Programming Praxis – The Seven Bridges of Königsberg

In today’s Programming Praxis exercise, our goal is to write a function that determines whether a given graph is a eulerian circuit, path, or neither and if so, to produce that path. Let’s get started, shall we?

import Data.List
import qualified Data.Map as M

A graph that has 1 or more than 2 vertices with an odd amount of neighbours will never be a eulerian path. To determine whether a path is a circuit we simply check if it loops around. If possible, we start at a vertex with an odd amount of neighbours, since this is required for paths and optional for circuits.

check :: Ord a => M.Map a [a] -> Maybe (String, [a])
check graph | notElem (length . filter (odd . length) $ M.elems graph) [0,2] = Nothing
            | head path == last path = Just ("Circuit", path)
            | otherwise              = Just ("Path", path)
    where path  = walk [] graph start
          start = maybe (last $ M.keys graph) id $

To actually walk the graph we use the algorithm provided in the problem description.

walk :: Ord a => [(a, [a])] -> M.Map a [a] -> a -> [a]
walk stack g v = case (g M.! v, stack) of
    (n:_,_)        -> walk ((v, g' M.! v):stack) g' n where
                      g' = M.adjust (delete n) v $ M.adjust (delete v) n g
    ([] ,(s,_):ss) -> v : walk ss g s
    ([] ,[])       -> [v]

Some tests to see if everything is working properly:

main :: IO ()
main = do
    let square   = M.fromList [('A',"BC"), ('B',"AD"), ('C',"AD"), ('D',"BC")]
    let envelope = M.fromList [('A',"BCD"), ('B',"ACD"), ('C',"ABDE"), ('D',"ABCE"), ('E',"CD")]
    let seven    = M.fromList [('A',"BBC"), ('B',"AACDD"), ('C',"ABD"), ('D',"BBC")]
    let five     = M.fromList [('A',"BC"), ('B',"ACD"), ('C',"ABD"), ('D',"BC")]
    let star     = M.fromList [('A',"B"), ('B',"ACD"), ('C',"B"), ('D',"B")]
    print $ check square
    print $ check envelope
    print $ check star == Nothing
    print $ check seven == Nothing
    print $ check five
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