Documentation / Algorithms

The Welsh-Powell Algorithm

This class is intended to implement the Welsh-Powell algorithm for the problem of graph coloring. It provides a greedy algorithm that runs on a static graph.

This is an iterative greedy algorithm:

  • Step 1: All vertices are sorted according to the decreasing value of their degree in a list V.
  • Step 2: Colors are ordered in a list C.
  • Step 3: The first non colored vertex v in V is colored with the first available color in C. available means a color that was not previously used by the algorithm.
  • Step 4: The remaining part of the ordered list V is traversed and the same color is allocated to every vertex for which no adjacent vertex has the same color.
  • Step 5: Steps 3 and 4 are applied iteratively until all the vertices have been colored.

Note that the given colors are not real colors. Instead they are positive integers starting 0. So, for instance, if a colored graph’s chromatic number is 3, then nodes will be “colored” with one of 0, 1 or 2.

After computation using compute(), the algorithm result for the computation, the chromatic number, is accessible with the getChromaticNumber() method. Colors (of “Integer” type) are stored in the graph as attributes (one for each node). By default the attribute name is “WelshPowell.color”, but you can optional choose the attribute name.

Example

 import java.io.IOException;
 import java.io.StringReader;
 
 import org.graphstream.algorithm.coloring.WelshPowell;
 import org.graphstream.graph.ElementNotFoundException;
 import org.graphstream.graph.Graph;
 import org.graphstream.graph.Node;
 import org.graphstream.graph.implementations.DefaultGraph;
 import org.graphstream.stream.GraphParseException;
 import org.graphstream.stream.file.FileSourceDGS;
 
 public class WelshPowellTest {
 	//     B-(1)-C
 	//    /       \
 	//  (1)       (10)
 	//  /           \
 	// A             F
 	//  \           /
 	//  (1)       (1)
 	//    \       /
 	//     D-(1)-E
 	static String my_graph = 
 		"DGS004\n" 
 		+ "my 0 0\n" 
 		+ "an A \n" 
 		+ "an B \n"
 		+ "an C \n"
 		+ "an D \n"
 		+ "an E \n"
 		+ "an F \n"
 		+ "ae AB A B weight:1 \n"
 		+ "ae AD A D weight:1 \n"
 		+ "ae BC B C weight:1 \n"
 		+ "ae CF C F weight:10 \n"
 		+ "ae DE D E weight:1 \n"
 		+ "ae EF E F weight:1 \n"
 		;
 	public static void main(String[] args) throws IOException, ElementNotFoundException, GraphParseException {
 		Graph graph = new DefaultGraph("Welsh Powell Test");
 		StringReader reader  = new StringReader(my_graph);
 		
 		FileSourceDGS source = new FileSourceDGS();
 		source.addSink(graph);
 		source.readAll(reader);
 		
 		WelshPowell wp = new WelshPowell("color");
 		wp.init(graph);
 		wp.compute();
 		
 		System.out.println("The chromatic number of this graph is : "+wp.getChromaticNumber());
 		for(Node n : graph){
 			System.out.println("Node "+n.getId()+ " : color " +n.getAttribute("color"));
 		}
 	}
 }

This shall return:

The chromatic number of this graph is : 3
Node D : color 0
Node E : color 2 
Node F : color 1
Node A : color 2
Node B : color 1
Node C : color 0

Display Colors

Consider you what to display the result of the coloring algorithm on a displayed graph, then adding the following code to the previous example may help you:

 Color[] cols = new Color[wp.getChromaticNumber()];
 for(int i=0;i< wp.getChromaticNumber();i++){
 	cols[i]=Color.getHSBColor((float) (Math.random()), 0.8f, 0.9f);
 }
 for(Node n : graph){ 
 	int col = (int) n.getNumber("color");
 	n.addAttribute("ui.style", "fill-color:rgba("+cols[col].getRed()+","+cols[col].getGreen()+","+cols[col].getBlue()+",200);" );
 }
 
 graph.display();

Complexity

This algorithm is known to use at most d(G)+1 colors where d(G) represents the largest value of the degree in the graph G.

Reference

  • Welsh, D. J. A.; Powell, M. B. (1967), “An upper bound for the chromatic number of a graph and its application to timetabling problems”, The Computer Journal 10 (1): 85–86, doi:10.1093/comjnl/10.1.85