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99 questions/80 to 89

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This is part of Ninety-Nine Haskell Problems, based on Ninety-Nine Prolog Problems.

If you want to work on one of these, put your name in the block so we know someone's working on it. Then, change n in your block to the appropriate problem number, and fill in the <Problem description>,<example in lisp>,<example in Haskell>,<solution in haskell> and <description of implementation> fields.

1 Graphs

A graph is defined as a set of nodes and a set of edges, where each edge is a pair of nodes.

graph1.gif

There are several ways to represent graphs in Prolog. One method is to represent each edge separately as one clause (fact). In this form, the graph depicted below is represented as the following predicate:

edge(h,g).
edge(k,f).
edge(f,b).
...

We call this edge-clause form. Obviously, isolated nodes cannot be represented. Another method is to represent the whole graph as one data object. According to the definition of the graph as a pair of two sets (nodes and edges), we may use the following Prolog term to represent the example graph:

graph([b,c,d,f,g,h,k],[e(b,c),e(b,f),e(c,f),e(f,k),e(g,h)])

We call this graph-term form. Note, that the lists are kept sorted, they are really sets, without duplicated elements. Each edge appears only once in the edge list; i.e. an edge from a node x to another node y is represented as e(x,y), the term e(y,x) is not present. The graph-term form is our default representation. In SWI-Prolog there are predefined predicates to work with sets.

A third representation method is to associate with each node the set of nodes that are adjacent to that node. We call this the adjacency-list form. In our example:

[n(b,[c,f]), n(c,[b,f]), n(d,[]), n(f,[b,c,k]), ...]

The representations we introduced so far are Prolog terms and therefore well suited for automated processing, but their syntax is not very user-friendly. Typing the terms by hand is cumbersome and error-prone. We can define a more compact and "human-friendly" notation as follows: A graph is represented by a list of atoms and terms of the type X-Y (i.e. functor '-' and arity 2). The atoms stand for isolated nodes, the X-Y terms describe edges. If an X appears as an endpoint of an edge, it is automatically defined as a node. Our example could be written as:

[b-c, f-c, g-h, d, f-b, k-f, h-g]

We call this the human-friendly form. As the example shows, the list does not have to be sorted and may even contain the same edge multiple times. Notice the isolated node d. (Actually, isolated nodes do not even have to be atoms in the Prolog sense, they can be compound terms, as in d(3.75,blue) instead of d in the example).

graph2.gif

When the edges are directed we call them arcs. These are represented by ordered pairs. Such a graph is called directed graph. To represent a directed graph, the forms discussed above are slightly modified. The example graph above is represented as follows:

Arc-clause form

arc(s,u).
arc(u,r).
...

Graph-term form

digraph([r,s,t,u,v],[a(s,r),a(s,u),a(u,r),a(u,s),a(v,u)])

Adjacency-list form

[n(r,[]),n(s,[r,u]),n(t,[]),n(u,[r]),n(v,[u])]

Note that the adjacency-list does not have the information on whether it is a graph or a digraph.

Human-friendly form

[s > r, t, u > r, s > u, u > s, v > u] 

Finally, graphs and digraphs may have additional information attached to nodes and edges (arcs). For the nodes, this is no problem, as we can easily replace the single character identifiers with arbitrary compound terms, such as city('London',4711). On the other hand, for edges we have to extend our notation. Graphs with additional information attached to edges are called labelled graphs.

graph3.gif

Arc-clause form

arc(m,q,7).
arc(p,q,9).
arc(p,m,5).

Graph-term form

digraph([k,m,p,q],[a(m,p,7),a(p,m,5),a(p,q,9)])

Adjacency-list form

[n(k,[]),n(m,[q/7]),n(p,[m/5,q/9]),n(q,[])]

Notice how the edge information has been packed into a term with functor '/' and arity 2, together with the corresponding node.

Human-friendly form

[p>q/9, m>q/7, k, p>m/5]

The notation for labelled graphs can also be used for so-called multi-graphs, where more than one edge (or arc) are allowed between two given nodes.

2 Problem 80

<Problem description>

Example:
<example in lisp>

Example in Haskell:
<example in Haskell>

Solution:

<solution in haskell>

<description of implementation>


3 Problem 81

Path from one node to another one

   Write a function that, given two nodes a and b in a graph, returns all the acyclic paths from a to b.
Example:
<example in lisp>

Example in Haskell:
paths 1 4 [(1,2),(2,3),(1,3),(3,4),(4,2),(5,6)]
[[1,2,3,4],[1,3,4]]
paths 2 6 [(1,2),(2,3),(1,3),(3,4),(4,2),(5,6)]
[]

Solution:

import List (elem)
 
paths :: Eq a => a -> a -> [(a,a)] -> [[a]]
paths a b g = paths1 a b g []
 
paths1 :: Eq a => a -> a -> [(a,a)] -> [a] -> [[a]]
paths1 a b g current = paths2 a b g current [ y | (x,y) <- g, x == a ]
 
paths2 :: Eq a => a -> a -> [(a,a)] -> [a] -> [a] -> [[a]]
paths2 a b g current []	| a == b = [current++[b]]
			| otherwise = []
paths2 a b g current (x:xs) | a == b = [current++[b]] 
			    | elem a current = []
			    | otherwise = (paths1 x b g (current++[a])) ++ (paths2 a b g current xs)

This solution uses a representation of a (directed) graph as a list of arcs (a,b).

4 Problem 82

<Problem description>

Example:
<example in lisp>

Example in Haskell:
<example in Haskell>

Solution:

<solution in haskell>

<description of implementation>

5 Problem 83

<Problem description>

Example:
<example in lisp>

Example in Haskell:
<example in Haskell>

Solution:

<solution in haskell>

<description of implementation>

6 Problem 84

<Problem description>

Example:
<example in lisp>

Example in Haskell:
<example in Haskell>

Solution:

<solution in haskell>

<description of implementation>

7 Problem 85

<Problem description>

Example:
<example in lisp>

Example in Haskell:
<example in Haskell>

Solution:

<solution in haskell>

<description of implementation>

8 Problem 86

<Problem description>

Example:
<example in lisp>

Example in Haskell:
<example in Haskell>

Solution:

<solution in haskell>

<description of implementation>

9 Problem 87

<Problem description>

Example:
<example in lisp>

Example in Haskell:
<example in Haskell>

Solution:

<solution in haskell>

<description of implementation>

10 Problem 88

<Problem description>

Example:
<example in lisp>

Example in Haskell:
<example in Haskell>

Solution:

<solution in haskell>

<description of implementation>

11 Problem 89

<Problem description>

Example:
<example in lisp>

Example in Haskell:
<example in Haskell>

Solution:

<solution in haskell>

<description of implementation>