Graph colouring: Difference between revisions
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colors=. (#y)#a: |
colors=. (#y)#a: |
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for_node. |
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color=. <{.(-.~ [:i.1+#) ~.;node#colors |
color=. <{.(-.~ [:i.1+#) ~.;node#colors |
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colors=. color node_index} colors |
colors=. color node_index} colors |
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Revision as of 15:25, 3 February 2022
You are encouraged to solve this task according to the task description, using any language you may know.
A Graph is a collection of nodes
(or vertices), connected by edges (or not).
Nodes directly connected by edges are called neighbours.
In our representation of graphs, nodes are numbered and edges are represented by the two node numbers connected by the edge separated by a dash. Edges define the nodes being connected. Only unconnected nodes need a separate description.
For example,
0-1 1-2 2-0 3
Describes the following graph. Note that node 3 has no neighbours
- Example graph
+---+ | 3 | +---+ +-------------------+ | | +---+ +---+ +---+ | 0 | --- | 1 | --- | 2 | +---+ +---+ +---+
A useful internal datastructure for a graph and for later graph algorithms is as a mapping between each node and the set/list of its neighbours.
In the above example:
0 maps-to 1 and 2 1 maps to 2 and 0 2 maps-to 1 and 0 3 maps-to <nothing>
- Graph colouring task
Colour the vertices of a given graph so that no edge is between verticies of the same colour.
- Integers may be used to denote different colours.
- Algorithm should do better than just assigning each vertex a separate colour. The idea is to minimise the number of colours used, although no algorithm short of exhaustive search for the minimum is known at present, (and exhaustive search is not a requirement).
- Show for each edge, the colours assigned on each vertex.
- Show the total number of nodes, edges, and colours used for each graph.
- Use the following graphs
- Ex1
0-1 1-2 2-0 3
+---+ | 3 | +---+ +-------------------+ | | +---+ +---+ +---+ | 0 | --- | 1 | --- | 2 | +---+ +---+ +---+
- Ex2
The wp articles left-side graph
1-6 1-7 1-8 2-5 2-7 2-8 3-5 3-6 3-8 4-5 4-6 4-7
+----------------------------------+
| |
| +---+ |
| +-----------------| 3 | ------+----+
| | +---+ | |
| | | | |
| | | | |
| | | | |
| +---+ +---+ +---+ +---+ |
| | 8 | --- | 1 | --- | 6 | --- | 4 | |
| +---+ +---+ +---+ +---+ |
| | | | |
| | | | |
| | | | |
| | +---+ +---+ +---+ |
+----+------ | 7 | --- | 2 | --- | 5 | -+
| +---+ +---+ +---+
| |
+-------------------+
- Ex3
The wp articles right-side graph which is the same graph as Ex2, but with different node orderings and namings.
1-4 1-6 1-8 3-2 3-6 3-8 5-2 5-4 5-8 7-2 7-4 7-6
+----------------------------------+
| |
| +---+ |
| +-----------------| 5 | ------+----+
| | +---+ | |
| | | | |
| | | | |
| | | | |
| +---+ +---+ +---+ +---+ |
| | 8 | --- | 1 | --- | 4 | --- | 7 | |
| +---+ +---+ +---+ +---+ |
| | | | |
| | | | |
| | | | |
| | +---+ +---+ +---+ |
+----+------ | 6 | --- | 3 | --- | 2 | -+
| +---+ +---+ +---+
| |
+-------------------+
- Ex4
This is the same graph, node naming, and edge order as Ex2 except some of the edges x-y are flipped to y-x. This might alter the node order used in the greedy algorithm leading to differing numbers of colours.
1-6 7-1 8-1 5-2 2-7 2-8 3-5 6-3 3-8 4-5 4-6 4-7
+-------------------------------------------------+
| |
| |
+-------------------+---------+ |
| | | |
+---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+
| 4 | --- | 5 | --- | 2 | --- | 7 | --- | 1 | --- | 6 | --- | 3 | --- | 8 |
+---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+
| | | | | |
+---------+-----------------------------+---------+ | |
| | | |
| | | |
+-----------------------------+-------------------+ |
| |
| |
+-----------------------------+
- References
- Greedy coloring Wikipedia.
- Graph Coloring : Greedy Algorithm & Welsh Powell Algorithm by Priyank Jain.
11l
<lang 11l>F first_avail_int(data)
‘return lowest int 0... not in data’
V d = Set(data)
L(i) 0..
I i !C d
R i
F greedy_colour(name, connections)
DefaultDict[Int, [Int]] graph
L(connection) connections.split(‘ ’)
I ‘-’ C connection
V (n1, n2) = connection.split(‘-’).map(Int)
graph[n1].append(n2)
graph[n2].append(n1)
E
graph[Int(connection)] = [Int]()
// Greedy colourisation algo
V order = sorted(graph.keys())
[Int = Int] colour
V neighbours = graph
L(node) order
V used_neighbour_colours = neighbours[node].filter(nbr -> nbr C @colour).map(nbr -> @colour[nbr])
colour[node] = first_avail_int(used_neighbour_colours)
print("\n"name)
V canonical_edges = Set[(Int, Int)]()
L(n1, neighbours) sorted(graph.items())
I !neighbours.empty
L(n2) neighbours
V edge = tuple_sorted((n1, n2))
I edge !C canonical_edges
print(‘ #.-#.: Colour: #., #.’.format(n1, n2, colour[n1], colour[n2]))
canonical_edges.add(edge)
E
print(‘ #.: Colour: #.’.format(n1, colour[n1]))
V lc = Set(colour.values()).len
print(" #Nodes: #.\n #Edges: #.\n #Colours: #.".format(colour.len, canonical_edges.len, lc))
L(name, connections) [
(‘Ex1’, ‘0-1 1-2 2-0 3’),
(‘Ex2’, ‘1-6 1-7 1-8 2-5 2-7 2-8 3-5 3-6 3-8 4-5 4-6 4-7’),
(‘Ex3’, ‘1-4 1-6 1-8 3-2 3-6 3-8 5-2 5-4 5-8 7-2 7-4 7-6’),
(‘Ex4’, ‘1-6 7-1 8-1 5-2 2-7 2-8 3-5 6-3 3-8 4-5 4-6 4-7’)]
greedy_colour(name, connections)</lang>
- Output:
Ex1
0-1: Colour: 0, 1
0-2: Colour: 0, 2
1-2: Colour: 1, 2
3: Colour: 0
#Nodes: 4
#Edges: 3
#Colours: 3
Ex2
1-6: Colour: 0, 1
1-7: Colour: 0, 1
1-8: Colour: 0, 1
2-5: Colour: 0, 1
2-7: Colour: 0, 1
2-8: Colour: 0, 1
3-5: Colour: 0, 1
3-6: Colour: 0, 1
3-8: Colour: 0, 1
4-5: Colour: 0, 1
4-6: Colour: 0, 1
4-7: Colour: 0, 1
#Nodes: 8
#Edges: 12
#Colours: 2
Ex3
1-4: Colour: 0, 1
1-6: Colour: 0, 2
1-8: Colour: 0, 3
2-3: Colour: 0, 1
2-5: Colour: 0, 2
2-7: Colour: 0, 3
3-6: Colour: 1, 2
3-8: Colour: 1, 3
4-5: Colour: 1, 2
4-7: Colour: 1, 3
5-8: Colour: 2, 3
6-7: Colour: 2, 3
#Nodes: 8
#Edges: 12
#Colours: 4
Ex4
1-6: Colour: 0, 1
1-7: Colour: 0, 1
1-8: Colour: 0, 1
2-5: Colour: 0, 1
2-7: Colour: 0, 1
2-8: Colour: 0, 1
3-5: Colour: 0, 1
3-6: Colour: 0, 1
3-8: Colour: 0, 1
4-5: Colour: 0, 1
4-6: Colour: 0, 1
4-7: Colour: 0, 1
#Nodes: 8
#Edges: 12
#Colours: 2
Go
As mentioned in the task description, there is no known efficient algorithm which can guarantee that a minimum number of colors is used for a given graph. The following uses both the so-called 'greedy' algorithm (as described here) and the Welsh-Powell algorithm (as described here), suitably adjusted to the needs of this task.
The results are exactly the same for both algorithms. Whilst one would normally expect Welsh-Powell to give better results overall, the last three examples are not well suited to it as each node has exactly the same number of neighbors i.e. the valences are equal.
The results agree with the Python entry for examples 1 and 2 but, for example 3, Python gives 2 colors compared to my 4 and, for example 4, Python gives 3 colors compared to my 2. <lang go>package main
import (
"fmt" "sort"
)
type graph struct {
nn int // number of nodes st int // node numbering starts from nbr [][]int // neighbor list for each node
}
type nodeval struct {
n int // number of node v int // valence of node i.e. number of neighbors
}
func contains(s []int, n int) bool {
for _, e := range s {
if e == n {
return true
}
}
return false
}
func newGraph(nn, st int) graph {
nbr := make([][]int, nn)
return graph{nn, st, nbr}
}
// Note that this creates a single 'virtual' edge for an isolated node. func (g graph) addEdge(n1, n2 int) {
n1, n2 = n1-g.st, n2-g.st // adjust to starting node number
g.nbr[n1] = append(g.nbr[n1], n2)
if n1 != n2 {
g.nbr[n2] = append(g.nbr[n2], n1)
}
}
// Uses 'greedy' algorithm. func (g graph) greedyColoring() []int {
// create a slice with a color for each node, starting with color 0
cols := make([]int, g.nn) // all zero by default including the first node
for i := 1; i < g.nn; i++ {
cols[i] = -1 // mark all nodes after the first as having no color assigned (-1)
}
// create a bool slice to keep track of which colors are available
available := make([]bool, g.nn) // all false by default
// assign colors to all nodes after the first
for i := 1; i < g.nn; i++ {
// iterate through neighbors and mark their colors as available
for _, j := range g.nbr[i] {
if cols[j] != -1 {
available[cols[j]] = true
}
}
// find the first available color
c := 0
for ; c < g.nn; c++ {
if !available[c] {
break
}
}
cols[i] = c // assign it to the current node
// reset the neighbors' colors to unavailable
// before the next iteration
for _, j := range g.nbr[i] {
if cols[j] != -1 {
available[cols[j]] = false
}
}
}
return cols
}
// Uses Welsh-Powell algorithm. func (g graph) wpColoring() []int {
// create nodeval for each node
nvs := make([]nodeval, g.nn)
for i := 0; i < g.nn; i++ {
v := len(g.nbr[i])
if v == 1 && g.nbr[i][0] == i { // isolated node
v = 0
}
nvs[i] = nodeval{i, v}
}
// sort the nodevals in descending order by valence
sort.Slice(nvs, func(i, j int) bool {
return nvs[i].v > nvs[j].v
})
// create colors slice with entries for each node
cols := make([]int, g.nn)
for i := range cols {
cols[i] = -1 // set all nodes to no color (-1) initially
}
currCol := 0 // start with color 0
for f := 0; f < g.nn-1; f++ {
h := nvs[f].n
if cols[h] != -1 { // already assigned a color
continue
}
cols[h] = currCol
// assign same color to all subsequent uncolored nodes which are
// not connected to a previous colored one
outer:
for i := f + 1; i < g.nn; i++ {
j := nvs[i].n
if cols[j] != -1 { // already colored
continue
}
for k := f; k < i; k++ {
l := nvs[k].n
if cols[l] == -1 { // not yet colored
continue
}
if contains(g.nbr[j], l) {
continue outer // node j is connected to an earlier colored node
}
}
cols[j] = currCol
}
currCol++
}
return cols
}
func main() {
fns := [](func(graph) []int){graph.greedyColoring, graph.wpColoring}
titles := []string{"'Greedy'", "Welsh-Powell"}
nns := []int{4, 8, 8, 8}
starts := []int{0, 1, 1, 1}
edges1 := [][2]int{{0, 1}, {1, 2}, {2, 0}, {3, 3}}
edges2 := [][2]int{{1, 6}, {1, 7}, {1, 8}, {2, 5}, {2, 7}, {2, 8},
{3, 5}, {3, 6}, {3, 8}, {4, 5}, {4, 6}, {4, 7}}
edges3 := [][2]int{{1, 4}, {1, 6}, {1, 8}, {3, 2}, {3, 6}, {3, 8},
{5, 2}, {5, 4}, {5, 8}, {7, 2}, {7, 4}, {7, 6}}
edges4 := [][2]int{{1, 6}, {7, 1}, {8, 1}, {5, 2}, {2, 7}, {2, 8},
{3, 5}, {6, 3}, {3, 8}, {4, 5}, {4, 6}, {4, 7}}
for j, fn := range fns {
fmt.Println("Using the", titles[j], "algorithm:\n")
for i, edges := range [][][2]int{edges1, edges2, edges3, edges4} {
fmt.Println(" Example", i+1)
g := newGraph(nns[i], starts[i])
for _, e := range edges {
g.addEdge(e[0], e[1])
}
cols := fn(g)
ecount := 0 // counts edges
for _, e := range edges {
if e[0] != e[1] {
fmt.Printf(" Edge %d-%d -> Color %d, %d\n", e[0], e[1],
cols[e[0]-g.st], cols[e[1]-g.st])
ecount++
} else {
fmt.Printf(" Node %d -> Color %d\n", e[0], cols[e[0]-g.st])
}
}
maxCol := 0 // maximum color number used
for _, col := range cols {
if col > maxCol {
maxCol = col
}
}
fmt.Println(" Number of nodes :", nns[i])
fmt.Println(" Number of edges :", ecount)
fmt.Println(" Number of colors :", maxCol+1)
fmt.Println()
}
}
}</lang>
- Output:
Using the 'Greedy' algorithm:
Example 1
Edge 0-1 -> Color 0, 1
Edge 1-2 -> Color 1, 2
Edge 2-0 -> Color 2, 0
Node 3 -> Color 0
Number of nodes : 4
Number of edges : 3
Number of colors : 3
Example 2
Edge 1-6 -> Color 0, 1
Edge 1-7 -> Color 0, 1
Edge 1-8 -> Color 0, 1
Edge 2-5 -> Color 0, 1
Edge 2-7 -> Color 0, 1
Edge 2-8 -> Color 0, 1
Edge 3-5 -> Color 0, 1
Edge 3-6 -> Color 0, 1
Edge 3-8 -> Color 0, 1
Edge 4-5 -> Color 0, 1
Edge 4-6 -> Color 0, 1
Edge 4-7 -> Color 0, 1
Number of nodes : 8
Number of edges : 12
Number of colors : 2
Example 3
Edge 1-4 -> Color 0, 1
Edge 1-6 -> Color 0, 2
Edge 1-8 -> Color 0, 3
Edge 3-2 -> Color 1, 0
Edge 3-6 -> Color 1, 2
Edge 3-8 -> Color 1, 3
Edge 5-2 -> Color 2, 0
Edge 5-4 -> Color 2, 1
Edge 5-8 -> Color 2, 3
Edge 7-2 -> Color 3, 0
Edge 7-4 -> Color 3, 1
Edge 7-6 -> Color 3, 2
Number of nodes : 8
Number of edges : 12
Number of colors : 4
Example 4
Edge 1-6 -> Color 0, 1
Edge 7-1 -> Color 1, 0
Edge 8-1 -> Color 1, 0
Edge 5-2 -> Color 1, 0
Edge 2-7 -> Color 0, 1
Edge 2-8 -> Color 0, 1
Edge 3-5 -> Color 0, 1
Edge 6-3 -> Color 1, 0
Edge 3-8 -> Color 0, 1
Edge 4-5 -> Color 0, 1
Edge 4-6 -> Color 0, 1
Edge 4-7 -> Color 0, 1
Number of nodes : 8
Number of edges : 12
Number of colors : 2
Using the Welsh-Powell algorithm:
Example 1
Edge 0-1 -> Color 0, 1
Edge 1-2 -> Color 1, 2
Edge 2-0 -> Color 2, 0
Node 3 -> Color 0
Number of nodes : 4
Number of edges : 3
Number of colors : 3
Example 2
Edge 1-6 -> Color 0, 1
Edge 1-7 -> Color 0, 1
Edge 1-8 -> Color 0, 1
Edge 2-5 -> Color 0, 1
Edge 2-7 -> Color 0, 1
Edge 2-8 -> Color 0, 1
Edge 3-5 -> Color 0, 1
Edge 3-6 -> Color 0, 1
Edge 3-8 -> Color 0, 1
Edge 4-5 -> Color 0, 1
Edge 4-6 -> Color 0, 1
Edge 4-7 -> Color 0, 1
Number of nodes : 8
Number of edges : 12
Number of colors : 2
Example 3
Edge 1-4 -> Color 0, 1
Edge 1-6 -> Color 0, 2
Edge 1-8 -> Color 0, 3
Edge 3-2 -> Color 1, 0
Edge 3-6 -> Color 1, 2
Edge 3-8 -> Color 1, 3
Edge 5-2 -> Color 2, 0
Edge 5-4 -> Color 2, 1
Edge 5-8 -> Color 2, 3
Edge 7-2 -> Color 3, 0
Edge 7-4 -> Color 3, 1
Edge 7-6 -> Color 3, 2
Number of nodes : 8
Number of edges : 12
Number of colors : 4
Example 4
Edge 1-6 -> Color 0, 1
Edge 7-1 -> Color 1, 0
Edge 8-1 -> Color 1, 0
Edge 5-2 -> Color 1, 0
Edge 2-7 -> Color 0, 1
Edge 2-8 -> Color 0, 1
Edge 3-5 -> Color 0, 1
Edge 6-3 -> Color 1, 0
Edge 3-8 -> Color 0, 1
Edge 4-5 -> Color 0, 1
Edge 4-6 -> Color 0, 1
Edge 4-7 -> Color 0, 1
Number of nodes : 8
Number of edges : 12
Number of colors : 2
Haskell
<lang haskell>import Data.Maybe import Data.List import Control.Monad.State import qualified Data.Map as M import Text.Printf
-- Primitive graph representation
type Node = Int type Color = Int type Graph = M.Map Node [Node]
nodes :: Graph -> [Node] nodes = M.keys
adjacentNodes :: Graph -> Node -> [Node] adjacentNodes g n = fromMaybe [] $ M.lookup n g
degree :: Graph -> Node -> Int degree g = length . adjacentNodes g
fromList :: [(Node, [Node])] -> Graph fromList = foldr add M.empty
where add (a, bs) g = foldr (join [a]) (join bs a g) bs join = flip (M.insertWith (++))
readGraph :: String -> Graph readGraph = fromList . map interprete . words
where
interprete s = case span (/= '-') s of
(a, "") -> (read a, [])
(a, '-':b) -> (read a, [read b])
-- Graph coloring functions
uncolored :: Node -> State [(Node, Color)] Bool uncolored n = isNothing <$> colorOf n
colorOf :: Node -> State [(Node, Color)] (Maybe Color) colorOf n = gets (lookup n)
greedyColoring :: Graph -> [(Node, Color)] greedyColoring g = mapM_ go (nodes g) `execState` []
where
go n = do
c <- colorOf n
when (isNothing c) $ do
adjacentColors <- nub . catMaybes <$> mapM colorOf (adjacentNodes g n)
let newColor = head $ filter (`notElem` adjacentColors) [1..]
modify ((n, newColor) :)
filterM uncolored (adjacentNodes g n) >>= mapM_ go
wpColoring :: Graph -> [(Node, Color)] wpColoring g = go [1..] nodesList `execState` []
where nodesList = sortOn (negate . degree g) (nodes g)
go _ [] = pure ()
go (c:cs) ns = do
mark c ns
filterM uncolored ns >>= go cs
mark c [] = pure () :: State [(Node, Color)] ()
mark c (n:ns) = do
modify ((n, c) :)
mark c (filter (`notElem` adjacentNodes g n) ns)</lang>
Simple usage examples:
λ> let g = fromList [(1,[2,3]),(2,[3,4]),(4,[3,5])] λ> g fromList [(1,[2,3]),(2,[1,3,4]),(3,[1,2,4]),(4,[2,3,5]),(5,[4])] λ> greedyColoring g [(5,2),(4,1),(3,3),(2,2),(1,1)] λ> wpColoring g [(1,3),(4,3),(3,2),(5,1),(2,1)]
The task: <lang haskell>ex1 = "0-1 1-2 2-0 3" ex2 = "1-6 1-7 1-8 2-5 2-7 2-8 3-5 3-6 3-8 4-5 4-6 4-7" ex3 = "1-4 1-6 1-8 3-2 3-6 3-8 5-2 5-4 5-8 7-2 7-4 7-6" ex4 = "1-6 7-1 8-1 5-2 2-7 2-8 3-5 6-3 3-8 4-5 4-6 4-7"
task method ex =
let g = readGraph ex
cs = sortOn fst $ method g
color n = fromJust $ lookup n cs
ns = nodes g
mkLine n = printf "%d\t%d\t%s\n" n (color n) (show (color <$> adjacentNodes g n))
in do
print ex
putStrLn $ "nodes:\t" ++ show ns
putStrLn $ "colors:\t" ++ show (nub (snd <$> cs))
putStrLn "node\tcolor\tadjacent colors"
mapM_ mkLine ns
putStrLn ""</lang>
Runing greedy algorithm:
λ> mapM_ (task greedyColoring) [ex1,ex2,ex3,ex4] "0-1 1-2 2-0 3" nodes: [0,1,2,3] colors: [1,2,3] node color adjacent colors 0 1 [2,3] 1 2 [1,3] 2 3 [2,1] 3 1 [] "1-6 1-7 1-8 2-5 2-7 2-8 3-5 3-6 3-8 4-5 4-6 4-7" nodes: [1,2,3,4,5,6,7,8] colors: [1,2] node color adjacent colors 1 1 [2,2,2] 2 1 [2,2,2] 3 1 [2,2,2] 4 1 [2,2,2] 5 2 [1,1,1] 6 2 [1,1,1] 7 2 [1,1,1] 8 2 [1,1,1] "1-4 1-6 1-8 3-2 3-6 3-8 5-2 5-4 5-8 7-2 7-4 7-6" nodes: [1,2,3,4,5,6,7,8] colors: [1,2] node color adjacent colors 1 1 [2,2,2] 2 2 [1,1,1] 3 1 [2,2,2] 4 2 [1,1,1] 5 1 [2,2,2] 6 2 [1,1,1] 7 1 [2,2,2] 8 2 [1,1,1] "1-6 7-1 8-1 5-2 2-7 2-8 3-5 6-3 3-8 4-5 4-6 4-7" nodes: [1,2,3,4,5,6,7,8] colors: [1,2] node color adjacent colors 1 1 [2,2,2] 2 1 [2,2,2] 3 1 [2,2,2] 4 1 [2,2,2] 5 2 [1,1,1] 6 2 [1,1,1] 7 2 [1,1,1] 8 2 [1,1,1]
Runing Welsh-Powell algorithm:
λ> mapM_ (task wpColoring) [ex1,ex2,ex3,ex4] "0-1 1-2 2-0 3" nodes: [0,1,2,3] colors: [1,2,3] node color adjacent colors 0 1 [2,3] 1 2 [1,3] 2 3 [2,1] 3 1 [] "1-6 1-7 1-8 2-5 2-7 2-8 3-5 3-6 3-8 4-5 4-6 4-7" nodes: [1,2,3,4,5,6,7,8] colors: [1,2] node color adjacent colors 1 1 [2,2,2] 2 1 [2,2,2] 3 1 [2,2,2] 4 1 [2,2,2] 5 2 [1,1,1] 6 2 [1,1,1] 7 2 [1,1,1] 8 2 [1,1,1] "1-4 1-6 1-8 3-2 3-6 3-8 5-2 5-4 5-8 7-2 7-4 7-6" nodes: [1,2,3,4,5,6,7,8] colors: [1,2,3,4] node color adjacent colors 1 1 [2,3,4] 2 1 [2,3,4] 3 2 [1,3,4] 4 2 [1,3,4] 5 3 [1,2,4] 6 3 [1,2,4] 7 4 [1,2,3] 8 4 [1,2,3] "1-6 7-1 8-1 5-2 2-7 2-8 3-5 6-3 3-8 4-5 4-6 4-7" nodes: [1,2,3,4,5,6,7,8] colors: [1,2] node color adjacent colors 1 1 [2,2,2] 2 1 [2,2,2] 3 1 [2,2,2] 4 1 [2,2,2] 5 2 [1,1,1] 6 2 [1,1,1] 7 2 [1,1,1] 8 2 [1,1,1]
J
Greedy coloring satisfies the task requirements. <lang J>parse=: {{
ref=: 2$L:1(;:each cut y) -.L:1 ;:'-' labels=: /:~~.;ref sparse=. (*:#labels) > 20*#ref graph=: (+.|:) 1 (labels i.L:1 ref)} $.^:sparse 0$~2##labels
}}
greedy=: Template:Colors=. (</lang>
To display colors for edges, we show the graph specification and beneath the representation of each edge A-B we show colorA:colorB.
<lang J> ex1=:'0-1 1-2 2-0 3' ex2=:'1-6 1-7 1-8 2-5 2-7 2-8 3-5 3-6 3-8 4-5 4-6 4-7' ex3=:'1-4 1-6 1-8 3-2 3-6 3-8 5-2 5-4 5-8 7-2 7-4 7-6' ex4=:'1-6 7-1 8-1 5-2 2-7 2-8 3-5 6-3 3-8 4-5 4-6 4-7' require'format/printf' require'format/printf' task=: {{
colors=. greedy parse y
echo'nodes: %d, edges: %d, colours: %d' sprintf #each graph;ref;~.colors
edgecolors=. <@(,&":':',&":])/"1(>labels i.L:1 ref){colors
,./>' ',.~each^:2(cut y),:each edgecolors
}}
task ex1
nodes: 4, edges: 4, colours: 3 0-1 1-2 2-0 3 0:1 1:2 2:0 0:0
task ex2
nodes: 8, edges: 12, colours: 2 1-6 1-7 1-8 2-5 2-7 2-8 3-5 3-6 3-8 4-5 4-6 4-7 0:1 0:1 0:1 0:1 0:1 0:1 0:1 0:1 0:1 0:1 0:1 0:1
task ex3
nodes: 8, edges: 12, colours: 4 1-4 1-6 1-8 3-2 3-6 3-8 5-2 5-4 5-8 7-2 7-4 7-6 0:1 0:2 0:3 1:0 1:2 1:3 2:0 2:1 2:3 3:0 3:1 3:2
task ex4
nodes: 8, edges: 12, colours: 2 1-6 7-1 8-1 5-2 2-7 2-8 3-5 6-3 3-8 4-5 4-6 4-7 0:1 1:0 1:0 1:0 0:1 0:1 0:1 1:0 0:1 0:1 0:1 0:1 </lang>
Julia
Uses a repeated randomization of node color ordering to seek a minimum number of colors needed. <lang julia>using Random
"""Useful constants for the colors to be selected for nodes of the graph""" const colors4 = ["blue", "red", "green", "yellow"] const badcolor = "black" @assert(!(badcolor in colors4))
"""
struct graph
undirected simple graph constructed from its name and a string listing of point to point connections """ mutable struct Graph
name::String
g::Dict{Int, Vector{Int}}
nodecolor::Dict{Int, String}
function Graph(nam::String, s::String)
gdic = Dict{Int, Vector{Int}}()
for p in eachmatch(r"(\d+)-(\d+)|(\d+)(?!\s*-)" , s)
if p != nothing
if p[3] != nothing
n3 = parse(Int, p[3])
get!(gdic, n3, [])
else
n1, n2 = parse(Int, p[1]), parse(Int, p[2])
p1vec = get!(gdic, n1, [])
!(n2 in p1vec) && push!(p1vec, n2)
p2vec = get!(gdic, n2, [])
!(n1 in p2vec) && push!(p2vec, n1)
end
end
end
new(nam, gdic, Dict{Int, String}())
end
end
"""
tryNcolors!(gr::Graph, N, maxtrials)
Try up to maxtrials to get a coloring with <= N colors """ function tryNcolors!(gr::Graph, N, maxtrials)
t, mintrial, minord = N, N + 1, Dict()
for _ in 1:maxtrials
empty!(gr.nodecolor)
ordering = shuffle(collect(keys(gr.g)))
for node in ordering
usedneighborcolors = [gr.nodecolor[c] for c in gr.g[node] if haskey(gr.nodecolor, c)]
gr.nodecolor[node] = badcolor
for c in colors4[1:N]
if !(c in usedneighborcolors)
gr.nodecolor[node] = c
break
end
end
end
t = length(unique(values(gr.nodecolor)))
if t < mintrial
mintrial = t
minord = deepcopy(gr.nodecolor)
end
end
if length(minord) > 0
gr.nodecolor = minord
end
end
"""
prettyprintcolors(gr::graph)
print out the colored nodes in graph """ function prettyprintcolors(gr::Graph)
println("\nColors for the graph named ", gr.name, ":")
edgesdone = Vector{Vector{Int}}()
for (node, neighbors) in gr.g
if !isempty(neighbors)
for n in neighbors
edge = node < n ? [node, n] : [n, node]
if !(edge in edgesdone)
println(" ", edge[1], "-", edge[2], " Color: ",
gr.nodecolor[edge[1]], ", ", gr.nodecolor[edge[2]])
push!(edgesdone, edge)
end
end
else
println(" ", node, ": ", gr.nodecolor[node])
end
end
println("\n", length(unique(keys(gr.nodecolor))), " nodes, ",
length(edgesdone), " edges, ",
length(unique(values(gr.nodecolor))), " colors.")
end
for (name, txt) in [("Ex1", "0-1 1-2 2-0 3"),
("Ex2", "1-6 1-7 1-8 2-5 2-7 2-8 3-5 3-6 3-8 4-5 4-6 4-7"),
("Ex3", "1-4 1-6 1-8 3-2 3-6 3-8 5-2 5-4 5-8 7-2 7-4 7-6"),
("Ex4", "1-6 7-1 8-1 5-2 2-7 2-8 3-5 6-3 3-8 4-5 4-6 4-7")]
exgraph = Graph(name, txt)
tryNcolors!(exgraph, 4, 100)
prettyprintcolors(exgraph)
end
</lang>
- Output:
Colors for the graph named Ex1:
0-1 Color: red, blue
0-2 Color: red, green
1-2 Color: blue, green
3: blue
4 nodes, 3 edges, 3 colors.
Colors for the graph named Ex2:
1-7 Color: blue, red
2-7 Color: blue, red
4-7 Color: blue, red
4-5 Color: blue, red
4-6 Color: blue, red
2-5 Color: blue, red
2-8 Color: blue, red
3-5 Color: blue, red
3-6 Color: blue, red
3-8 Color: blue, red
1-8 Color: blue, red
1-6 Color: blue, red
8 nodes, 12 edges, 2 colors.
Colors for the graph named Ex3:
2-7 Color: red, blue
4-7 Color: red, blue
6-7 Color: red, blue
1-4 Color: blue, red
4-5 Color: red, blue
2-3 Color: red, blue
2-5 Color: red, blue
3-6 Color: blue, red
3-8 Color: blue, red
1-8 Color: blue, red
5-8 Color: blue, red
1-6 Color: blue, red
8 nodes, 12 edges, 2 colors.
Colors for the graph named Ex4:
1-7 Color: blue, red
2-7 Color: blue, red
4-7 Color: blue, red
4-5 Color: blue, red
4-6 Color: blue, red
2-5 Color: blue, red
2-8 Color: blue, red
3-5 Color: blue, red
3-6 Color: blue, red
3-8 Color: blue, red
1-8 Color: blue, red
1-6 Color: blue, red
8 nodes, 12 edges, 2 colors.
Mathematica / Wolfram Language
<lang Mathematica>ClearAll[ColorGroups] ColorGroups[g_Graph] := Module[{h, cols, indset, diffcols},
h = g;
cols = {};
While[! EmptyGraphQ[h], maxd = RandomChoice[VertexList[h]];
indset = Flatten[FindIndependentVertexSet[{h, maxd}]];
AppendTo[cols, indset];
h = VertexDelete[h, indset];
];
AppendTo[cols, VertexList[h]];
diffcols = Length[cols];
cols = Catenate[Map[Thread][Rule @@@ Transpose[{cols, Range[Length[cols]]}]]];
Print[Column[Row[{"Edge ", #1, " to ", #2, " has colors ", #1 /. cols, " and ", #2 /. cols }] & @@@ EdgeList[g]]];
Print[Grid[{{"Vertices: ", VertexCount[g]}, {"Edges: ", EdgeCount[g]}, {"Colors used: ", diffcols}}]]
]
ColorGroups[Graph[Range[0, 3], {0 \[UndirectedEdge] 1, 1 \[UndirectedEdge] 2,
2 \[UndirectedEdge] 0}]]
ColorGroups[Graph[Range[8], {1 \[UndirectedEdge] 6, 1 \[UndirectedEdge] 7,
1 \[UndirectedEdge] 8, 2 \[UndirectedEdge] 5, 2 \[UndirectedEdge] 7, 2 \[UndirectedEdge] 8, 3 \[UndirectedEdge] 5, 3 \[UndirectedEdge] 6, 3 \[UndirectedEdge] 8, 4 \[UndirectedEdge] 5, 4 \[UndirectedEdge] 6, 4 \[UndirectedEdge] 7}]]
ColorGroups[Graph[Range[8], {1 \[UndirectedEdge] 4, 1 \[UndirectedEdge] 6,
1 \[UndirectedEdge] 8, 3 \[UndirectedEdge] 2, 3 \[UndirectedEdge] 6, 3 \[UndirectedEdge] 8, 5 \[UndirectedEdge] 2, 5 \[UndirectedEdge] 4, 5 \[UndirectedEdge] 8, 7 \[UndirectedEdge] 2, 7 \[UndirectedEdge] 4, 7 \[UndirectedEdge] 6}]]
ColorGroups[Graph[Range[8], {1 \[UndirectedEdge] 6, 7 \[UndirectedEdge] 1,
8 \[UndirectedEdge] 1, 5 \[UndirectedEdge] 2, 2 \[UndirectedEdge] 7, 2 \[UndirectedEdge] 8, 3 \[UndirectedEdge] 5, 6 \[UndirectedEdge] 3, 3 \[UndirectedEdge] 8, 4 \[UndirectedEdge] 5, 4 \[UndirectedEdge] 6, 4 \[UndirectedEdge] 7}]]</lang>
- Output:
Edge 0 to 1 has colors 1 and 3 Edge 1 to 2 has colors 3 and 2 Edge 2 to 0 has colors 2 and 1 Vertices: 4 Edges: 3 Colors used: 3 Edge 1 to 6 has colors 1 and 2 Edge 1 to 7 has colors 1 and 2 Edge 1 to 8 has colors 1 and 2 Edge 2 to 5 has colors 1 and 2 Edge 2 to 7 has colors 1 and 2 Edge 2 to 8 has colors 1 and 2 Edge 3 to 5 has colors 1 and 2 Edge 3 to 6 has colors 1 and 2 Edge 3 to 8 has colors 1 and 2 Edge 4 to 5 has colors 1 and 2 Edge 4 to 6 has colors 1 and 2 Edge 4 to 7 has colors 1 and 2 Vertices: 8 Edges: 12 Colors used: 2 Edge 1 to 4 has colors 1 and 2 Edge 1 to 6 has colors 1 and 2 Edge 1 to 8 has colors 1 and 2 Edge 3 to 2 has colors 1 and 2 Edge 3 to 6 has colors 1 and 2 Edge 3 to 8 has colors 1 and 2 Edge 5 to 2 has colors 1 and 2 Edge 5 to 4 has colors 1 and 2 Edge 5 to 8 has colors 1 and 2 Edge 7 to 2 has colors 1 and 2 Edge 7 to 4 has colors 1 and 2 Edge 7 to 6 has colors 1 and 2 Vertices: 8 Edges: 12 Colors used: 2 Edge 1 to 6 has colors 2 and 1 Edge 7 to 1 has colors 1 and 2 Edge 8 to 1 has colors 1 and 2 Edge 5 to 2 has colors 1 and 2 Edge 2 to 7 has colors 2 and 1 Edge 2 to 8 has colors 2 and 1 Edge 3 to 5 has colors 2 and 1 Edge 6 to 3 has colors 1 and 2 Edge 3 to 8 has colors 2 and 1 Edge 4 to 5 has colors 2 and 1 Edge 4 to 6 has colors 2 and 1 Edge 4 to 7 has colors 2 and 1 Vertices: 8 Edges: 12 Colors used: 2
Nim
We use the “DSatur” algorithm described here: https://en.wikipedia.org/wiki/DSatur.
For the four examples, it gives the minimal number of colors.
<lang Nim>import algorithm, sequtils, strscans, strutils, tables
const NoColor = 0
type
Color = range[0..63]
Node = ref object num: Natural # Node number. color: Color # Node color. degree: Natural # Node degree. dsat: Natural # Node Dsaturation. neighbors: seq[Node] # List of neighbors.
Graph = seq[Node] # List of nodes ordered by number.
- ---------------------------------------------------------------------------------------------------
proc initGraph(graphRepr: string): Graph =
## Initialize the graph from its string representation.
var mapping: Table[Natural, Node] # Temporary mapping.
for elem in graphRepr.splitWhitespace():
var num1, num2: int
if elem.scanf("$i-$i", num1, num2):
let node1 = mapping.mgetOrPut(num1, Node(num: num1))
let node2 = mapping.mgetOrPut(num2, Node(num: num2))
node1.neighbors.add node2
node2.neighbors.add node1
elif elem.scanf("$i", num1):
discard mapping.mgetOrPut(num1, Node(num: num1))
else:
raise newException(ValueError, "wrong description: " & elem)
for node in mapping.values: node.degree = node.neighbors.len result = sortedByIt(toSeq(mapping.values), it.num)
- ---------------------------------------------------------------------------------------------------
proc numbers(nodes: seq[Node]): string =
## Return the numbers of a list of nodes.
for node in nodes:
result.addSep(" ")
result.add($node.num)
- ---------------------------------------------------------------------------------------------------
proc `$`(graph: Graph): string =
## Return the description of the graph.
var maxColor = NoColor
for node in graph:
stdout.write "Node ", node.num, ": color = ", node.color
if node.color > maxColor: maxColor = node.color
if node.neighbors.len > 0:
echo " neighbors = ", sortedByIt(node.neighbors, it.num).numbers
else:
echo ""
echo "Number of colors: ", maxColor
- ---------------------------------------------------------------------------------------------------
proc `<`(node1, node2: Node): bool =
## Comparison of nodes, by dsaturation first, then by degree. if node1.dsat == node2.dsat: node1.degree < node2.degree else: node1.dsat < node2.dsat
- ---------------------------------------------------------------------------------------------------
proc getMaxDsatNode(nodes: var seq[Node]): Node =
## Return the node with the greatest dsaturation. let idx = nodes.maxIndex() result = nodes[idx] nodes.delete(idx)
- ---------------------------------------------------------------------------------------------------
proc minColor(node: Node): COLOR =
## Return the minimum available color for a node. var colorUsed: set[Color] for neighbor in node.neighbors: colorUsed.incl neighbor.color for color in 1..Color.high: if color notin colorUsed: return color
- ---------------------------------------------------------------------------------------------------
proc distinctColorsCount(node: Node): Natural =
## Return the number of distinct colors of the neighbors of a node. var colorUsed: set[Color] for neighbor in node.neighbors: colorUsed.incl neighbor.color result = colorUsed.card
- ---------------------------------------------------------------------------------------------------
proc updateDsats(node: Node) =
## Update the dsaturations of the neighbors of a node.
for neighbor in node.neighbors:
if neighbor.color == NoColor:
neighbor.dsat = neighbor.distinctColorsCount()
- ---------------------------------------------------------------------------------------------------
proc colorize(graphRepr: string) =
## Colorize a graph.
let graph = initGraph(graphRepr) var nodes = sortedByIt(graph, -it.degree) # Copy or graph sorted by decreasing degrees. while nodes.len > 0: let node = nodes.getMaxDsatNode() node.color = node.minColor() node.updateDsats()
echo "Graph: ", graphRepr echo graph
- ---------------------------------------------------------------------------------------------------
when isMainModule:
colorize("0-1 1-2 2-0 3")
colorize("1-6 1-7 1-8 2-5 2-7 2-8 3-5 3-6 3-8 4-5 4-6 4-7")
colorize("1-4 1-6 1-8 3-2 3-6 3-8 5-2 5-4 5-8 7-2 7-4 7-6")
colorize("1-6 7-1 8-1 5-2 2-7 2-8 3-5 6-3 3-8 4-5 4-6 4-7")</lang>
- Output:
Graph: 0-1 1-2 2-0 3 Node 0: color = 1 neighbors = 1 2 Node 1: color = 2 neighbors = 0 2 Node 2: color = 3 neighbors = 0 1 Node 3: color = 1 Number of colors: 3 Graph: 1-6 1-7 1-8 2-5 2-7 2-8 3-5 3-6 3-8 4-5 4-6 4-7 Node 1: color = 1 neighbors = 6 7 8 Node 2: color = 1 neighbors = 5 7 8 Node 3: color = 1 neighbors = 5 6 8 Node 4: color = 1 neighbors = 5 6 7 Node 5: color = 2 neighbors = 2 3 4 Node 6: color = 2 neighbors = 1 3 4 Node 7: color = 2 neighbors = 1 2 4 Node 8: color = 2 neighbors = 1 2 3 Number of colors: 2 Graph: 1-4 1-6 1-8 3-2 3-6 3-8 5-2 5-4 5-8 7-2 7-4 7-6 Node 1: color = 1 neighbors = 4 6 8 Node 2: color = 2 neighbors = 3 5 7 Node 3: color = 1 neighbors = 2 6 8 Node 4: color = 2 neighbors = 1 5 7 Node 5: color = 1 neighbors = 2 4 8 Node 6: color = 2 neighbors = 1 3 7 Node 7: color = 1 neighbors = 2 4 6 Node 8: color = 2 neighbors = 1 3 5 Number of colors: 2 Graph: 1-6 7-1 8-1 5-2 2-7 2-8 3-5 6-3 3-8 4-5 4-6 4-7 Node 1: color = 1 neighbors = 6 7 8 Node 2: color = 1 neighbors = 5 7 8 Node 3: color = 1 neighbors = 5 6 8 Node 4: color = 1 neighbors = 5 6 7 Node 5: color = 2 neighbors = 2 3 4 Node 6: color = 2 neighbors = 1 3 4 Node 7: color = 2 neighbors = 1 2 4 Node 8: color = 2 neighbors = 1 2 3 Number of colors: 2
Perl
<lang perl>use strict; use warnings; no warnings 'uninitialized'; use feature 'say'; use constant True => 1; use List::Util qw(head uniq);
sub GraphNodeColor {
my(%OneMany, %NodeColor, %NodePool, @ColorPool); my(@data) = @_;
for (@data) {
my($a,$b) = @$_;
push @{$OneMany{$a}}, $b;
push @{$OneMany{$b}}, $a;
}
@ColorPool = 0 .. -1 + scalar %OneMany;
$NodePool{$_} = True for keys %OneMany;
if ($OneMany{}) { # skip islanders for now
delete $NodePool{$_} for @{$OneMany{}};
delete $NodePool{};
}
while (%NodePool) {
my $color = shift @ColorPool;
my %TempPool = %NodePool;
while (my $n = head 1, sort keys %TempPool) {
$NodeColor{$n} = $color;
delete $TempPool{$n};
delete $TempPool{$_} for @{$OneMany{$n}} ; # skip neighbors as well
delete $NodePool{$n};
}
if ($OneMany{}) { # islanders use an existing color
$NodeColor{$_} = head 1, sort values %NodeColor for @{$OneMany{}};
}
}
%NodeColor
}
my @DATA = (
[ [1,2],[2,3],[3,1],[4,undef],[5,undef],[6,undef] ], [ [1,6],[1,7],[1,8],[2,5],[2,7],[2,8],[3,5],[3,6],[3,8],[4,5],[4,6],[4,7] ], [ [1,4],[1,6],[1,8],[3,2],[3,6],[3,8],[5,2],[5,4],[5,8],[7,2],[7,4],[7,6] ], [ [1,6],[7,1],[8,1],[5,2],[2,7],[2,8],[3,5],[6,3],[3,8],[4,5],[4,6],[4,7] ]
);
for my $d (@DATA) {
my %result = GraphNodeColor @$d;
my($graph,$colors);
$graph .= '(' . join(' ', @$_) . '), ' for @$d;
$colors .= ' ' . $result{$$_[0]} . '-' . ($result{$$_[1]} // ) . ' ' for @$d;
say 'Graph : ' . $graph =~ s/,\s*$//r; say 'Colors : ' . $colors; say 'Nodes : ' . keys %result; say 'Edges : ' . @$d; say 'Unique : ' . uniq values %result; say ;
}</lang>
- Output:
Graph : (1 2), (2 3), (3 1), (4 ), (5 ), (6 ) Colors : 0-1 1-2 2-0 0- 0- 0- Nodes : 6 Edges : 6 Unique : 3 Graph : (1 6), (1 7), (1 8), (2 5), (2 7), (2 8), (3 5), (3 6), (3 8), (4 5), (4 6), (4 7) Colors : 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1 Nodes : 8 Edges : 12 Unique : 2 Graph : (1 4), (1 6), (1 8), (3 2), (3 6), (3 8), (5 2), (5 4), (5 8), (7 2), (7 4), (7 6) Colors : 0-1 0-2 0-3 1-0 1-2 1-3 2-0 2-1 2-3 3-0 3-1 3-2 Nodes : 8 Edges : 12 Unique : 4 Graph : (1 6), (7 1), (8 1), (5 2), (2 7), (2 8), (3 5), (6 3), (3 8), (4 5), (4 6), (4 7) Colors : 0-1 1-0 1-0 1-0 0-1 0-1 0-1 1-0 0-1 0-1 0-1 0-1 Nodes : 8 Edges : 12 Unique : 2
Phix
Exhaustive search, trims search space to < best so far, newused improves on unique().
Many more examples/testing would be needed before I would trust this the tiniest bit.
NB: As per talk page, when writing this I did not remotely imagine it might be used on over 400,000 nodes with over 3 million links...
-- demo\rosetta\Graph_colouring.exw with javascript_semantics constant tests = split(""" 0-1 1-2 2-0 3 1-6 1-7 1-8 2-5 2-7 2-8 3-5 3-6 3-8 4-5 4-6 4-7 1-4 1-6 1-8 3-2 3-6 3-8 5-2 5-4 5-8 7-2 7-4 7-6 1-6 7-1 8-1 5-2 2-7 2-8 3-5 6-3 3-8 4-5 4-6 4-7 ""","\n",true) function colour(sequence nodes, links, colours, soln, integer best, next, used=0) -- fill/try each colours[next], recursing as rqd and saving any improvements. -- nodes/links are read-only here, colours is the main workspace, soln/best are -- the results, next is 1..length(nodes), and used is length(unique(colours)). -- On really big graphs I might consider making nodes..best static, esp colours, -- in which case you will probably also want a "colours[next] = 0" reset below. integer c = 1 colours = deep_copy(colours) while c<best do bool avail = true for i=1 to length(links[next]) do if colours[links[next][i]]==c then avail = false exit end if end for if avail then colours[next] = c integer newused = used + (find(c,colours)==next) if next<length(nodes) then {best,soln} = colour(nodes,links,colours,soln,best,next+1,newused) elsif newused<best then {best,soln} = {newused,deep_copy(colours)} end if end if c += 1 end while return {best,soln} end function function add_node(sequence nodes, links, string n) integer rdx = find(n,nodes) if rdx=0 then nodes = append(nodes,n) links = append(links,{}) rdx = length(nodes) end if return {nodes, links, rdx} end function for t=1 to length(tests) do string tt = tests[t] sequence lt = split(tt," "), nodes = {}, links = {} integer linkcount = 0, left, right for l=1 to length(lt) do sequence ll = split(lt[l],"-") {nodes, links, left} = add_node(deep_copy(nodes),deep_copy(links),ll[1]) if length(ll)=2 then {nodes, links, right} = add_node(deep_copy(nodes),deep_copy(links),ll[2]) links[left] = deep_copy(links[left])&right links[right] = deep_copy(links[right])&left linkcount += 1 end if end for integer ln = length(nodes) printf(1,"test%d: %d nodes, %d edges, ",{t,ln,linkcount}) sequence colours = repeat(0,ln), soln = tagset(ln) -- fallback solution integer next = 1, best = ln printf(1,"%d colours:%v\n",colour(nodes,links,colours,soln,best,next)) end for
- Output:
test1: 4 nodes, 3 edges, 3 colours:{1,2,3,1}
test2: 8 nodes, 12 edges, 2 colours:{1,2,2,2,1,2,1,1}
test3: 8 nodes, 12 edges, 2 colours:{1,2,2,2,1,2,1,1}
test4: 8 nodes, 12 edges, 2 colours:{1,2,2,2,2,1,1,1}
Python
<lang python>import re from collections import defaultdict from itertools import count
connection_re = r"""
(?: (?P<N1>\d+) - (?P<N2>\d+) | (?P<N>\d+) (?!\s*-)) """
class Graph:
def __init__(self, name, connections):
self.name = name
self.connections = connections
g = self.graph = defaultdict(list) # maps vertex to direct connections
matches = re.finditer(connection_re, connections,
re.MULTILINE | re.VERBOSE)
for match in matches:
n1, n2, n = match.groups()
if n:
g[n] += []
else:
g[n1].append(n2) # Each the neighbour of the other
g[n2].append(n1)
def greedy_colour(self, order=None):
"Greedy colourisation algo."
if order is None:
order = self.graph # Choose something
colour = self.colour = {}
neighbours = self.graph
for node in order:
used_neighbour_colours = (colour[nbr] for nbr in neighbours[node]
if nbr in colour)
colour[node] = first_avail_int(used_neighbour_colours)
self.pp_colours()
return colour
def pp_colours(self):
print(f"\n{self.name}")
c = self.colour
e = canonical_edges = set()
for n1, neighbours in sorted(self.graph.items()):
if neighbours:
for n2 in neighbours:
edge = tuple(sorted([n1, n2]))
if edge not in canonical_edges:
print(f" {n1}-{n2}: Colour: {c[n1]}, {c[n2]}")
canonical_edges.add(edge)
else:
print(f" {n1}: Colour: {c[n1]}")
lc = len(set(c.values()))
print(f" #Nodes: {len(c)}\n #Edges: {len(e)}\n #Colours: {lc}")
def first_avail_int(data):
"return lowest int 0... not in data"
d = set(data)
for i in count():
if i not in d:
return i
if __name__ == '__main__':
for name, connections in [
('Ex1', "0-1 1-2 2-0 3"),
('Ex2', "1-6 1-7 1-8 2-5 2-7 2-8 3-5 3-6 3-8 4-5 4-6 4-7"),
('Ex3', "1-4 1-6 1-8 3-2 3-6 3-8 5-2 5-4 5-8 7-2 7-4 7-6"),
('Ex4', "1-6 7-1 8-1 5-2 2-7 2-8 3-5 6-3 3-8 4-5 4-6 4-7"),
]:
g = Graph(name, connections)
g.greedy_colour()</lang>
- Output:
Ex1
0-1: Colour: 0, 1
0-2: Colour: 0, 2
1-2: Colour: 1, 2
3: Colour: 0
#Nodes: 4
#Edges: 3
#Colours: 3
Ex2
1-6: Colour: 0, 1
1-7: Colour: 0, 1
1-8: Colour: 0, 1
2-5: Colour: 0, 1
2-7: Colour: 0, 1
2-8: Colour: 0, 1
3-5: Colour: 0, 1
3-6: Colour: 0, 1
3-8: Colour: 0, 1
4-5: Colour: 0, 1
4-6: Colour: 0, 1
4-7: Colour: 0, 1
#Nodes: 8
#Edges: 12
#Colours: 2
Ex3
1-4: Colour: 0, 1
1-6: Colour: 0, 1
1-8: Colour: 0, 1
2-3: Colour: 1, 0
2-5: Colour: 1, 0
2-7: Colour: 1, 0
3-6: Colour: 0, 1
3-8: Colour: 0, 1
4-5: Colour: 1, 0
4-7: Colour: 1, 0
5-8: Colour: 0, 1
6-7: Colour: 1, 0
#Nodes: 8
#Edges: 12
#Colours: 2
Ex4
1-6: Colour: 0, 1
1-7: Colour: 0, 1
1-8: Colour: 0, 1
2-5: Colour: 2, 0
2-7: Colour: 2, 1
2-8: Colour: 2, 1
3-5: Colour: 2, 0
3-6: Colour: 2, 1
3-8: Colour: 2, 1
4-5: Colour: 2, 0
4-6: Colour: 2, 1
4-7: Colour: 2, 1
#Nodes: 8
#Edges: 12
#Colours: 3
Python dicts preserve insertion order and Ex2/Ex3 edges are traced in a similar way which could be the cause of exactly the same colours used for Ex2 and Ex3. The wp article must use an earlier version of Python/different ordering of edge definitions.
Ex4 changes the order of nodes enough to affect the number of colours used.
Racket
<lang racket></lang>
- Output:
Raku
(formerly Perl 6) <lang perl6>sub GraphNodeColor(@RAW) {
my %OneMany = my %NodeColor;
for @RAW { %OneMany{$_[0]}.push: $_[1] ; %OneMany{$_[1]}.push: $_[0] }
my @ColorPool = "0", "1" … ^+%OneMany.elems; # as string
my %NodePool = %OneMany.BagHash; # this DWIM is nice
if %OneMany<NaN>:exists { %NodePool{$_}:delete for %OneMany<NaN>, NaN } # pending
while %NodePool.Bool {
my $color = @ColorPool.shift;
my %TempPool = %NodePool;
while (my \n = %TempPool.keys.sort.first) {
%NodeColor{n} = $color;
%TempPool{n}:delete;
%TempPool{$_}:delete for @(%OneMany{n}) ; # skip neighbors as well
%NodePool{n}:delete;
}
}
if %OneMany<NaN>:exists { # islanders use an existing color
%NodeColor{$_} = %NodeColor.values.sort.first for @(%OneMany<NaN>)
}
return %NodeColor
}
my \DATA = [
[<0 1>,<1 2>,<2 0>,<3 NaN>,<4 NaN>,<5 NaN>], [<1 6>,<1 7>,<1 8>,<2 5>,<2 7>,<2 8>,<3 5>,<3 6>,<3 8>,<4 5>,<4 6>,<4 7>], [<1 4>,<1 6>,<1 8>,<3 2>,<3 6>,<3 8>,<5 2>,<5 4>,<5 8>,<7 2>,<7 4>,<7 6>], [<1 6>,<7 1>,<8 1>,<5 2>,<2 7>,<2 8>,<3 5>,<6 3>,<3 8>,<4 5>,<4 6>,<4 7>],
];
for DATA {
say "DATA : ", $_;
say "Result : ";
my %out = GraphNodeColor $_;
say "$_[0]-$_[1]:\t Color %out{$_[0]} ",$_[1].isNaN??!!%out{$_[1]} for @$_;
say "Nodes : ", %out.keys.elems;
say "Edges : ", $_.elems;
say "Colors : ", %out.values.Set.elems;
}</lang>
- Output:
DATA : [(0 1) (1 2) (2 0) (3 NaN) (4 NaN) (5 NaN)] Result : 0-1: Color 0 1 1-2: Color 1 2 2-0: Color 2 0 3-NaN: Color 0 4-NaN: Color 0 5-NaN: Color 0 Nodes : 6 Edges : 6 Colors : 3 DATA : [(1 6) (1 7) (1 8) (2 5) (2 7) (2 8) (3 5) (3 6) (3 8) (4 5) (4 6) (4 7)] Result : 1-6: Color 0 1 1-7: Color 0 1 1-8: Color 0 1 2-5: Color 0 1 2-7: Color 0 1 2-8: Color 0 1 3-5: Color 0 1 3-6: Color 0 1 3-8: Color 0 1 4-5: Color 0 1 4-6: Color 0 1 4-7: Color 0 1 Nodes : 8 Edges : 12 Colors : 2 DATA : [(1 4) (1 6) (1 8) (3 2) (3 6) (3 8) (5 2) (5 4) (5 8) (7 2) (7 4) (7 6)] Result : 1-4: Color 0 1 1-6: Color 0 2 1-8: Color 0 3 3-2: Color 1 0 3-6: Color 1 2 3-8: Color 1 3 5-2: Color 2 0 5-4: Color 2 1 5-8: Color 2 3 7-2: Color 3 0 7-4: Color 3 1 7-6: Color 3 2 Nodes : 8 Edges : 12 Colors : 4 DATA : [(1 6) (7 1) (8 1) (5 2) (2 7) (2 8) (3 5) (6 3) (3 8) (4 5) (4 6) (4 7)] Result : 1-6: Color 0 1 7-1: Color 1 0 8-1: Color 1 0 5-2: Color 1 0 2-7: Color 0 1 2-8: Color 0 1 3-5: Color 0 1 6-3: Color 1 0 3-8: Color 0 1 4-5: Color 0 1 4-6: Color 0 1 4-7: Color 0 1 Nodes : 8 Edges : 12 Colors : 2
Wren
<lang ecmascript>import "/dynamic" for Struct import "/sort" for Sort import "/fmt" for Fmt
// (n)umber of node and its (v)alence i.e. number of neighbors var NodeVal = Struct.create("NodeVal", ["n", "v"])
class Graph {
construct new(nn, st) {
_nn = nn // number of nodes
_st = st // node numbering starts from
_nbr = List.filled(nn, null) // neighbor list for each node
for (i in 0...nn) _nbr[i] = []
}
nn { _nn }
st { _st }
// Note that this creates a single 'virtual' edge for an isolated node.
addEdge(n1, n2) {
// adjust to starting node number
n1 = n1 - _st
n2 = n2 - _st
_nbr[n1].add(n2)
if (n1 != n2) _nbr[n2].add(n1)
}
// Uses 'greedy' algorithm.
greedyColoring {
// create a list with a color for each node
var cols = List.filled(_nn, -1) // -1 denotes no color assigned
cols[0] = 0 // first node assigned color 0
// create a bool list to keep track of which colors are available
var available = List.filled(_nn, false)
// assign colors to all nodes after the first
for (i in 1..._nn) {
// iterate through neighbors and mark their colors as available
for (j in _nbr[i]) {
if (cols[j] != -1) available[cols[j]] = true
}
// find the first available color
var c = available.indexOf(false)
cols[i] = c // assign it to the current node
// reset the neighbors' colors to unavailable
// before the next iteration
for (j in _nbr[i]) {
if (cols[j] != -1) available[cols[j]] = false
}
}
return cols
}
// Uses Welsh-Powell algorithm.
wpColoring {
// create NodeVal for each node
var nvs = List.filled(_nn, null)
for (i in 0..._nn) {
var v = _nbr[i].count
if (v == 1 && _nbr[i][0] == i) { // isolated node
v = 0
}
nvs[i] = NodeVal.new(i, v)
}
// sort the NodeVals in descending order by valence
var cmp = Fn.new { |nv1, nv2| (nv2.v - nv1.v).sign }
Sort.insertion(nvs, cmp) // stable sort
// create colors list with entries for each node
var cols = List.filled(_nn, -1) // set all nodes to no color (-1) initially
var currCol = 0 // start with color 0
for (f in 0..._nn-1) {
var h = nvs[f].n
if (cols[h] != -1) { // already assigned a color
continue
}
cols[h] = currCol
// assign same color to all subsequent uncolored nodes which are
// not connected to a previous colored one
var i = f + 1
while (i < _nn) {
var outer = false
var j = nvs[i].n
if (cols[j] != -1) { // already colored
i = i + 1
continue
}
var k = f
while (k < i) {
var l = nvs[k].n
if (cols[l] == -1) { // not yet colored
k = k + 1
continue
}
if (_nbr[j].contains(l)) {
outer = true
break // node j is connected to an earlier colored node
}
k = k + 1
}
if (!outer) cols[j] = currCol
i = i + 1
}
currCol = currCol + 1
}
return cols
}
}
var fns = [Fn.new { |g| g.greedyColoring }, Fn.new { |g| g.wpColoring }] var titles = ["'Greedy'", "Welsh-Powell"] var nns = [4, 8, 8, 8] var starts = [0, 1, 1, 1] var edges1 = [[0, 1], [1, 2], [2, 0], [3, 3]] var edges2 = [[1, 6], [1, 7], [1, 8], [2, 5], [2, 7], [2, 8],
[3, 5], [3, 6], [3, 8], [4, 5], [4, 6], [4, 7]]
var edges3 = [[1, 4], [1, 6], [1, 8], [3, 2], [3, 6], [3, 8],
[5, 2], [5, 4], [5, 8], [7, 2], [7, 4], [7, 6]]
var edges4 = [[1, 6], [7, 1], [8, 1], [5, 2], [2, 7], [2, 8],
[3, 5], [6, 3], [3, 8], [4, 5], [4, 6], [4, 7]]
var j = 0 for (fn in fns) {
System.print("Using the %(titles[j]) algorithm:\n")
var i = 0
for (edges in [edges1, edges2, edges3, edges4]) {
System.print(" Example %(i+1)")
var g = Graph.new(nns[i], starts[i])
for (e in edges) g.addEdge(e[0], e[1])
var cols = fn.call(g)
var ecount = 0 // counts edges
for (e in edges) {
if (e[0] != e[1]) {
Fmt.print(" Edge $d-$d -> Color $d, $d", e[0], e[1],
cols[e[0]-g.st], cols[e[1]-g.st])
ecount = ecount + 1
} else {
Fmt.print(" Node $d -> Color $d\n", e[0], cols[e[0]-g.st])
}
}
var maxCol = 0 // maximum color number used
for (col in cols) {
if (col > maxCol) maxCol = col
}
System.print(" Number of nodes : %(nns[i])")
System.print(" Number of edges : %(ecount)")
System.print(" Number of colors : %(maxCol+1)")
System.print()
i = i + 1
}
j = j + 1
}</lang>
- Output:
Using the 'Greedy' algorithm:
Example 1
Edge 0-1 -> Color 0, 1
Edge 1-2 -> Color 1, 2
Edge 2-0 -> Color 2, 0
Node 3 -> Color 0
Number of nodes : 4
Number of edges : 3
Number of colors : 3
Example 2
Edge 1-6 -> Color 0, 1
Edge 1-7 -> Color 0, 1
Edge 1-8 -> Color 0, 1
Edge 2-5 -> Color 0, 1
Edge 2-7 -> Color 0, 1
Edge 2-8 -> Color 0, 1
Edge 3-5 -> Color 0, 1
Edge 3-6 -> Color 0, 1
Edge 3-8 -> Color 0, 1
Edge 4-5 -> Color 0, 1
Edge 4-6 -> Color 0, 1
Edge 4-7 -> Color 0, 1
Number of nodes : 8
Number of edges : 12
Number of colors : 2
Example 3
Edge 1-4 -> Color 0, 1
Edge 1-6 -> Color 0, 2
Edge 1-8 -> Color 0, 3
Edge 3-2 -> Color 1, 0
Edge 3-6 -> Color 1, 2
Edge 3-8 -> Color 1, 3
Edge 5-2 -> Color 2, 0
Edge 5-4 -> Color 2, 1
Edge 5-8 -> Color 2, 3
Edge 7-2 -> Color 3, 0
Edge 7-4 -> Color 3, 1
Edge 7-6 -> Color 3, 2
Number of nodes : 8
Number of edges : 12
Number of colors : 4
Example 4
Edge 1-6 -> Color 0, 1
Edge 7-1 -> Color 1, 0
Edge 8-1 -> Color 1, 0
Edge 5-2 -> Color 1, 0
Edge 2-7 -> Color 0, 1
Edge 2-8 -> Color 0, 1
Edge 3-5 -> Color 0, 1
Edge 6-3 -> Color 1, 0
Edge 3-8 -> Color 0, 1
Edge 4-5 -> Color 0, 1
Edge 4-6 -> Color 0, 1
Edge 4-7 -> Color 0, 1
Number of nodes : 8
Number of edges : 12
Number of colors : 2
Using the Welsh-Powell algorithm:
Example 1
Edge 0-1 -> Color 0, 1
Edge 1-2 -> Color 1, 2
Edge 2-0 -> Color 2, 0
Node 3 -> Color 0
Number of nodes : 4
Number of edges : 3
Number of colors : 3
Example 2
Edge 1-6 -> Color 0, 1
Edge 1-7 -> Color 0, 1
Edge 1-8 -> Color 0, 1
Edge 2-5 -> Color 0, 1
Edge 2-7 -> Color 0, 1
Edge 2-8 -> Color 0, 1
Edge 3-5 -> Color 0, 1
Edge 3-6 -> Color 0, 1
Edge 3-8 -> Color 0, 1
Edge 4-5 -> Color 0, 1
Edge 4-6 -> Color 0, 1
Edge 4-7 -> Color 0, 1
Number of nodes : 8
Number of edges : 12
Number of colors : 2
Example 3
Edge 1-4 -> Color 0, 1
Edge 1-6 -> Color 0, 2
Edge 1-8 -> Color 0, 3
Edge 3-2 -> Color 1, 0
Edge 3-6 -> Color 1, 2
Edge 3-8 -> Color 1, 3
Edge 5-2 -> Color 2, 0
Edge 5-4 -> Color 2, 1
Edge 5-8 -> Color 2, 3
Edge 7-2 -> Color 3, 0
Edge 7-4 -> Color 3, 1
Edge 7-6 -> Color 3, 2
Number of nodes : 8
Number of edges : 12
Number of colors : 4
Example 4
Edge 1-6 -> Color 0, 1
Edge 7-1 -> Color 1, 0
Edge 8-1 -> Color 1, 0
Edge 5-2 -> Color 1, 0
Edge 2-7 -> Color 0, 1
Edge 2-8 -> Color 0, 1
Edge 3-5 -> Color 0, 1
Edge 6-3 -> Color 1, 0
Edge 3-8 -> Color 0, 1
Edge 4-5 -> Color 0, 1
Edge 4-6 -> Color 0, 1
Edge 4-7 -> Color 0, 1
Number of nodes : 8
Number of edges : 12
Number of colors : 2
zkl
<lang zkl>fcn colorGraph(nodeStr){ // "0-1 1-2 2-0 3"
numEdges,graph := 0,Dictionary(); // ( 0:(1,2), 1:L(0,2), 2:(1,0), 3:() )
foreach n in (nodeStr.split(" ")){ // parse string to graph
n=n - " ";
if(n.holds("-")){
a,b := n.split("-"); // keep as string graph.appendV(a,b); graph.appendV(b,a); numEdges+=1;
}
else graph[n]=T; // island
}
colors,colorPool := Dictionary(), ["A".."Z"].walk();
graph.pump(Void,'wrap([(node,nbrs)]){ // ( "1",(0,2), "3",() )
clrs:=colorPool.copy(); // all colors are available, then remove neighbours
foreach i in (nbrs){ clrs.remove(colors.find(i)) } // if nbr has color, color not available
colors[node] = clrs[0]; // first available remaining color
});
return(graph,colors,numEdges)
}</lang> <lang zkl>fcn printColoredGraph(graphStr){
graph,colors,numEdges := colorGraph(graphStr);
nodes:=graph.keys.sort();
println("Graph: ",graphStr);
println("Node/color: ",
nodes.pump(List,'wrap(v){ String(v,"/",colors[v]) }).concat(", "));
println("Node : neighbours --> colors:");
foreach node in (nodes){
ns:=graph[node];
println(node," : ",ns.concat(" ")," --> ",
colors[node]," : ",ns.apply(colors.get).concat(" "));
}
println("Number nodes: ",nodes.len());
println("Number edges: ",numEdges);
println("Number colors: ",
colors.values.pump(Dictionary().add.fp1(Void)).len()); // create set, count
println();
}</lang> <lang zkl>graphs:=T(
"0-1 1-2 2-0 3", "1-6 1-7 1-8 2-5 2-7 2-8 3-5 3-6 3-8 4-5 4-6 4-7", "1-4 1-6 1-8 3-2 3-6 3-8 5-2 5-4 5-8 7-2 7-4 7-6", "1-6 7-1 8-1 5-2 2-7 2-8 3-5 6-3 3-8 4-5 4-6 4-7"
); graphs.apply2(printColoredGraph);</lang>
- Output:
Graph: 0-1 1-2 2-0 3 Node/color: 0/A, 1/B, 2/C, 3/A Node : neighbours --> colors: 0 : 1 2 --> A : B C 1 : 0 2 --> B : A C 2 : 1 0 --> C : B A 3 : --> A : Number nodes: 4 Number edges: 3 Number colors: 3 Graph: 1-6 1-7 1-8 2-5 2-7 2-8 3-5 3-6 3-8 4-5 4-6 4-7 Node/color: 1/A, 2/A, 3/A, 4/A, 5/B, 6/B, 7/B, 8/B Node : neighbours --> colors: 1 : 6 7 8 --> A : B B B 2 : 5 7 8 --> A : B B B 3 : 5 6 8 --> A : B B B 4 : 5 6 7 --> A : B B B 5 : 2 3 4 --> B : A A A 6 : 1 3 4 --> B : A A A 7 : 1 2 4 --> B : A A A 8 : 1 2 3 --> B : A A A Number nodes: 8 Number edges: 12 Number colors: 2 Graph: 1-4 1-6 1-8 3-2 3-6 3-8 5-2 5-4 5-8 7-2 7-4 7-6 Node/color: 1/A, 2/A, 3/B, 4/B, 5/C, 6/C, 7/D, 8/D Node : neighbours --> colors: 1 : 4 6 8 --> A : B C D 2 : 3 5 7 --> A : B C D 3 : 2 6 8 --> B : A C D 4 : 1 5 7 --> B : A C D 5 : 2 4 8 --> C : A B D 6 : 1 3 7 --> C : A B D 7 : 2 4 6 --> D : A B C 8 : 1 3 5 --> D : A B C Number nodes: 8 Number edges: 12 Number colors: 4 Graph: 1-6 7-1 8-1 5-2 2-7 2-8 3-5 6-3 3-8 4-5 4-6 4-7 Node/color: 1/A, 2/A, 3/A, 4/A, 5/B, 6/B, 7/B, 8/B Node : neighbours --> colors: 1 : 6 7 8 --> A : B B B 2 : 5 7 8 --> A : B B B 3 : 5 6 8 --> A : B B B 4 : 5 6 7 --> A : B B B 5 : 2 3 4 --> B : A A A 6 : 1 3 4 --> B : A A A 7 : 1 2 4 --> B : A A A 8 : 1 2 3 --> B : A A A Number nodes: 8 Number edges: 12 Number colors: 2