Graph Coloring | Set 2 (Greedy Algorithm)

We introduced graph coloring and applications in previous post. As discussed in the previous post, graph coloring is widely used. Unfortunately, there is no efficient algorithm available for coloring a graph with minimum number of colors as the problem is a known NP Complete problem. There are approximate algorithms to solve the problem though. Following is the basic Greedy Algorithm to assign colors. It doesn’t guarantee to use minimum colors, but it guarantees an upper bound on the number of colors. The basic algorithm never uses more than d+1 colors where d is the maximum degree of a vertex in the given graph.

Basic Greedy Coloring Algorithm:

1. Color first vertex with first color.
2. Do following for remaining V-1 vertices.
          a) Consider the currently picked vertex and color it with the 
             lowest numbered color that has not been used on any previously
             colored vertices adjacent to it. If all previously used colors 
             appear on vertices adjacent to v, assign a new color to it.

Following are C++ and Java implementations of the above Greedy Algorithm.

C++

// A C++ program to implement greedy algorithm for graph coloring
#include <iostream>
#include <list>
using namespace std;

// A class that represents an undirected graph
class Graph
{
    int V;    // No. of vertices
    list<int> *adj;    // A dynamic array of adjacency lists
public:
    // Constructor and destructor
    Graph(int V)   { this->V = V; adj = new list<int>[V]; }
    ~Graph()       { delete [] adj; }

    // function to add an edge to graph
    void addEdge(int v, int w);

    // Prints greedy coloring of the vertices
    void greedyColoring();
};

void Graph::addEdge(int v, int w)
{
    adj[v].push_back(w);
    adj[w].push_back(v);  // Note: the graph is undirected
}

// Assigns colors (starting from 0) to all vertices and prints
// the assignment of colors
void Graph::greedyColoring()
{
    int result[V];

    // Assign the first color to first vertex
    result[0]  = 0;

    // Initialize remaining V-1 vertices as unassigned
    for (int u = 1; u < V; u++)
        result[u] = -1;  // no color is assigned to u

    // A temporary array to store the available colors. True
    // value of available[cr] would mean that the color cr is
    // assigned to one of its adjacent vertices
    bool available[V];
    for (int cr = 0; cr < V; cr++)
        available[cr] = false;

    // Assign colors to remaining V-1 vertices
    for (int u = 1; u < V; u++)
    {
        // Process all adjacent vertices and flag their colors
        // as unavailable
        list<int>::iterator i;
        for (i = adj[u].begin(); i != adj[u].end(); ++i)
            if (result[*i] != -1)
                available[result[*i]] = true;

        // Find the first available color
        int cr;
        for (cr = 0; cr < V; cr++)
            if (available[cr] == false)
                break;

        result[u] = cr; // Assign the found color

        // Reset the values back to false for the next iteration
        for (i = adj[u].begin(); i != adj[u].end(); ++i)
            if (result[*i] != -1)
                available[result[*i]] = false;
    }

    // print the result
    for (int u = 0; u < V; u++)
        cout << "Vertex " << u << " --->  Color "
             << result[u] << endl;
}

// Driver program to test above function
int main()
{
    Graph g1(5);
    g1.addEdge(0, 1);
    g1.addEdge(0, 2);
    g1.addEdge(1, 2);
    g1.addEdge(1, 3);
    g1.addEdge(2, 3);
    g1.addEdge(3, 4);
    cout << "Coloring of graph 1 \n";
    g1.greedyColoring();

    Graph g2(5);
    g2.addEdge(0, 1);
    g2.addEdge(0, 2);
    g2.addEdge(1, 2);
    g2.addEdge(1, 4);
    g2.addEdge(2, 4);
    g2.addEdge(4, 3);
    cout << "\nColoring of graph 2 \n";
    g2.greedyColoring();

    return 0;
}

Java

// A Java program to implement greedy algorithm for graph coloring
import java.io.*;
import java.util.*;
import java.util.LinkedList;

// This class represents an undirected graph using adjacency list
class Graph
{
    private int V;   // No. of vertices
    private LinkedList<Integer> adj[]; //Adjacency List

    //Constructor
    Graph(int v)
    {
        V = v;
        adj = new LinkedList[v];
        for (int i=0; i<v; ++i)
            adj[i] = new LinkedList();
    }

    //Function to add an edge into the graph
    void addEdge(int v,int w)
    {
        adj[v].add(w);
        adj[w].add(v); //Graph is undirected
    }

    // Assigns colors (starting from 0) to all vertices and
    // prints the assignment of colors
    void greedyColoring()
    {
        int result[] = new int[V];

        // Assign the first color to first vertex
        result[0]  = 0;

        // Initialize remaining V-1 vertices as unassigned
        for (int u = 1; u < V; u++)
            result[u] = -1;  // no color is assigned to u

        // A temporary array to store the available colors. True
        // value of available[cr] would mean that the color cr is
        // assigned to one of its adjacent vertices
        boolean available[] = new boolean[V];
        for (int cr = 0; cr < V; cr++)
            available[cr] = false;

        // Assign colors to remaining V-1 vertices
        for (int u = 1; u < V; u++)
        {
            // Process all adjacent vertices and flag their colors
            // as unavailable
            Iterator<Integer> it = adj[u].iterator() ;
            while (it.hasNext())
            {
                int i = it.next();
                if (result[i] != -1)
                    available[result[i]] = true;
            }

            // Find the first available color
            int cr;
            for (cr = 0; cr < V; cr++)
                if (available[cr] == false)
                    break;

            result[u] = cr; // Assign the found color

            // Reset the values back to false for the next iteration
            it = adj[u].iterator() ;
            while (it.hasNext())
            {
                int i = it.next();
                if (result[i] != -1)
                    available[result[i]] = false;
            }
        }

        // print the result
        for (int u = 0; u < V; u++)
            System.out.println("Vertex " + u + " --->  Color "
                                + result[u]);
    }

    // Driver method
    public static void main(String args[])
    {
        Graph g1 = new Graph(5);
        g1.addEdge(0, 1);
        g1.addEdge(0, 2);
        g1.addEdge(1, 2);
        g1.addEdge(1, 3);
        g1.addEdge(2, 3);
        g1.addEdge(3, 4);
        System.out.println("Coloring of graph 1");
        g1.greedyColoring();

        System.out.println();
        Graph g2 = new Graph(5);
        g2.addEdge(0, 1);
        g2.addEdge(0, 2);
        g2.addEdge(1, 2);
        g2.addEdge(1, 4);
        g2.addEdge(2, 4);
        g2.addEdge(4, 3);
        System.out.println("Coloring of graph 2 ");
        g2.greedyColoring();
    }
}
// This code is contributed by Aakash Hasija


Output:

Coloring of graph 1
Vertex 0 --->  Color 0
Vertex 1 --->  Color 1
Vertex 2 --->  Color 2
Vertex 3 --->  Color 0
Vertex 4 --->  Color 1

Coloring of graph 2
Vertex 0 --->  Color 0
Vertex 1 --->  Color 1
Vertex 2 --->  Color 2
Vertex 3 --->  Color 0
Vertex 4 --->  Color 3

Time Complexity: O(V^2 + E) in worst case.

Analysis of Basic Algorithm
The above algorithm doesn’t always use minimum number of colors. Also, the number of colors used sometime depend on the order in which vertices are processed. For example, consider the following two graphs. Note that in graph on right side, vertices 3 and 4 are swapped. If we consider the vertices 0, 1, 2, 3, 4 in left graph, we can color the graph using 3 colors. But if we consider the vertices 0, 1, 2, 3, 4 in right graph, we need 4 colors.

graph_coloring2

graph_coloring2

 

 

 

 
So the order in which the vertices are picked is important. Many people have suggested different ways to find an ordering that work better than the basic algorithm on average. The most common is Welsh–Powell Algorithm which considers vertices in descending order of degrees.

How does the basic algorithm guarantee an upper bound of d+1?
Here d is the maximum degree in the given graph. Since d is maximum degree, a vertex cannot be attached to more than d vertices. When we color a vertex, at most d colors could have already been used by its adjacent. To color this vertex, we need to pick the smallest numbered color that is not used by the adjacent vertices. If colors are numbered like 1, 2, …., then the value of such smallest number must be between 1 to d+1 (Note that d numbers are already picked by adjacent vertices).
This can also be proved using induction. See this video lecture for proof.

We will soon be discussing some interesting facts about chromatic number and graph coloring.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

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