where the cycle is broken to form a linear list is arbitrary, so each cycle is merely one representative of the equivalence class generated by the relation identifying (v:vs) with vs++[v]. In particular, this function chooses the representative with the lowest numbered starting vertex. -} cycles g = concatMap cycles' (vertices g) where cycles' v = build [] v v build p s v = let p' = p ++ [v] local = [ p' | x <- (g!v), x == s ] good w = w > s && not (w `elem` p') ws = filter good (g ! v) extensions = concatMap (build p' s) ws in local ++ extensions {- Computes a table of the number of loops at each vertex. -} loopdegree g = array (bounds g) [ (v, count v (g!v)) | v <- vertices g] where count v [] = 0 count v (x:xs) | x == v = 1 + count v xs | otherwise = count v xs {- Tests two graphs to see if they are isomorphic. This is a worst-case exponential algorithm, but should perform acceptably in practice on small graphs. It uses backtracking, associating vertices one by one until it either finds a complete isomorphism, or reaches a vertex that can't be matched with a remaining vertex in the other graph. -} isIsomorphic g1 g2 = let v1 = vertices g1 v2 = vertices g2 in (length v1 == length v2) && test [] v1 v2 where {- Takes the first vertex v of g1, looks for vertices w of g2 that work "so far", and tries to construct an isomorphism that maps v to w. -} test m [] [] = True test m (v:vs) ws = let cs = filter (similar m v) ws in any (\w -> test ((v,w):m) vs (delete w ws)) cs {- Tests whether a given mapping v -> w for v in g1, w in g2 works "so far". In order for the mapping to work, the vertices must agree on their in-degree, out-degree, and number of loops; and already mapped adjacent vertices via in- and out-edges must correspond. -} similar m v w = (in1 ! v) == (in2 ! w) && (out1 ! v) == (out2 ! w) && (loop1 ! v) == (loop2 ! w) && match ((v,w):m) (g1 !v) (g2 !w) && match ((v,w):m) (g1'!v) (g2'!w) {- Tests whether a list of vertices agrees in those edges that are already mapped to each other via the association list m. -} match m vs ws = let kvs = mapMaybe (\v -> find ((== v) . fst) m) vs kws = mapMaybe (\w -> find ((== w) . snd) m) ws in sort kvs == sort kws {- Some global information about the graphs that can be calculated only once to save time. This includes the degrees and number of loops at each vertex, and the transpose of the graphs (used to find in-edges). -} g1' = transposeG g1 g2' = transposeG g2 in1 = outdegree g1' out1 = outdegree g1 in2 = outdegree g2' out2 = outdegree g2 loop1 = loopdegree g1 loop2 = loopdegree g2 {- Returns the degree sequence of a given graph. The degree sequence is a sorted sequence of tuples representing the in-degree, out-degree, and loop-degree, respectively, of each vertex in the graph. The degree sequence has the desirable properties that it is: (a) a graph property (that is, invariant under isomorphism) (b) relatively cheap to compute, and (c) classifies non-isomorphic graphs very effectively into small groups. -} degsequence g = sort (zip3 (elems (indegree g)) (elems (outdegree g)) (elems (loopdegree g))) {- Given a list of graphs, removes the duplicates up to isomorphism. The implementation takes advantage of the fact that the degree sequence is the same for any pair of isomorphic graphs. Therefore, the process maintains a map from degree sequences found so far to their respective graphs. This removes the need to compare most sets of graphs in a normal list. If the list contains only graphs with the same degree sequence, then this function will be very slow, as it will perform O(n^2) isomorphism tests, each of which are worst-case exponential. -} isonub :: [Graph] -> [Graph] isonub gs = go M.empty gs where go hs [] = [] go hs (g:gs) = let dseq = degsequence g poss = M.findWithDefault [] dseq hs in if any (isIsomorphic g) poss then go hs gs else g : go (M.insert dseq (g:poss) hs) gs {- Finds the shortest paths from the given vertex to any other vertex of the graph. The paths returned are lists of vertices, so if there are parallel edges in the graph, each result may actually correspond to multiple paths. The algorithm is essentially breadth first search, but done in bulk for each increase in depth, ensuring that all shortest paths are found for each vertex. -} shortestPaths g v = go (M.singleton v [[v]]) [v] where go ans [] = M.map (map reverse) ans go ans ws = let new = map (step ans) ws next = foldl (M.unionWith (++)) ans new in go next (M.keys next \\ M.keys ans) step ans w = let new = filter (not . (`M.member` ans)) (g!w) soFar = ans M.! w paths = map (\x -> map (x:) soFar) new in M.fromList (zip new paths) {- Computes all possible directed subforests of a given graph. A subforest is a subgraph with the property that there is at most one edge entering any vertex. The approach is to first choose the set of vertices, which may be any subset of the vertices of the given graph, and then choose the edge (if any) entering each vertex. Parallel edges are ignored, since they would lead to isomorphic subforests. -} subforests :: Graph -> [Graph] subforests g = filter ((== []) . cycles) $ concatMap subsAt $ subsequences (vertices g) where g' = transposeG g subsAt vs = let vmap = zip vs [1..] tr v = fromJust (lookup v vmap) tre (v,w) = (tr v, tr w) inAt v = filter (`elem` vs) $ nub $ (g' ! v) in [ buildG (1, length vs) fixedEdges | edges <- [ [] : [ [(w,v)] | w <- inAt v ] | v <- vs ], let fixedEdges = map tre (concat edges) ]

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Would you know how your isIsomorphic and isonub functions compare to Jean-Philippe Bernardy’s HGAL package (available via Hackage)? And how do you define “small” graphs? (I’m needing to remove isomorphic duplicates from potentially millions of graphs of various orders…)

I have little doubt that the HGAL implementation of isIsomorphic is considerably faster in most cases, though you could give it a shot. I don’t see anything like isonub in the hgal package, though if I understand correctly, you could map canonicGraph from the Data.Graph.Automorphism package over the list, and then just sort the list and remove consecutive duplicates using the standard Ord and Eq instances.

I was unable to get hgal installed quickly when I wrote this (and again now, I get errors from cabal install, related to building an old version of the array package)… and since my application only had to deal with tens of thousands of graphs of at most about 5 vertices, this worked just as well.

I’ve been using HGAL basically by doing “map fst . nubBy ((==) `on` snd) . map (\f -> (f,canonicGraph . toGraph f))” (where toGraph is function on my data type I’m using). I’m probably going to try re-implementing HGAL on my own directly from Brendan McKay’s paper to see if I can improve performance somehow, as my app spends more time doing canonicGraph than anything else :s

Ivan, if you have a large enough number of graphs, using nubBy is unwise. You’re actually better off sorting (Graph is an instance of Ord), and then removing consecutive duplicates. (map head . groupBy ((==) `on` snd . sort) would replace your nubBy in that case.

Actually, I’m using a Set to store the current canonical graphs found, after much urging by the people on #haskell.

Even so, however, I beg to differ that nubBy is unwise: about 2/3 of my time is spent running canonicGraph (which I do _before_ removing duplicates). Considering that for one test I ran, I had over 2.5 million values out of which only around 140 were unique, nubBy isn’t that much of a problem.

The big advantage of nubBy as opposed to using “map head . group . sort” or “Set.toList . Set.fromList” is that nubBy is _lazy_, so that my program can start spitting out values as soon as it finds them as opposed to having to sort the entire list (which gets _very_ large) before it can start printing values.