M着色问题



什么是M着色问题?

M着色问题中,我们的任务是确定是否可以为给定图的节点分配m种不同的颜色,使得图中没有两个相邻的顶点具有相同的颜色。如果存在解决方案,则显示每个顶点分配了哪种颜色。M着色问题实际上用于解决诸如聚类、调度、作业分配等问题。

图是一种抽象数据类型(ADT),由一组通过链接连接的对象组成。

输入输出场景

假设给定的无向图G(V, E)及其邻接矩阵如下所示:

M-Coloring

设最大颜色m = 3,表示可使用的最大颜色数。回溯算法可用于解决上述图的M着色问题。此算法将返回哪个节点将分配哪个颜色。如果解决方案不可行,则将返回false。

对于这种情况,输出应为节点0 -> 颜色1,节点1 -> 颜色2,节点2 -> 颜色3,节点3 -> 颜色2。下图说明了这一点:

M-Coloring output

使用回溯法解决M着色问题

解决M着色问题的简单方法是生成所有可能的顶点与颜色组合,并检查是否有任何组合满足给定的约束条件。但是,对于较大的图,这种方法效率低下。

要使用回溯法解决M着色问题,请按照以下步骤操作:

  • 从顶点0开始,我们将尝试逐个将颜色分配给不同的节点。

  • 但是,在分配之前,我们必须检查颜色是否安全。当相邻顶点包含相同的颜色时,颜色不安全。

  • 接下来,我们将检查是否存在任何满足约束条件的颜色分配。如果存在,我们将该分配标记为M着色问题的解决方案。

示例

在以下示例中,我们将说明如何在给定的无向图中解决M着色问题。

Open Compiler
#include <stdio.h> #include <stdlib.h> #include <stdbool.h> #define V 4 bool graph[V][V] = { {0, 1, 1, 1}, {1, 0, 1, 0}, {1, 1, 0, 1}, {1, 0, 1, 0} }; void showColors(int color[]) { printf("Assigned Colors are:\n"); for (int i = 0; i < V; i++) printf("%d ", color[i]); printf("\n"); } //check whether putting a color valid for v bool isValid(int v, int color[], int c) { for (int i = 0; i < V; i++) if (graph[v][i] && c == color[i]) return false; return true; } bool graphColoring(int colors, int color[], int vertex) { //when all vertices are considered if (vertex == V) return true; for (int col = 1; col <= colors; col++) { //check whether color is valid or not if (isValid(vertex, color, col)) { color[vertex] = col; // go for additional vertices if (graphColoring(colors, color, vertex + 1)) return true; color[vertex] = 0; } } //when no colors can be assigned return false; } bool checkSolution(int m) { //make color matrix for each vertex int *color = (int *)malloc(V * sizeof(int)); for (int i = 0; i < V; i++) //initially set to 0 color[i] = 0; //for vertex 0 check graph coloring if (graphColoring(m, color, 0) == false) { printf("Solution does not exist.\n"); free(color); return false; } showColors(color); free(color); return true; } int main() { // Number of colors int colors = 3; checkSolution(colors); return 0; }
Open Compiler
#include<iostream> #define V 4 using namespace std; bool graph[V][V] = { {0, 1, 1, 1}, {1, 0, 1, 0}, {1, 1, 0, 1}, {1, 0, 1, 0}, }; void showColors(int color[]) { cout << "Assigned Colors are: " <<endl; for (int i = 0; i < V; i++) cout << color[i] << " "; cout << endl; } //check whether putting a color valid for v bool isValid(int v,int color[], int c) { for (int i = 0; i < V; i++) if (graph[v][i] && c == color[i]) return false; return true; } bool graphColoring(int colors, int color[], int vertex) { //when all vertices are considered if (vertex == V) return true; for (int col = 1; col <= colors; col++) { //check whether color is valid or not if (isValid(vertex,color, col)) { color[vertex] = col; // go for additional vertices if (graphColoring (colors, color, vertex+1) == true) return true; color[vertex] = 0; } } //when no colors can be assigned return false; } bool checkSolution(int m) { //make color matrix for each vertex int *color = new int[V]; for (int i = 0; i < V; i++) //initially set to 0 color[i] = 0; //for vertex 0 check graph coloring if (graphColoring(m, color, 0) == false) { cout << "Solution does not exist."; return false; } showColors(color); return true; } int main() { // Number of colors int colors = 3; checkSolution (colors); }
Open Compiler
public class GraphColoring { static final int V = 4; static boolean[][] graph = { {false, true, true, true}, {true, false, true, false}, {true, true, false, true}, {true, false, true, false} }; static void showColors(int[] color) { System.out.println("Assigned Colors are:"); for (int i = 0; i < V; i++) { System.out.print(color[i] + " "); } System.out.println(); } //check whether putting a color valid for v static boolean isValid(int v, int[] color, int c) { for (int i = 0; i < V; i++) { if (graph[v][i] && c == color[i]) { return false; } } return true; } static boolean graphColoring(int colors, int[] color, int vertex) { //when all vertices are considered if (vertex == V) { return true; } for (int col = 1; col <= colors; col++) { //check whether color is valid or not if (isValid(vertex, color, col)) { color[vertex] = col; // go for additional vertices if (graphColoring(colors, color, vertex + 1)) { return true; } color[vertex] = 0; } } //when no colors can be assigned return false; } static boolean checkSolution(int m) { //make color matrix for each vertex int[] color = new int[V]; for (int i = 0; i < V; i++) { //initially set to 0 color[i] = 0; } //for vertex 0 check graph coloring if (!graphColoring(m, color, 0)) { System.out.println("Solution does not exist."); return false; } showColors(color); return true; } public static void main(String[] args) { // Number of colors int colors = 3; checkSolution(colors); } }
Open Compiler
V = 4 graph = [ [0, 1, 1, 1], [1, 0, 1, 0], [1, 1, 0, 1], [1, 0, 1, 0] ] def show_colors(color): print("Assigned Colors are:") for i in range(V): print(color[i], end=" ") print() def is_valid(v, color, c): for i in range(V): if graph[v][i] and c == color[i]: return False return True def graph_coloring(colors, color, vertex): if vertex == V: return True for col in range(1, colors + 1): if is_valid(vertex, color, col): color[vertex] = col if graph_coloring(colors, color, vertex + 1): return True color[vertex] = 0 return False def check_solution(m): color = [0] * V if not graph_coloring(m, color, 0): print("Solution does not exist.") return False show_colors(color) return True if __name__ == "__main__": colors = 3 check_solution(colors)

输出

Assigned Colors are: 
1 2 3 2
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