Graph colouring: Difference between revisions
m (→{{header|Racket}}: stub) |
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8 nodes, 12 edges, 2 colors. |
8 nodes, 12 edges, 2 colors. |
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</pre> |
</pre> |
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=={{header|Nim}}== |
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We use the “DSatur” algorithm described here: https://en.wikipedia.org/wiki/DSatur. |
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For the four examples, it gives the minimal number of colors. |
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<lang Nim>import algorithm, sequtils, strscans, strutils, tables |
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const NoColor = 0 |
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type |
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Color = range[0..63] |
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Node = ref object |
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num: Natural # Node number. |
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color: Color # Node color. |
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degree: Natural # Node degree. |
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dsat: Natural # Node Dsaturation. |
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neighbors: seq[Node] # List of neighbors. |
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Graph = seq[Node] # List of nodes ordered by number. |
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#--------------------------------------------------------------------------------------------------- |
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proc initGraph(graphRepr: string): Graph = |
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## Initialize the graph from its string representation. |
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var mapping: Table[Natural, Node] # Temporary mapping. |
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for elem in graphRepr.splitWhitespace(): |
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var num1, num2: int |
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if elem.scanf("$i-$i", num1, num2): |
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let node1 = mapping.mgetOrPut(num1, Node(num: num1)) |
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let node2 = mapping.mgetOrPut(num2, Node(num: num2)) |
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node1.neighbors.add node2 |
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node2.neighbors.add node1 |
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elif elem.scanf("$i", num1): |
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discard mapping.mgetOrPut(num1, Node(num: num1)) |
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else: |
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raise newException(ValueError, "wrong description: " & elem) |
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for node in mapping.values: |
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node.degree = node.neighbors.len |
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result = sortedByIt(toSeq(mapping.values), it.num) |
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#--------------------------------------------------------------------------------------------------- |
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proc numbers(nodes: seq[Node]): string = |
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## Return the numbers of a list of nodes. |
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for node in nodes: |
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result.addSep(" ") |
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result.add($node.num) |
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#--------------------------------------------------------------------------------------------------- |
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proc `$`(graph: Graph): string = |
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## Return the description of the graph. |
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var maxColor = NoColor |
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for node in graph: |
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stdout.write "Node ", node.num, ": color = ", node.color |
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if node.color > maxColor: maxColor = node.color |
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if node.neighbors.len > 0: |
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echo " neighbors = ", sortedByIt(node.neighbors, it.num).numbers |
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else: |
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echo "" |
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echo "Number of colors: ", maxColor |
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#--------------------------------------------------------------------------------------------------- |
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proc `<`(node1, node2: Node): bool = |
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## Comparison of nodes, by dsaturation first, then by degree. |
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if node1.dsat == node2.dsat: node1.degree < node2.degree |
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else: node1.dsat < node2.dsat |
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#--------------------------------------------------------------------------------------------------- |
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proc getMaxDsatNode(nodes: var seq[Node]): Node = |
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## Return the node with the greatest dsaturation. |
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let idx = nodes.maxIndex() |
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result = nodes[idx] |
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nodes.delete(idx) |
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#--------------------------------------------------------------------------------------------------- |
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proc minColor(node: Node): COLOR = |
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## Return the minimum available color for a node. |
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var colorUsed: set[Color] |
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for neighbor in node.neighbors: |
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colorUsed.incl neighbor.color |
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for color in 1..Color.high: |
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if color notin colorUsed: return color |
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#--------------------------------------------------------------------------------------------------- |
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proc distinctColorsCount(node: Node): Natural = |
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## Return the number of distinct colors of the neighbors of a node. |
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var colorUsed: set[Color] |
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for neighbor in node.neighbors: |
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colorUsed.incl neighbor.color |
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result = colorUsed.card |
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#--------------------------------------------------------------------------------------------------- |
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proc updateDsats(node: Node) = |
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## Update the dsaturations of the neighbors of a node. |
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for neighbor in node.neighbors: |
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if neighbor.color == NoColor: |
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neighbor.dsat = neighbor.distinctColorsCount() |
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#--------------------------------------------------------------------------------------------------- |
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proc colorize(graphRepr: string) = |
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## Colorize a graph. |
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let graph = initGraph(graphRepr) |
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var nodes = sortedByIt(graph, -it.degree) # Copy or graph sorted by decreasing degrees. |
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while nodes.len > 0: |
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let node = nodes.getMaxDsatNode() |
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node.color = node.minColor() |
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node.updateDsats() |
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echo "Graph: ", graphRepr |
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echo graph |
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#--------------------------------------------------------------------------------------------------- |
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when isMainModule: |
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colorize("0-1 1-2 2-0 3") |
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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") |
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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") |
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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> |
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{{out}} |
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<pre>Graph: 0-1 1-2 2-0 3 |
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Node 0: color = 1 neighbors = 1 2 |
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Node 1: color = 2 neighbors = 0 2 |
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Node 2: color = 3 neighbors = 0 1 |
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Node 3: color = 1 |
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Number of colors: 3 |
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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 |
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Node 1: color = 1 neighbors = 6 7 8 |
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Node 2: color = 1 neighbors = 5 7 8 |
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Node 3: color = 1 neighbors = 5 6 8 |
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Node 4: color = 1 neighbors = 5 6 7 |
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Node 5: color = 2 neighbors = 2 3 4 |
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Node 6: color = 2 neighbors = 1 3 4 |
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Node 7: color = 2 neighbors = 1 2 4 |
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Node 8: color = 2 neighbors = 1 2 3 |
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Number of colors: 2 |
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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 |
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Node 1: color = 1 neighbors = 4 6 8 |
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Node 2: color = 2 neighbors = 3 5 7 |
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Node 3: color = 1 neighbors = 2 6 8 |
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Node 4: color = 2 neighbors = 1 5 7 |
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Node 5: color = 1 neighbors = 2 4 8 |
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Node 6: color = 2 neighbors = 1 3 7 |
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Node 7: color = 1 neighbors = 2 4 6 |
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Node 8: color = 2 neighbors = 1 3 5 |
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Number of colors: 2 |
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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 |
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Node 1: color = 1 neighbors = 6 7 8 |
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Node 2: color = 1 neighbors = 5 7 8 |
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Node 3: color = 1 neighbors = 5 6 8 |
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Node 4: color = 1 neighbors = 5 6 7 |
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Node 5: color = 2 neighbors = 2 3 4 |
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Node 6: color = 2 neighbors = 1 3 4 |
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Node 7: color = 2 neighbors = 1 2 4 |
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Node 8: color = 2 neighbors = 1 2 3 |
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Number of colors: 2</pre> |
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=={{header|Phix}}== |
=={{header|Phix}}== |
Revision as of 17:13, 9 January 2021
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.
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
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.
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
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...
<lang Phix>-- demo\rosetta\Graph_colouring.exw
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 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,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(nodes,links,ll[1]) if length(ll)=2 then {nodes, links, right} = add_node(nodes,links,ll[2]) links[left] &= right 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</lang>
- 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
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