1. Advanced Functional Programming Tim Sheard 1 Lecture 17 Advanced Functional Programming Tim Sheard Oregon Graduate Institute of Science & Technology Lecture 17: Erwig style - Functional Graphs DFS – style of depth first search

2. Advanced Functional Programming Tim Sheard 2 Lecture 17 References The style of representing graphs, and most of the examples, come from the work of Martin Erwig http://www.cs.orst.edu/~erwig/ Functional Programming with Graphs 2nd ACM SIGPLAN Int. Conf. on Functional Programming (ICFP'97), 52-65, 1997(ICFP'97) Inductive Graphs and Functional Graph AlgorithmsInductive Graphs and Functional Graph Algorithms, Martin Erwig Journal of Functional Programming,

3. Advanced Functional Programming Tim Sheard 3 Lecture 17 Graphs Nodes = 1,2,3 Edges = (1,2),(2,1),(3,1),(2,3) Node Labels = a,b,c Edge Labels = right,left,up,down 1 a c b 2 3 right left down up

4. Advanced Functional Programming Tim Sheard 4 Lecture 17 Types type Node = Int type LNode a = (Node,a) type UNode = LNode () type Edge = (Node,Node) type LEdge b = (Node,Node,b) type UEdge = LEdge ()

5. Advanced Functional Programming Tim Sheard 5 Lecture 17 More Types data Graph a b -- Abstract Type --unlabled Graph type UGraph = Graph () () type Path = [Node] type LPath a = [LNode a] type UPath = [UNode]

6. Advanced Functional Programming Tim Sheard 6 Lecture 17 Making Graphs ex1 = mkGraph [(1,"a"),(2,"b"),(3,"c")] [(1,2,"right"),(2,1,"left"), (2,3,"down"),(3,1,"up")] Main> ex1 1:"a"->[("right",2)] 2:"b"->[("left",1),("down",3)] 3:"c"->[("up",1)] 1 a c b 2 3 right left down up

7. Advanced Functional Programming Tim Sheard 7 Lecture 17 Looking inside noNodes :: Graph a b -> Int nodes :: Graph a b -> [Node] labNodes :: Graph a b -> [LNode a] -- [(Node,a)] edges :: Graph a b -> [Edge] --[(Node,Node)] labEdges :: Graph a b -> [LEdge b] -– [(Node,Node,b)]

8. Advanced Functional Programming Tim Sheard 8 Lecture 17 Examples Main> noNodes ex1 3 Main> nodes ex1 [1,2,3] Main> labEdges ex1 [(1,2,"right"),(2,1,"left"), (2,3,"down"),(3,1,"up")] 1 a c b 2 3 right left down up

9. Advanced Functional Programming Tim Sheard 9 Lecture 17 Decomposing Graphs type Adj b = [(b,Node)] type Context a b = (Adj b,Node,a,Adj b) type MContext a b = Maybe (Context a b) type Decomp a b = (MContext a b,Graph a b) type GDecomp a b = (Context a b,Graph a b) type UContext = ([Node],Node,[Node]) type UDecomp = (Maybe UContext,UGraph) match :: Node -> Graph a b -> Decomp a b edge label Its predecessors Its succesors The node Its label

10. Advanced Functional Programming Tim Sheard 10 Lecture 17 (Just(ps,n,lab,ss),ex2) = match 2 ex1 Main> ps [("right",1)] --predecessors Main> ss --successors [("left",1),("down",3)] Main> n –- the node 2 Main> lab -- its label "b" Main> ex2 1:"a"->[] 3:"c"->[("up",1)] 1 a c b 2 3 right left down up

11. Advanced Functional Programming Tim Sheard 11 Lecture 17 (Just(ps2,n2,lab2,ss2),ex3) = match 3 ex2 Main> ps2 [] Main> n2 3 Main> lab2 "c" Main> ss2 [("up",1)] Main> ex3 1:"a"->[] 1 a c 3 up

12. Advanced Functional Programming Tim Sheard 12 Lecture 17 Constructing Graphs Inductively empty :: Graph a b embed :: Context a b -> Graph a b -> Graph a b infixr & c & g = embed c g ex4 = (ps2,n2,lab2,ss2) & ex3 Main> ex4 1:"a"->[] 3:"c"->[("up",1)] 1 a c 3 up

13. Advanced Functional Programming Tim Sheard 13 Lecture 17 Operations on graphs -- maps on nodes nmap :: (a->c) -> Graph a b -> Graph c b -- maps on edges emap :: (b->c) -> Graph a b -> Graph a c -- graph reversal grev :: Graph a b -> Graph a b

14. Advanced Functional Programming Tim Sheard 14 Lecture 17 Example ex5 = nmap (ord. head) ex1 Main> ex5 1:97->[("right",2)] 2:98->[("down",3),("left",1)] 3:99->[("up",1)] Main> ex1 1:"a"->[("right",2)] 2:"b"->[("left",1),("down",3)] 3:"c"->[("up",1)]

15. Advanced Functional Programming Tim Sheard 15 Lecture 17 Example 2 ex6 = emap length ex1 Main> ex6 1:"a"->[(5,2)] 2:"b"->[(4,3),(4,1)] 3:"c"->[(2,1)] Main> ex1 1:"a"->[("right",2)] 2:"b"->[("left",1),("down",3)] 3:"c"->[("up",1)]

16. Advanced Functional Programming Tim Sheard 16 Lecture 17 Roll your own mymap f g | isEmpty g = empty | otherwise = let ns = nodes g (Just(ps,n,lab,ss),g2) = match (head ns) g in (ps,n,f lab,ss) & (mymap f g2)

17. Advanced Functional Programming Tim Sheard 17 Lecture 17 Depth First Search data Tree a = Branch a [Tree a] deriving Show df :: Node -> Graph b a -> (Tree b,Graph b a) df root g = let (Just(_,v,lab,ss), g') = match root g (r,g1) = dff ss g' in (Branch lab r,g1) dff :: [(c,Node)] -> Graph b c -> ([Tree b],Graph b c) dff [] g = ([],g) dff ((lab,v):l) g = let (x,g1) = df v g (y,g2) = dff l g1 in (x:y,g2)

18. Advanced Functional Programming Tim Sheard 18 Lecture 17 Example ex7 = fst (df 1 ex1) Main> ex7 Branch "c" [Branch "b" [Branch "a" []]]

19. Advanced Functional Programming Tim Sheard 19 Lecture 17 Launchbury & King “Structuring Depth-First Search Algorithms in Haskell” David King and John Launchbury - POPL ’95 – pp 344--354 Graph algorithms based upon Depth First Search. Asymtotically efficient – uses lazy state

20. Advanced Functional Programming Tim Sheard 20 Lecture 17 Representing Graphs import ST import qualified Array as A type Vertex = Int -- Representing graphs: type Table a = A.Array Vertex a type Graph = Table [Vertex] -- Array Int [Int] Index for each node Edges (out of) that index 1 23 45 7 8 9 10 6 1[2,3] 2[7,4] 3[5] 4[6,9,7] 5[8] 6[9] 7 8[10] 9 10[]

21. Advanced Functional Programming Tim Sheard 21 Lecture 17 Functions on graphs type Vertex = Int type Edge = (Vertex,Vertex) vertices :: Graph -> [Vertex] indices :: Graph -> [Int] edges :: Graph -> [Edge]

22. Advanced Functional Programming Tim Sheard 22 Lecture 17 Building Graphs buildG :: Bounds -> [Edge] -> Graph graph = buildG (1,10) [ (1, 2), (1, 6), (2, 3), (2, 5), (3, 1), (3, 4), (5, 4), (7, 8), (7, 10), (8, 6), (8, 9), (8, 10) ] 1 2 5 6 4 7 8 9 10 3

23. Advanced Functional Programming Tim Sheard 23 Lecture 17 DFS and Forests data Tree a = Node a (Forest a) type Forest a = [Tree a] nodesTree (Node a f) ans = nodesForest f (a:ans) nodesForest [] ans = ans nodesForest (t : f) ans = nodesTree t (nodesForest f ans) Note how any tree can be spanned by a Forest. The Forest is not always unique. 1 2 5 6 4 7 8 9 10 3

24. Advanced Functional Programming Tim Sheard 24 Lecture 17 DFS The DFS algorithm finds a spanning forest for a graph, from a set of roots. dfs :: Graph -> [Vertex] -> Forest Vertex dfs g vs = prune (A.bounds g) (map (generate g) vs) generate :: Graph -> Vertex -> Tree Vertex generate g v = Node v (map (generate g) (g `aat` v))

25. Advanced Functional Programming Tim Sheard 25 Lecture 17 Sets of nodes already visited type Set s = STArray s Vertex Bool mkEmpty :: Bounds -> ST s (Set s) mkEmpty bnds = newSTArray bnds False contains :: Set s -> Vertex -> ST s Bool contains m v = readSTArray m v include :: Set s -> Vertex -> ST s () include m v = writeSTArray m v True

26. Advanced Functional Programming Tim Sheard 26 Lecture 17 Pruning already visited paths prune :: Bounds -> Forest Vertex -> Forest Vertex prune bnds ts = runST (do { m <- mkEmpty bnds; chop m ts }) chop :: Set s -> Forest Vertex -> ST s (Forest Vertex) chop m [] = return [] chop m (Node v ts : us) do { visited <- contains m v ; if visited then chop m us else do { include m v ; as <- chop m ts ; bs <- chop m us ; return(Node v as : bs) }

27. Advanced Functional Programming Tim Sheard 27 Lecture 17 Topological Sort postorder :: Tree a -> [a] postorder (Node a ts) = postorderF ts ++ [a] postorderF :: Forest a -> [a] postorderF ts = concat (map postorder ts) postOrd :: Graph -> [Vertex] postOrd = postorderF. Dff dff :: Graph -> Forest Vertex dff g = dfs g (vertices g)