Another week, another problem.
Problem 85: Graph isomorphism
Uhm, OK, the problems have begun delving too much into weird-for-tutorials territory, just like Project Euler goes too far into math:
import Data.List (permutations) data Graph a = Graph [a] [(a, a)] deriving (Show, Eq) -- We're going to need our adjacency-list representation: data Adjacency a = Adj [(a, [a])] deriving (Show, Eq) -- Two graphs are isomorphic if they have the same canonical representation: iso :: (Ord a, Enum a, Show a) => Graph a -> Graph a -> Bool iso g@(Graph xs ys) h@(Graph xs' ys') = length xs == length xs' && length ys == length ys' && canon (graph_to_adj g) == canon (graph_to_adj h)
Now to the meat of the solution. Our relabeling strategy in order to find a canonical form
will work through all possible permutations of
[1..(length g)] and will therefore be
very expensive. I’ve come across some algorithm(s) which exchange speed for probabilistic
guarantees, but I felt they were overkill for this exercise.
I also noticed most of said algorithms were behind paywalls, e.g. Springer.
canon :: (Ord a, Enum a, Show a) => Adjacency a -> String canon (Adj g) = minimum $ map f $ permutations [1..(length g)] where -- Graph vertices: vs = map fst g -- Find, via brute force on all possible orderings (permutations) of vs, -- a mapping of vs to [1..(length g)] which is minimal. -- For example, map [1, 5, 6, 7] to [1, 2, 3, 4]. -- Minimal is defined lexicographically, since `f` returns strings: f p = let n = zip vs p in (show [(snd x, sort id $ map (\x -> snd $ head $ snd $ break ((==) x . fst) n) $ snd $ take_edge g x) | x <- sort snd n]) -- Sort elements of N in ascending order of (map f N): sort f n = foldr (\x xs -> let (lt, gt) = break ((<) (f x) . f) xs in lt ++ [x] ++ gt)  n -- Get the first entry from the adjacency list G that starts from the given node X -- (actually, the vertex is the first entry of the pair, hence `(fst x)`): take_edge g x = head $ dropWhile ((/=) (fst x) . fst) g graphG1 = Graph [1, 2, 3, 4, 5, 6, 7, 8] [(1, 5), (1, 6), (1, 7), (2, 5), (2, 6), (2, 8), (3, 5), (3, 7), (3, 8), (4, 6), (4, 7), (4, 8)] graphH1 = Graph [1, 2, 3, 4, 5, 6, 7, 8] [(1, 2), (1, 4), (1, 5), (6, 2), (6, 5), (6, 7), (8, 4), (8, 5), (8, 7), (3, 2), (3, 4), (3, 7)] -- Should be `True`: main = do print $ iso graphG1 graphH1
Problem 86: Node degree and graph coloring.
The problem statement gives us the Welsh-Powell algorithm for graph coloring, which is relatively simple to implement. First let’s take care of the boilerplate and subproblems (a) and (b):
import Data.List (find, sortBy) import Data.Ord (comparing) data Graph a = Graph [a] [(a, a)] deriving (Show, Eq) data Adjacency a = Adj [(a, [a])] deriving (Show, Eq) petersen = Graph ['a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i', 'j'] [('a', 'b'), ('a', 'e'), ('a', 'f'), ('b', 'c'), ('b', 'g'), ('c', 'd'), ('c', 'h'), ('d', 'e'), ('d', 'i'), ('e', 'j'), ('f', 'h'), ('f', 'i'), ('g', 'i'), ('g', 'j'), ('h', 'j')] degree :: (Eq a, Ord a, Show a) => Graph a -> a -> Int degree (Graph _ es) n = length $ filter ((==) n . fst) es sort_degree :: (Eq a, Ord a, Show a) => Graph a -> Adjacency a sort_degree g = Adj $ sortBy (flip $ comparing $ length . snd) l where Adj l = graph_to_adj g
Alright, that was the easy part. Now on to implement WP.
wpcolor will map each vertex to
a color, signified by an integer value:
wpcolor :: (Eq a, Ord a, Show a) => Graph a -> [(a, Int)] wpcolor g = wpcolor' l  1 where -- Step 1: All vertices are sorted according to decreasing degree Adj l = sort_degree g wpcolor'  ys _ = ys wpcolor' xs ys n = let ys' = color xs ys n in wpcolor' [x | x <- xs, notElem (fst x, n) ys'] ys' (n+1) -- Color will take care of steps 3 & 4, by coloring vertices not connected to -- already colored vertices: color  ys n = ys color ((v, e) : xs) ys n = if any (\x -> (x, n) `elem` ys) e then color xs ys n else color xs ((v, n) : ys) n main = do print $ wpcolor petersen
Problem 87: Depth-first traversal revisited
DFS is not a particularly hard problem, but the stack implementation took me some time to get right.
import Data.List data Graph a = Graph [a] [(a, a)] deriving (Show, Eq) dfs :: (Eq a, Show a) => Graph a -> a -> [a] dfs (Graph vs es) n | n `notElem` vs =  | otherwise = dfs' (Graph vs es) [n] dfs' :: (Eq a) => Graph a -> [a] -> [a] dfs' (Graph  _) _ =  dfs' (Graph _ _)  =  dfs' (Graph vs es) (top:stack) | top `notElem` vs = dfs' (Graph remaining es) stack | otherwise = top : dfs' (Graph remaining es) (adjacent ++ stack) where adjacent = [x | (y, x) <- es, y == top] -- Remove the below if the graph is considered directed: ++ [x | (x, y) <- es, y == top] remaining = [x | x <- vs, x /= top] main = do print $ dfs (Graph [1,2,3,4,5] [(1,2),(2,3),(1,4),(3,4),(5,2),(5,4)]) 1
There we go, that is way simpler than a bunch of
if-else-then, but it’s working in the
same way. The quadratic edge lookup is ugly. For linear graph algorithms including DFS,