# The Monad.Reader/Issue5/Practical Graph Handling

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'''Abstract.''' | '''Abstract.''' | ||

− | Tree-based data structures are easy to deal with in | + | Tree-based data structures are easy to deal with in Haskell. |

However, working with graph-like structures in practice is much less obvious. | However, working with graph-like structures in practice is much less obvious. | ||

In this article I present a solution that has worked for me in many cases. | In this article I present a solution that has worked for me in many cases. | ||

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For example, the fold operation on lists can be typed as follows: | For example, the fold operation on lists can be typed as follows: | ||

− | + | ||

+ | <haskell> | ||

foldr :: (a -> b -> b) -> -- ^ operation to apply | foldr :: (a -> b -> b) -> -- ^ operation to apply | ||

b -> -- ^ initial value | b -> -- ^ initial value | ||

[a] -> -- ^ input list | [a] -> -- ^ input list | ||

b -- ^ result | b -- ^ result | ||

− | + | </haskell> | |

Conversely, "unfold" builds a complex structure out of a building | Conversely, "unfold" builds a complex structure out of a building | ||

function, applying it iteratively. | function, applying it iteratively. | ||

− | + | <haskell> | |

unfoldr :: (b -> Maybe (a, b)) -> -- ^ building function (Nothing => end of list) | unfoldr :: (b -> Maybe (a, b)) -> -- ^ building function (Nothing => end of list) | ||

b -> -- ^ seed value | b -> -- ^ seed value | ||

[a] -- ^ result | [a] -- ^ result | ||

− | + | </haskell> | |

The second argument is the initial value from which the | The second argument is the initial value from which the | ||

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1. Any number of children for a node; | 1. Any number of children for a node; | ||

− | + | 2. "Backwards" arcs (cycles); | |

− | + | 3. Labelled edges. | |

The most relevant point being 2, of course. | The most relevant point being 2, of course. | ||

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From the above, we can deduce that the type of unfoldG will be: | From the above, we can deduce that the type of unfoldG will be: | ||

− | + | <haskell> | |

unfoldG :: (Ord s) => (s -> (n, [(e, s)])) -> s -> (Vertex, LabGraph n e) | unfoldG :: (Ord s) => (s -> (n, [(e, s)])) -> s -> (Vertex, LabGraph n e) | ||

unfoldG f r = (r', res) | unfoldG f r = (r', res) | ||

where ([r'], res) = unfoldGMany f [r] | where ([r'], res) = unfoldGMany f [r] | ||

− | + | </haskell> | |

− | where | + | where <code>s</code> is the seed type, <code>n</code> is the node labels, <code>e</code> the edges labels. |

− | The | + | The <code>Ord s</code> constraint reflects point 2 above. |

It is needed because the unfoldG function must record every | It is needed because the unfoldG function must record every | ||

seed value encountered. | seed value encountered. | ||

− | Whenever a seed is seen a second time, | + | Whenever a seed is seen a second time, <code>unfoldG</code> will recognize |

it and create a "backward arc". | it and create a "backward arc". | ||

− | We use | + | We use <code>Ord</code> instead of <code>Eq</code> because a mere equality test rules out using <code>Data.Map</code>. |

The attentive reader will note that we return an additional | The attentive reader will note that we return an additional | ||

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seed corresponds to. | seed corresponds to. | ||

− | In order to get an intuitive feeling of how | + | In order to get an intuitive feeling of how <code>unfoldG</code> works, |

let's examine a simple example. | let's examine a simple example. | ||

− | + | <haskell> | |

gr1 :: LabGraph Int Char | gr1 :: LabGraph Int Char | ||

(_,gr1) = unfoldG gen (0::Int) | (_,gr1) = unfoldG gen (0::Int) | ||

where gen x = (x,[('a',(x+1) `mod` 10), ('b', (x+2) `mod` 10)]) | where gen x = (x,[('a',(x+1) `mod` 10), ('b', (x+2) `mod` 10)]) | ||

− | + | </haskell> | |

− | + | <code>gr1</code> being defined as above, its structure is: | |

attachment:gr1.png | attachment:gr1.png | ||

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Because we might want to build a graph from a set of seeds | Because we might want to build a graph from a set of seeds | ||

instead of a single one, we will also need the following function: | instead of a single one, we will also need the following function: | ||

− | + | <haskell> | |

unfoldGMany :: (Ord s) => (s -> (n, [(e, s)])) -> [s] -> ([Vertex], LabGraph n e) | unfoldGMany :: (Ord s) => (s -> (n, [(e, s)])) -> [s] -> ([Vertex], LabGraph n e) | ||

unfoldGMany f roots = runST ( unfoldGManyST f roots ) -- detailed later | unfoldGMany f roots = runST ( unfoldGManyST f roots ) -- detailed later | ||

− | + | </haskell> | |

− | + | <code>unfoldG</code>, alone, is already very a practical tool, because it | |

− | lets you reify a function ( | + | lets you reify a function (<code>a -> a</code>) graph. It then can be examined, |

processed, etc. whereas the function can only be evaluated. | processed, etc. whereas the function can only be evaluated. | ||

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On a graph, the catamorphism (fold) type will become: | On a graph, the catamorphism (fold) type will become: | ||

− | + | <haskell> | |

foldG :: (Eq r) => r -> (Vertex -> [(e, r)] -> r) -> Graph e -> Vertex -> r | foldG :: (Eq r) => r -> (Vertex -> [(e, r)] -> r) -> Graph e -> Vertex -> r | ||

foldG i f g v = foldGAll i f g ! v | foldG i f g v = foldGAll i f g ! v | ||

− | + | </haskell> | |

− | As for | + | As for <code>unfoldG</code>, the <code>foldG</code> |

function must include a special mechanism to handle cycles. | function must include a special mechanism to handle cycles. | ||

The idea is to apply the operation iteratively until the result | The idea is to apply the operation iteratively until the result | ||

converges. It's the purpose of the first | converges. It's the purpose of the first | ||

− | parameter is to " | + | parameter is to "bootstrap" the process: |

it will be used as an initial value. | it will be used as an initial value. | ||

− | Thus, | + | Thus, <code>foldG i f g ! v</code> will iteratively |

− | apply | + | apply <code>f</code> on nodes of graph <code>g</code>, |

− | using | + | using <code>i</code> as "bottom" value. It will return |

− | the value computed at vertex | + | the value computed at vertex <code>v</code>. |

− | Of course, this will work only if | + | Of course, this will work only if <code>f</code> is well-behaved: |

it must converge at some point. | it must converge at some point. | ||

I won't dwelve in to the theoretical details | I won't dwelve in to the theoretical details | ||

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formal explanation. | formal explanation. | ||

− | Notice that | + | Notice that <code>foldG</code> can work on a graph without node labels. |

If the parameter function needs to access node labels, it can | If the parameter function needs to access node labels, it can | ||

− | do so without | + | do so without <code>foldG</code> needing to know. |

It's also worth noticing that, in our implementation, the | It's also worth noticing that, in our implementation, the | ||

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hence the need for : | hence the need for : | ||

− | + | <haskell> | |

foldGAll :: (Eq r) => r -> (Vertex -> [(e, r)] -> r) -> Graph e -> Table r | foldGAll :: (Eq r) => r -> (Vertex -> [(e, r)] -> r) -> Graph e -> Table r | ||

− | + | </haskell> | |

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− | + | <haskell> | |

type Vertex = Int | type Vertex = Int | ||

type Table a = Array Vertex a | type Table a = Array Vertex a | ||

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type Bounds = (Vertex, Vertex) | type Bounds = (Vertex, Vertex) | ||

type Edge e = (Vertex, e, Vertex) | type Edge e = (Vertex, e, Vertex) | ||

− | + | </haskell> | |

A graph is a mere adjacency list table, tagged with edge labels. | A graph is a mere adjacency list table, tagged with edge labels. | ||

The above structure lacks labels for nodes. | The above structure lacks labels for nodes. | ||

This is easily fixed by adding a labeling (or coloring) function. | This is easily fixed by adding a labeling (or coloring) function. | ||

− | + | <haskell> | |

type Labeling a = Vertex -> a | type Labeling a = Vertex -> a | ||

data LabGraph n e = LabGraph (Graph e) (Labeling n) | data LabGraph n e = LabGraph (Graph e) (Labeling n) | ||

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labels (LabGraph gr l) = map l (indices gr) | labels (LabGraph gr l) = map l (indices gr) | ||

− | + | </haskell> | |

The above departs slightly from what's prescribed in [[#cycle-therapy 1]]. Instead of | The above departs slightly from what's prescribed in [[#cycle-therapy 1]]. Instead of | ||

− | a ''true graph'' built by knot-tying, we chose to use an | + | a ''true graph'' built by knot-tying, we chose to use an <code>Array</code> |

with integers as explicit vertex references. | with integers as explicit vertex references. | ||

This is closely follows | This is closely follows | ||

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most of the algorithms from Data.Graph with only minor changes: | most of the algorithms from Data.Graph with only minor changes: | ||

− | + | <haskell> | |

-- | Build a graph from a list of edges. | -- | Build a graph from a list of edges. | ||

buildG :: Bounds -> [Edge e] -> Graph e | buildG :: Bounds -> [Edge e] -> Graph e | ||

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reverseE :: Graph e -> [Edge e] | reverseE :: Graph e -> [Edge e] | ||

reverseE g = [ (w, l, v) | (v, l, w) <- edges g ] | reverseE g = [ (w, l, v) | (v, l, w) <- edges g ] | ||

− | + | </haskell> | |

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For example, here's the function to output a graph as a GraphViz file: | For example, here's the function to output a graph as a GraphViz file: | ||

− | + | <haskell> | |

showGraphViz (LabGraph gr lab) = | showGraphViz (LabGraph gr lab) = | ||

"digraph name {\n" ++ | "digraph name {\n" ++ | ||

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edges :: Graph e -> [Edge e] | edges :: Graph e -> [Edge e] | ||

edges g = [ (v, l, w) | v <- indices g, (l, w) <- g!v ] | edges g = [ (v, l, w) | v <- indices g, (l, w) <- g!v ] | ||

− | + | </haskell> | |

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computation of the transitive closure of a non-deterministic function. | computation of the transitive closure of a non-deterministic function. | ||

− | + | <haskell> | |

closure :: Ord a => (a -> [a]) -> (a -> [a]) | closure :: Ord a => (a -> [a]) -> (a -> [a]) | ||

closure f i = labels $ snd $ unfoldG f' i | closure f i = labels $ snd $ unfoldG f' i | ||

where f' x = (x, [((), fx) | fx <- f x]) | where f' x = (x, [((), fx) | fx <- f x]) | ||

− | + | </haskell> | |

In this context, "non deterministic" means that it yields many | In this context, "non deterministic" means that it yields many | ||

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For example, if we define | For example, if we define | ||

− | + | <haskell> | |

interleave (x1:x2:xs) = (x1:x2:xs) : (map (x2:) (interleave (x1:xs))) | interleave (x1:x2:xs) = (x1:x2:xs) : (map (x2:) (interleave (x1:xs))) | ||

interleave xs = [xs] | interleave xs = [xs] | ||

interleave "abcd" ==> ["abcd","bacd","bcad","bcda"] | interleave "abcd" ==> ["abcd","bacd","bcad","bcda"] | ||

− | + | </haskell> | |

a very bad way to compute the permutations of list can be | a very bad way to compute the permutations of list can be | ||

− | + | <haskell> | |

permutations = closure interleave | permutations = closure interleave | ||

Line 307: | Line 308: | ||

"dcab","cdab","cadb","acdb","cdba","dcba", | "dcab","cdab","cadb","acdb","cdba","dcba", | ||

"cbda","bcda","bdca","dbca","bcad","cbad"] | "cbda","bcda","bdca","dbca","bcad","cbad"] | ||

− | + | </haskell> | |

− | But sometimes the function to 'close' is more complicated than | + | But sometimes the function to 'close' is more complicated than <code>interleave</code> and |

− | then | + | then <code>closure</code> becomes really useful. |

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Most readers probably know the Dijkstra's algorithm to | Most readers probably know the Dijkstra's algorithm to | ||

compute the solution to the problem. We will not try | compute the solution to the problem. We will not try | ||

− | to reproduce it here, instead we will define the computation in terms of | + | to reproduce it here, instead we will define the computation in terms of <code>foldG</code>. |

Here it goes: | Here it goes: | ||

− | + | <haskell> | |

-- | Compute the distance to v for every vertex of gr. | -- | Compute the distance to v for every vertex of gr. | ||

distsTo :: Vertex -> Graph Float -> Table Float | distsTo :: Vertex -> Graph Float -> Table Float | ||

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| v == v' = 0 | | v == v' = 0 | ||

| otherwise = minimum [distV+arcWeight | (distV, arcWeight) <- neighbours] | | otherwise = minimum [distV+arcWeight | (distV, arcWeight) <- neighbours] | ||

− | + | </haskell> | |

So clear that it barely needs to be explained. :) | So clear that it barely needs to be explained. :) | ||

Just notice how the minimize function assumes that the | Just notice how the minimize function assumes that the | ||

distance is already computed for all its neighbours. | distance is already computed for all its neighbours. | ||

− | This works because | + | This works because <code>foldG</code> will iterate until it finds the fixed point. |

On this simple graph, | On this simple graph, | ||

− | + | <haskell> | |

grDist = buildG (1,5) [(1,5.0,2), (2,5.0,3), (2,7.0,4), (3,5.0,4), (4,5.0,5), (4,3.0,1)] | grDist = buildG (1,5) [(1,5.0,2), (2,5.0,3), (2,7.0,4), (3,5.0,4), (4,5.0,5), (4,3.0,1)] | ||

− | + | </haskell> | |

attachment:grdist.png | attachment:grdist.png | ||

− | the result of | + | the result of <haskell> |

dists = distsTo 5 grDist | dists = distsTo 5 grDist | ||

− | + | </haskell> is | |

attachment:grdist2.png | attachment:grdist2.png | ||

− | (labeling each node with the its result, ie. distance to vertex | + | (labeling each node with the its result, ie. distance to vertex <code>5</code>) |

=== Finite Automaton === | === Finite Automaton === | ||

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of states/transitions, some of the states being marked as initial or final. | of states/transitions, some of the states being marked as initial or final. | ||

− | + | <haskell> | |

type Automaton t = (Vertex, Graph t, Set Vertex) -- ^ Initial, transitions, finals | type Automaton t = (Vertex, Graph t, Set Vertex) -- ^ Initial, transitions, finals | ||

− | + | </haskell> | |

− | For starters, here is how the | + | For starters, here is how the <code>showGraphViz</code> function can be applied to automaton display: |

− | + | <haskell> | |

automatonToGraphviz (i, gr, fs) = showGraphViz (LabGraph gr lab) | automatonToGraphviz (i, gr, fs) = showGraphViz (LabGraph gr lab) | ||

where lab :: Labeling String | where lab :: Labeling String | ||

lab v = (if v == i then (">"++) else id) $ | lab v = (if v == i then (">"++) else id) $ | ||

(if v `Set.member` fs then (++"|") else id) [] | (if v `Set.member` fs then (++"|") else id) [] | ||

− | + | </haskell> | |

Nothing ground breaking. We only label the nodes accordingly to | Nothing ground breaking. We only label the nodes accordingly to | ||

their final or initial status. | their final or initial status. | ||

− | + | <haskell> | |

aut1 = (1, buildG (1,3) [(1,'a',2),(2,'a',2),(2,'b',2),(2,'c',3),(1,'a',3)], Set.fromList [3]) | aut1 = (1, buildG (1,3) [(1,'a',2),(2,'a',2),(2,'b',2),(2,'c',3),(1,'a',3)], Set.fromList [3]) | ||

− | + | </haskell> | |

attachment:aut1.png | attachment:aut1.png | ||

Line 387: | Line 388: | ||

is that non-deterministic execution of the automaton is equivalent | is that non-deterministic execution of the automaton is equivalent | ||

to deterministic execution on all possible transitions at once. | to deterministic execution on all possible transitions at once. | ||

− | Refer to [[#hop&ull 6]] for details. This is relatively easily done using | + | Refer to [[#hop&ull 6]] for details. This is relatively easily done using <code>unfoldG</code>. |

− | + | <haskell> | |

simpleGenerator f x = (x, f x) | simpleGenerator f x = (x, f x) | ||

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finals2 = Set.fromList $ filter isFinal $ indices aut2 | finals2 = Set.fromList $ filter isFinal $ indices aut2 | ||

setAny f = any f . Set.toList | setAny f = any f . Set.toList | ||

− | + | </haskell> | |

The 'build' function is the tricky part. Yet, it's not as complicated as it seems: all it does is | The 'build' function is the tricky part. Yet, it's not as complicated as it seems: all it does is | ||

Line 408: | Line 409: | ||

#Build target state-sets accordingly. | #Build target state-sets accordingly. | ||

− | + | <haskell> | |

aut2 = nfaToDfa aut1 | aut2 = nfaToDfa aut1 | ||

− | + | </haskell> | |

attachment:aut2.png | attachment:aut2.png | ||

Line 417: | Line 418: | ||

strings accepted by the automaton, (aka. the language it | strings accepted by the automaton, (aka. the language it | ||

defines). Most of the time this will be infinite, so | defines). Most of the time this will be infinite, so | ||

− | we will limit ourselves to strings of length | + | we will limit ourselves to strings of length <code>n</code> maximum. |

− | We need finiteness because otherwise | + | We need finiteness because otherwise <code>foldG</code> would not find |

a fixed point: string sets would keep growing idefinitely. | a fixed point: string sets would keep growing idefinitely. | ||

− | + | <haskell> | |

accepted n (initial1, aut1, finals1) = Set.unions [resultTable ! v | v <- Set.toList finals1] | accepted n (initial1, aut1, finals1) = Set.unions [resultTable ! v | v <- Set.toList finals1] | ||

-- gather what's accepted at all final states | -- gather what's accepted at all final states | ||

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step v trans = Set.unions ((if v == initial1 then Set.singleton [] else Set.empty) : | step v trans = Set.unions ((if v == initial1 then Set.singleton [] else Set.empty) : | ||

[Set.map ((++[t]) . take (n-1) ) s | (t,s) <- trans]) | [Set.map ((++[t]) . take (n-1) ) s | (t,s) <- trans]) | ||

− | + | </haskell> | |

Notice that we need to reverse the graph arcs, otherwise the information propagates in the wrong direction. | Notice that we need to reverse the graph arcs, otherwise the information propagates in the wrong direction. | ||

With | With | ||

− | + | <haskell> | |

accAut1 = accepted 4 aut1 | accAut1 = accepted 4 aut1 | ||

accAut2 = accepted 4 aut2 | accAut2 = accepted 4 aut2 | ||

− | + | </haskell> | |

we have | we have | ||

− | + | <haskell> | |

accAut1 == accAut2 == {"a","aaac","aabc","aac","abac","abbc","abc","ac"} | accAut1 == accAut2 == {"a","aaac","aabc","aac","abac","abbc","abc","ac"} | ||

− | + | </haskell> | |

=== LALR Automaton === | === LALR Automaton === | ||

Line 450: | Line 451: | ||

In the process of generating tables for a LALR automaton, | In the process of generating tables for a LALR automaton, | ||

− | there are three steps amenable to implementation by | + | there are three steps amenable to implementation by <code>foldG</code> and <code>unfoldG</code>. |

− | + | 1. Construction of the closure of a LR-items kernel. This one is very similar to the <code>closure</code> function described above, except that we don't discard the graph structure. It'll be of use for step 3. | |

− | + | 2. LR(0) automaton generation. Then again a use for <code>unfoldG</code>. | |

− | + | 3. Propagation of the lookahead. It is a fold over the whole graph of LR-items, basically using set union as coalescing operation. It is very similar to computation of acceptable strings above. | |

Line 461: | Line 462: | ||

=== UnfoldG === | === UnfoldG === | ||

− | For the sake of completeness, here's how to implement the | + | For the sake of completeness, here's how to implement the <code>unfoldG</code> function. |

The algorithm effectively a depth-first search, written in imperative style. | The algorithm effectively a depth-first search, written in imperative style. | ||

The only difference is that the search graph is remembered and returned as result. | The only difference is that the search graph is remembered and returned as result. | ||

− | + | <haskell> | |

unfoldGManyST :: (Ord a) => (a -> (c, [(b, a)])) | unfoldGManyST :: (Ord a) => (a -> (c, [(b, a)])) | ||

Line 501: | Line 502: | ||

memTabBind key val mt = modifySTRef mt (Map.insert key val) | memTabBind key val mt = modifySTRef mt (Map.insert key val) | ||

− | + | </haskell> | |

Notice how every time a seed is encountered, its corresponding vertex number stored. | Notice how every time a seed is encountered, its corresponding vertex number stored. | ||

Line 509: | Line 510: | ||

=== FoldG === | === FoldG === | ||

− | + | <haskell> | |

foldGAllImplementation bot f gr = finalTbl | foldGAllImplementation bot f gr = finalTbl | ||

where finalTbl = fixedPoint updateTbl initialTbl | where finalTbl = fixedPoint updateTbl initialTbl | ||

Line 520: | Line 521: | ||

where recompute v = f v [(b, tbl!k) | (b, k) <- gr!v] | where recompute v = f v [(b, tbl!k) | (b, k) <- gr!v] | ||

bnds = bounds gr | bnds = bounds gr | ||

− | + | </haskell> | |

Line 552: | Line 553: | ||

== References == | == References == | ||

− | + | * [[Anchor(cycle-therapy)]] [1] ''Cycle Therapy: A Prescription for Fold and Unfold on Regular Trees'', F. Turbak and J.B. Wells, http://cs.wellesley.edu/~fturbak/pubs/ppdp01.pdf | |

− | + | * [[Anchor(king-thesis)]] [2] ''Functional Programming and Graph Algorithms'', D. J. King, http://www.macs.hw.ac.uk/~gnik/publications | |

− | + | * [[Anchor(induct)]] [3] ''Inductive Graphs and Functional Graph Algorithms'', Martin Erwig, http://web.engr.oregonstate.edu/~erwig/papers/abstracts.html | |

− | + | * [[Anchor(dfs)]] [4] , D. J. King and John Launchbury, http://www.cse.ogi.edu/~jl/Papers/dfs.ps | |

− | + | * [[Anchor(bananas-lenses)]] [5] ''Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire'', Erik Meijer, Maarten Fokkinga, Ross Paterson. http://citeseer.ist.psu.edu/meijer91functional.html | |

− | + | * [[Anchor(hop&ull)]] [6] ''Introduction to Automata Theory, Languages, and Computation'', JE Hopcroft, and JD Ullman, http://www-db.stanford.edu/~ullman/ialc.html | |

− | + | * [[Anchor(dragon)]] [7] ''Compilers: Principles, Techniques and Tools'', Alfred V. Aho, Ravi Sethi, and Jeffrey D. Ullman. (Addison-Wesley 1986; ISBN 0-201-10088-6) | |

---- | ---- | ||

CategoryArticle | CategoryArticle |

## Latest revision as of 18:18, 27 October 2011

**This article needs reformatting! Please help tidy it up.**--WouterSwierstra 14:26, 9 May 2008 (UTC)

## Contents |

# [edit] 1 A Practical Approach to Graph Manipulation

*by JeanPhilippeBernardy for The Monad.Reader Issue 5*
BR
*Date(2005-07-08T20:48:51Z)*

**Abstract.**

Tree-based data structures are easy to deal with in Haskell. However, working with graph-like structures in practice is much less obvious. In this article I present a solution that has worked for me in many cases.

## [edit] 1.1 Introduction

I always found that dealing with graphs in Haskell is a tricky subject. Even something like a implementing a depth-first search, which is trivially achieved in an imperative language, deserves an article on its own for Haskell #dfs 4. A PhD thesis has been written on the subject of graphs and functional programming #king-thesis 2, and it seems that it still doesn't exhaust the design-space: radically different ideas have been proposed afterwards #induct 3.

In this article I'll present (a simplified version of) a solution that I think deserves more coverage #cycle-therapy 1. The idea is to abstract graph manipulation by anamorphisms and catamorphisms. This approach features "separation of concerns" and "composability", hence it can be readily applied to practical problems.

- Section 2 shows how anamorphisms and catamorphisms can be generalised to graphs.
- Section 3 details the data structures used to represent graphs
- Section 4 discusses various problems where cata/anamorphisms can be applied
- Section 5 gives a sample implementation for the catamorphism and anamorphism
- Section 6 concludes

### [edit] 1.1.1 Nota

This article has been generated from a literate haskell source. So, although the text of this wiki page will not compile, all the examples are real and run. The source can be accessed here: attachment:PracticalGraphHandling.lhs

We will assume you know the hierarchical libraries. Refer to http://haskell.org/ghc/docs/latest/html/libraries/index.html in case of doubt.

## [edit] 1.2 Origami with Graphs

### [edit] 1.2.1 Fold & Unfold (the big deal)

Most of you probably know what a "fold" (also known as catamorphism) is. For those who don't, intuitively, it's an higher-order operation that reduces a complex structure to a single value. It applies a function given as parameter to each node, propagating the results up to the root. This is a highly imprecise definition, for more details please read #bananas-lenses 5.

For example, the fold operation on lists can be typed as follows:

foldr :: (a -> b -> b) -> -- ^ operation to apply b -> -- ^ initial value [a] -> -- ^ input list b -- ^ result

Conversely, "unfold" builds a complex structure out of a building function, applying it iteratively.

unfoldr :: (b -> Maybe (a, b)) -> -- ^ building function (Nothing => end of list) b -> -- ^ seed value [a] -- ^ result

The second argument is the initial value from which the whole resulting list will be derived, by applying the 1st argument. In the following we'll refer to it as the "seed".

The catamorphism/anamorphism abstractions have proven to be
very useful in practise. They're ubiquitous to any haskell
programming, either explicitly, or implicitly (hidden in
higher-level operations). In this article I'll show how
those abstractions can be generalised to graph structures,
and argue that they are equally useful in this case.

The rest of the article assumes the reader is fairly familiar with fold and unfold. Fortunately there are many articles on the subject. For example you can refer to #bananas-lenses 5 if you ever feel uncomfortable.

### [edit] 1.2.2 Generalisation

Let's examine how fold/unfold can be generalized for graphs. Since we are working on graphs instead of lists, we must account for

1. Any number of children for a node; 2. "Backwards" arcs (cycles); 3. Labelled edges.

The most relevant point being 2, of course.

#### [edit] 1.2.2.1 unfoldG

From the above, we can deduce that the type of unfoldG will be:

unfoldG :: (Ord s) => (s -> (n, [(e, s)])) -> s -> (Vertex, LabGraph n e) unfoldG f r = (r', res) where ([r'], res) = unfoldGMany f [r]

where `s`

is the seed type, `n`

is the node labels, `e`

the edges labels.

The `Ord s`

constraint reflects point 2 above.
It is needed because the unfoldG function must record every
seed value encountered.
Whenever a seed is seen a second time, `unfoldG`

will recognize
it and create a "backward arc".
We use `Ord`

instead of `Eq`

because a mere equality test rules out using `Data.Map`

.

The attentive reader will note that we return an additional Vertex value. This is needed to identifty which node the root seed corresponds to.

In order to get an intuitive feeling of how `unfoldG`

works,
let's examine a simple example.

gr1 :: LabGraph Int Char (_,gr1) = unfoldG gen (0::Int) where gen x = (x,[('a',(x+1) `mod` 10), ('b', (x+2) `mod` 10)])

`gr1`

being defined as above, its structure is:

attachment:gr1.png

Because we might want to build a graph from a set of seeds instead of a single one, we will also need the following function:

unfoldGMany :: (Ord s) => (s -> (n, [(e, s)])) -> [s] -> ([Vertex], LabGraph n e) unfoldGMany f roots = runST ( unfoldGManyST f roots ) -- detailed later

`unfoldG`

, alone, is already very a practical tool, because it
lets you reify a function (`a -> a`

) graph. It then can be examined,
processed, etc. whereas the function can only be evaluated.

#### [edit] 1.2.2.2 foldG

On a graph, the catamorphism (fold) type will become:

foldG :: (Eq r) => r -> (Vertex -> [(e, r)] -> r) -> Graph e -> Vertex -> r foldG i f g v = foldGAll i f g ! v

As for `unfoldG`

, the `foldG`

function must include a special mechanism to handle cycles.
The idea is to apply the operation iteratively until the result
converges. It's the purpose of the first
parameter is to "bootstrap" the process:
it will be used as an initial value.

Thus, `foldG i f g ! v`

will iteratively
apply `f`

on nodes of graph `g`

,
using `i`

as "bottom" value. It will return
the value computed at vertex `v`

.
Of course, this will work only if `f`

is well-behaved:
it must converge at some point.
I won't dwelve in to the theoretical details
here, see #cycle-therapy 1 for a
formal explanation.

Notice that `foldG`

can work on a graph without node labels.
If the parameter function needs to access node labels, it can
do so without `foldG`

needing to know.

It's also worth noticing that, in our implementation, the information will be propagated in the reverse direction of arcs.

It's very common to need the result value for each vertex, hence the need for :

foldGAll :: (Eq r) => r -> (Vertex -> [(e, r)] -> r) -> Graph e -> Table r

The implementation of these functions doesn't matter much. The point of the article is not how these can be implemented, but how they can be used for daily programming tasks. For completeness though, we'll provide a sample implemenation at the end of the article.

## [edit] 1.3 Data Structure & Accessors

Without further ado, let's define the data structures we'll work on.

type Vertex = Int type Table a = Array Vertex a type Graph e = Table [(e, Vertex)] type Bounds = (Vertex, Vertex) type Edge e = (Vertex, e, Vertex)

A graph is a mere adjacency list table, tagged with edge labels.

The above structure lacks labels for nodes. This is easily fixed by adding a labeling (or coloring) function.

type Labeling a = Vertex -> a data LabGraph n e = LabGraph (Graph e) (Labeling n) vertices (LabGraph gr _) = indices gr labels (LabGraph gr l) = map l (indices gr)

The above departs slightly from what's prescribed in #cycle-therapy 1. Instead of
a *true graph* built by knot-tying, we chose to use an `Array`

with integers as explicit vertex references.
This is closely follows
Data.Graph in the hierarchical libraries,
the only difference being that we have labelled edges.

Not only this is simpler, but it has the advantage that we can reuse most of the algorithms from Data.Graph with only minor changes:

-- | Build a graph from a list of edges. buildG :: Bounds -> [Edge e] -> Graph e buildG bounds0 edges0 = accumArray (flip (:)) [] bounds0 [(v, (l,w)) | (v,l,w) <- edges0] -- | The graph obtained by reversing all edges. transposeG :: Graph e -> Graph e transposeG g = buildG (bounds g) (reverseE g) reverseE :: Graph e -> [Edge e] reverseE g = [ (w, l, v) | (v, l, w) <- edges g ]

However, as previously said, we'll try to abstract
away from the details of the structure.
This is not always possible, but in such cases,
I believe the array representation to be
a good choice, because it's easy to work with.
If anything, one can readily use all the
standard array functions.

For example, here's the function to output a graph as a GraphViz file:

showGraphViz (LabGraph gr lab) = "digraph name {\n" ++ "rankdir=LR;\n" ++ (concatMap showNode $ indices gr) ++ (concatMap showEdge $ edges gr) ++ "}\n" where showEdge (from, t, to) = show from ++ " -> " ++ show to ++ " [label = \"" ++ show t ++ "\"];\n" showNode v = show v ++ " [label = " ++ (show $ lab v) ++ "];\n" edges :: Graph e -> [Edge e] edges g = [ (v, l, w) | v <- indices g, (l, w) <- g!v ]

## [edit] 1.4 Applications

I'll now enumerate a few problems where the "origami" approach can be applied successfully.

### [edit] 1.4.1 Closure

A simple application (special case) of "unfoldG" the computation of the transitive closure of a non-deterministic function.

closure :: Ord a => (a -> [a]) -> (a -> [a]) closure f i = labels $ snd $ unfoldG f' i where f' x = (x, [((), fx) | fx <- f x])

In this context, "non deterministic" means that it yields many values, as a list. As noted before, this will work only when everything remains finite in size.

For example, if we define

interleave (x1:x2:xs) = (x1:x2:xs) : (map (x2:) (interleave (x1:xs))) interleave xs = [xs] interleave "abcd" ==> ["abcd","bacd","bcad","bcda"]

a very bad way to compute the permutations of list can be

permutations = closure interleave permutations "abcd" ==> ["abcd","bacd","acbd","cabd","abdc","badc", "adbc","dabc","dbac","bdac","dacb","adcb", "dcab","cdab","cadb","acdb","cdba","dcba", "cbda","bcda","bdca","dbca","bcad","cbad"]

But sometimes the function to 'close' is more complicated than `interleave`

and
then `closure`

becomes really useful.

### [edit] 1.4.2 Shortest Path

Let us now examine the toy problem of finding the distance
to a given node from all the other nodes of the graph.
Most readers probably know the Dijkstra's algorithm to
compute the solution to the problem. We will not try
to reproduce it here, instead we will define the computation in terms of `foldG`

.

Here it goes:

-- | Compute the distance to v for every vertex of gr. distsTo :: Vertex -> Graph Float -> Table Float distsTo v gr = foldGAll infinite distance gr where infinite = 10000000 -- well, you get the idea distance v' neighbours | v == v' = 0 | otherwise = minimum [distV+arcWeight | (distV, arcWeight) <- neighbours]

So clear that it barely needs to be explained. :)
Just notice how the minimize function assumes that the
distance is already computed for all its neighbours.
This works because `foldG`

will iterate until it finds the fixed point.

On this simple graph,

grDist = buildG (1,5) [(1,5.0,2), (2,5.0,3), (2,7.0,4), (3,5.0,4), (4,5.0,5), (4,3.0,1)]

attachment:grdist.png

the result ofdists = distsTo 5 grDist

attachment:grdist2.png

(labeling each node with the its result, ie. distance to vertex `5`

)

### [edit] 1.4.3 Finite Automaton

Finite automatons are basically graphs, so let's see how we can apply the framework to their analysis.

First, let's define an automaton. For our purposes, it is a graph of states/transitions, some of the states being marked as initial or final.

type Automaton t = (Vertex, Graph t, Set Vertex) -- ^ Initial, transitions, finals

For starters, here is how the `showGraphViz`

function can be applied to automaton display:

automatonToGraphviz (i, gr, fs) = showGraphViz (LabGraph gr lab) where lab :: Labeling String lab v = (if v == i then (">"++) else id) $ (if v `Set.member` fs then (++"|") else id) []

Nothing ground breaking. We only label the nodes accordingly to their final or initial status.

aut1 = (1, buildG (1,3) [(1,'a',2),(2,'a',2),(2,'b',2),(2,'c',3),(1,'a',3)], Set.fromList [3])

attachment:aut1.png

A more interesting example is how to transform a non-deterministic
automaton to an equivalent deterministic one. The underlying idea
is that non-deterministic execution of the automaton is equivalent
to deterministic execution on all possible transitions at once.
Refer to #hop&ull 6 for details. This is relatively easily done using `unfoldG`

.

simpleGenerator f x = (x, f x) nfaToDfa :: Ord e => Automaton e -> Automaton e nfaToDfa (initial1, aut1, finals1) = (initial2, aut2, finals2) where (initial2, LabGraph aut2 mapping) = unfoldG (simpleGenerator build) seed seed = Set.singleton initial1 build state = Map.toList $ Map.fromListWith Set.union $ map lift $ concat $ map (aut1 !) $ Set.toList state lift (t,s) = (t, Set.singleton s) isFinal = setAny (`Set.member` finals1) . mapping finals2 = Set.fromList $ filter isFinal $ indices aut2 setAny f = any f . Set.toList

The 'build' function is the tricky part. Yet, it's not as complicated as it seems: all it does is

- Find all reachable nodes from a set of nodes;
- Classify them by transition label
- Build target state-sets accordingly.

`aut2 = nfaToDfa aut1`

attachment:aut2.png

Another thing we possibly wish to compute is the set of
strings accepted by the automaton, (aka. the language it
defines). Most of the time this will be infinite, so
we will limit ourselves to strings of length `n`

maximum.
We need finiteness because otherwise `foldG`

would not find
a fixed point: string sets would keep growing idefinitely.

accepted n (initial1, aut1, finals1) = Set.unions [resultTable ! v | v <- Set.toList finals1] -- gather what's accepted at all final states where resultTable = foldGAll Set.empty step (transposeG aut1) step v trans = Set.unions ((if v == initial1 then Set.singleton [] else Set.empty) : [Set.map ((++[t]) . take (n-1) ) s | (t,s) <- trans])

Notice that we need to reverse the graph arcs, otherwise the information propagates in the wrong direction.

With

accAut1 = accepted 4 aut1 accAut2 = accepted 4 aut2

we have

accAut1 == accAut2 == {"a","aaac","aabc","aac","abac","abbc","abc","ac"}

### [edit] 1.4.4 LALR Automaton

Another area where I applied graph (un)folding is LALR(1) parser generation. The detailed code depends on just too many things to fit in this paper, thus we will only sketch how pieces fit together. Also, since a course on parsing is clearly beyond the scope of this article, please refer to local copy of the dragon book #dragon 7 for details on the method.

In the process of generating tables for a LALR automaton,
there are three steps amenable to implementation by `foldG`

and `unfoldG`

.

1. Construction of the closure of a LR-items kernel. This one is very similar to the `closure`

function described above, except that we don't discard the graph structure. It'll be of use for step 3.
2. LR(0) automaton generation. Then again a use for `unfoldG`

.
3. Propagation of the lookahead. It is a fold over the whole graph of LR-items, basically using set union as coalescing operation. It is very similar to computation of acceptable strings above.

## [edit] 1.5 Implementation

### [edit] 1.5.1 UnfoldG

For the sake of completeness, here's how to implement the `unfoldG`

function.

The algorithm effectively a depth-first search, written in imperative style. The only difference is that the search graph is remembered and returned as result.

unfoldGManyST :: (Ord a) => (a -> (c, [(b, a)])) -> [a] -> ST s ([Vertex], LabGraph c b) unfoldGManyST gen seeds = do mtab <- newSTRef (Map.empty) allNodes <- newSTRef [] vertexRef <- newSTRef firstId let allocVertex = do vertex <- readSTRef vertexRef writeSTRef vertexRef (vertex + 1) return vertex let cyc src = do probe <- memTabFind mtab src case probe of Just result -> return result Nothing -> do v <- allocVertex memTabBind src v mtab let (lab, deps) = gen src ws <- mapM (cyc . snd) deps let res = (v, lab, [(fst d, w) | d <- deps | w <- ws]) modifySTRef allNodes (res:) return v mapM_ cyc seeds list <- readSTRef allNodes seedsResult <- (return . map fromJust) =<< mapM (memTabFind mtab) seeds lastId <- readSTRef vertexRef let cycamore = array (firstId, lastId-1) [(i, k) | (i, a, k) <- list] let labels = array (firstId, lastId-1) [(i, a) | (i, a, k) <- list] return (seedsResult, LabGraph cycamore (labels !)) where firstId = 0::Vertex memTabFind mt key = return . Map.lookup key =<< readSTRef mt memTabBind key val mt = modifySTRef mt (Map.insert key val)

Notice how every time a seed is encountered, its corresponding vertex number stored. Whenever the seed is encountered again, the stored is just returned.

### [edit] 1.5.2 FoldG

foldGAllImplementation bot f gr = finalTbl where finalTbl = fixedPoint updateTbl initialTbl initialTbl = listArray bnds (replicate (rangeSize bnds) bot) fixedPoint f x = fp x where fp z = if z == z' then z else fp z' where z' = f z updateTbl tbl = listArray bnds $ map recompute $ indices gr where recompute v = f v [(b, tbl!k) | (b, k) <- gr!v] bnds = bounds gr

The proposed implementation for foldG is rather bold.
It just applies the coalescing
function repeatedly till it converges.

While this is not an ideal situation, it's perfectly suited for a first-trial implementation, or when performance is not crucial.

If execution time becomes critical, then more specialized versions can be crafted. In the case of the shortest path algorithm, for example, it could take advantage of the nice properties of the coalescing function to use a priority queue and greedily find the fixed point. This would restore the optimal O(n * log n) complexity.

## [edit] 1.6 Conclusion

The approach presented may not be excellent for controlling details of implementation and tuning run-time performance, but I think that's not the point of haskell programming anyway. On the other hand, it is very good for quick implementation of a large range of graph algorithms. The fact that it's mostly based on a generalisation on fold and unfold should appeal to haskell programmers.

## [edit] 1.7 References

- Anchor(cycle-therapy) [1]
*Cycle Therapy: A Prescription for Fold and Unfold on Regular Trees*, F. Turbak and J.B. Wells, http://cs.wellesley.edu/~fturbak/pubs/ppdp01.pdf - Anchor(king-thesis) [2]
*Functional Programming and Graph Algorithms*, D. J. King, http://www.macs.hw.ac.uk/~gnik/publications - Anchor(induct) [3]
*Inductive Graphs and Functional Graph Algorithms*, Martin Erwig, http://web.engr.oregonstate.edu/~erwig/papers/abstracts.html - Anchor(dfs) [4] , D. J. King and John Launchbury, http://www.cse.ogi.edu/~jl/Papers/dfs.ps
- Anchor(bananas-lenses) [5]
*Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire*, Erik Meijer, Maarten Fokkinga, Ross Paterson. http://citeseer.ist.psu.edu/meijer91functional.html - Anchor(hop&ull) [6]
*Introduction to Automata Theory, Languages, and Computation*, JE Hopcroft, and JD Ullman, http://www-db.stanford.edu/~ullman/ialc.html - Anchor(dragon) [7]
*Compilers: Principles, Techniques and Tools*, Alfred V. Aho, Ravi Sethi, and Jeffrey D. Ullman. (Addison-Wesley 1986; ISBN 0-201-10088-6)

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