图着色

图着色问题是图标记的一种特殊情况。在这个问题中，每个节点都会被着上一些颜色。但着色有一些约束。相邻顶点不能使用相同的颜色。

为了解决这个问题，我们需要使用贪心算法，但它不能保证使用最少的颜色。

输入和输出

Input:
Adjacency matrix of the graph.
0 0 1 0 1
0 0 1 1 1
1 1 0 1 0
0 1 1 0 1
1 1 0 1 0

Output:
Node: 0, Assigned with Color: 0
Node: 1, Assigned with Color: 0
Node: 2, Assigned with Color: 1
Node: 3, Assigned with Color: 2
Node: 4, Assigned with Color: 1

算法

graphColoring(graph)

输入 − 给定的图。

输出 − 每个节点都分配了一些颜色。

Begin
   declare a list of colors
   initially set the color 0 for first node
   define an array colorUsed to track which color is used, and which colors have never used.

   for all vertices i except first one, do
      mark i as unassigned to any color
   done

   mark colorUsed to false for all vertices
   for all vertices u in the graph except 1st vertex, do
      for all vertex v adjacent with u, do
         if color[v] is unassigned, then
            mark colorUsed[color[v]] := true
      done

      for all colors col in the color list, do
         if color is not used, then
            stop the loop
      done

      color[u] := col
      for each vertex v which is adjacent with u, do
         if color[v] is unassigned, then
            colorUsed[color[v]] := false
      done
   done

   for all vertices u in the graph, do
      display the node and its color
   done
End

示例

#include<iostream>
#define NODE 6
using namespace std;

int graph[NODE][NODE] = {
   {0, 1, 1, 1, 0, 0},
   {1, 0, 0, 1, 1, 0},
   {1, 0, 0, 1, 0, 1},
   {1, 1, 1, 0, 1, 1},
   {0, 1, 0, 1, 0, 1},
   {0, 0, 1, 1, 1, 0}
};

void graphColoring() {
   int color[NODE];
   color[0] = 0;    //Assign first color for the first node
   bool colorUsed[NODE];    //Used to check whether color is used or not

   for(int i = 1; i<NODE; i++)
      color[i] = -1;    //initialize all other vertices are unassigned

   for(int i = 0; i<NODE; i++)
      colorUsed[i] = false;    //initially any colors are not chosen
         
   for(int u = 1; u<NODE; u++) {    //for all other NODE - 1 vertices
      for(int v = 0; v<NODE; v++) {
         if(graph[u][v]){
            if(color[v] != -1)    //when one color is assigned, make it unavailable
               colorUsed[color[v]] = true;
         }
     }

     int col;
     for(col = 0; col<NODE; col++)
        if(!colorUsed[col])    //find a color which is not assigned
           break;
         
     color[u] = col;    //assign found color in the list
         
     for(int v = 0; v<NODE; v++) {    //for next iteration make color availability to false
        if(graph[u][v]) {
           if(color[v] != -1)
              colorUsed[color[v]] = false;
        }
     }  
  }
   
  for(int u = 0; u<NODE; u++)
     cout <<"Color: " << u << ", Assigned with Color: " <<color[u] <<endl;
}

main() {
   graphColoring();
}

输出

Node: 0, Assigned with Color: 0
Node: 1, Assigned with Color: 0
Node: 2, Assigned with Color: 1
Node: 3, Assigned with Color: 2
Node: 4, Assigned with Color: 1

karthikeya Boyini

更新日期: 16-6 月-2020

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