# Graph colouring (2014/11/13)

Currently I am looking into register allocation using graph colouring (also see Chaitinâ€™s paper on the subject). The graph colouring problem also occurs when trying to colour countries in a political world map where adjacent countries must have different colours. I thought it would be interesting to try a minimal implementation in Scheme. Feel free to suggest improvements in the comment section below .

A (undirected) graph is usually represented by a list of nodes (vertices) and a list of edges. If there are no insular nodes, a list of edges is sufficient. In Scheme (here GNU Guile implementation) one can do this as follows.

'((b . a) (a . c) (d . c))

The graph can be visualised on the command line using Graphviz and ImageMagick as follows.

echo 'graph test { b -- a; a -- c; d -- c; }' | dot -Tpng | display -

The following helper function *graphviz* uses a system call to display a graph from within the REPL.

(define (dot graph colors) (apply string-append (append (list "graph g {") (map (lambda (color) (format #f " ~a [style=filled, fillcolor=~a];" (car color) (cdr color))) colors) (map (lambda (edge) (format #f " ~a -- ~a;" (car edge) (cdr edge))) graph) (list " }")))) (define (graphviz graph colors) (system (format #f "echo '~a' | dot -Tpng | display -" (dot graph colors)))) (graphviz '((b . a) (a . c) (d . c)) '())

One can get the nodes of the graph by extracting all elements and
suppressing any duplicates. The definition of delete-duplicates is part of SRFI-1
(*Scheme Request for Implementation 1*).

(use-modules (srfi srfi-1)) (define (nodes graph) (delete-duplicates (append (map car graph) (map cdr graph)))) (nodes '((b . a) (a . c) (d . c))) ; (b a d c)

The adjacency list of a node is simply the list of nodes of the sub-graph obtained by filtering for edges connecting to this node.

(use-modules (ice-9 curried-definitions)) (define ((has-node? node) edge) (or (eq? (car edge) node) (eq? (cdr edge) node))) (define (adjacent graph node) (nodes (filter (has-node? node) graph))) (adjacent '((b . a) (a . c) (d . c)) 'c) ; (a d c)

Chaitinâ€™s graph coloring algorithm works by successively removing nodes with a low adjacency count from the graph. Removing a node from our graph can be done as follows.

(define (remove-node graph node) (filter (compose not (has-node? node)) graph)) (remove-node '((b . a) (a . c) (d . c)) 'c) ; ((b . a))

Using the *argument of the minimum* one can determine the node with lowest adjacency count.

(define (argmin fun lst) (let* [(vals (map fun lst)) (minval (apply min vals))] (list-ref lst (- (length lst) (length (member minval vals))))))

Now one can recursively remove the node with lowest adjacency count and then assign colours starting with the last node and working backwards. If an adjacent node has a colour already, another colour must be used.

(use-modules (srfi srfi-26)) (define (assign-colors graph nodes colors) (if (null? nodes) '() (let* [(target (argmin (compose length (cut adjacent graph <>)) nodes)) (coloring (assign-colors (remove-node graph target) (delete target nodes) colors)) (blocked (map (cut assq-ref coloring <>) (adjacent graph target))) (available (lset-difference eq? colors blocked))] (cons (cons target (car available)) coloring)))) (define (coloring graph colors) (assign-colors graph (nodes graph) colors)) (coloring '((b . a) (a . c) (d . c)) '(red green blue)) ; ((b . red) (a . green) (d . green) (c . red)) (let [(graph '((b . a) (a . c) (d . c)))] (graphviz graph (coloring graph '(red green blue))))

And here is an example of coloring a graph with a few more nodes.

(let [(graph '((run . intr) (intr . runbl) (runbl . run) (run . kernel) (kernel . zombie) (kernel . sleep) (kernel . runmem) (sleep . swap) (swap . runswap) (runswap . new) (runswap . runmem) (new . runmem) (sleep . runmem)))] (graphviz graph (coloring graph '(red green blue yellow))))