使用的方法顺序顶点排序
最大第一个顶点排序
顺序顶点排序在第一种技术中，颜色按照顶点的度数降序排列后依次分配给顶点。这种技术确保通常有更多邻居的较大程度的顶点首先被着色。
算法确定每个图顶点的级别。
确定顶点的度数并按降序对它们进行排序。
为数组中每个顶点的位置设置分配的颜色。
按照此处确定的顺序对顶点重复步骤 2。
为每个顶点指定其相邻顶点尚未使用的最小颜色。
示例#include <iostream>#include <vector>#include <algorithm>using namespace std;// graph structurestruct graph { int v; // number of vertices vector<vector<int>> adj; // adjacency list // constructor graph(int v) : v(v), adj(v) {} // function to add an edge between two vertices void addedge(int u, int v) { adj[u].push_back(v); adj[v].push_back(u); }};// function to compare vertices based on weightbool compareweights(pair<int, int> a, pair<int, int> b) { return a.second > b.second;}// function to perform graph coloring using welsh-powell algorithmvoid graphcoloring(graph& graph) { int v = graph.v; vector<pair<int, int>> vertexweights; // assign weights to each vertex based on their degree for (int v = 0; v < v; v++) { int weight = graph.adj[v].size(); vertexweights.push_back(make_pair(v, weight)); } // sort vertices in descending order of weights sort(vertexweights.begin(), vertexweights.end(), compareweights); // array to store colors assigned to vertices vector<int> color(v, -1); // assign colors to vertices in the sorted order for (int i = 0; i < v; i++) { int v = vertexweights[i].first; // find the smallest unused color for the current vertex vector<bool> usedcolors(v, false); for (int adjvertex : graph.adj[v]) { if (color[adjvertex] != -1) usedcolors[color[adjvertex]] = true; } // assign the smallest unused color to the current vertex for (int c = 0; c < v; c++) { if (!usedcolors[c]) { color[v] = c; break; } } } // print the coloring result for (int v = 0; v < v; v++) { cout << vertex << v << is assigned color << color[v] << endl; }}int main() { // create a sample graph graph graph(6); graph.addedge(0, 1); graph.addedge(0, 2); graph.addedge(1, 2); graph.addedge(1, 3); graph.addedge(2, 3); graph.addedge(3, 4); graph.addedge(4, 5); // perform graph coloring graphcoloring(graph); return 0;}
输出vertex 0 is assigned color 2vertex 1 is assigned color 0vertex 2 is assigned color 1vertex 3 is assigned color 2vertex 4 is assigned color 0vertex 5 is assigned color 1
最大第一个顶点排序与方式一类似，第二种方式包括根据顶点的度数以降序排列顶点。这种方法首先对最高程度的顶点进行着色，然后递归地为其未着色的邻居着色，而不是顺序分配颜色。
算法确定每个图顶点的度数。
确定顶点的度数并按降序对它们进行排序。
为数组中每个顶点的位置设置分配的颜色。
从最大度数的顶点开始着色。
选择当前未着色顶点的每个邻居可用的最少颜色。
示例#include <iostream>#include <vector>#include <algorithm>#include <unordered_set>using namespace std;class graph {private: int numvertices; vector<unordered_set<int>> adjacencylist;public: graph(int vertices) { numvertices = vertices; adjacencylist.resize(numvertices); } void addedge(int src, int dest) { adjacencylist[src].insert(dest); adjacencylist[dest].insert(src); } int getnumvertices() { return numvertices; } unordered_set<int>& getneighbors(int vertex) { return adjacencylist[vertex]; }};void welshpowelllargestfirst(graph graph) { int numvertices = graph.getnumvertices(); vector<int> colors(numvertices, -1); vector<pair<int, int>> largestfirst; for (int i = 0; i < numvertices; i++) { largestfirst.push_back(make_pair(graph.getneighbors(i).size(), i)); } sort(largestfirst.rbegin(), largestfirst.rend()); int numcolors = 0; for (const auto& vertexpair : largestfirst) { int vertex = vertexpair.second; if (colors[vertex] != -1) { continue; // vertex already colored } colors[vertex] = numcolors; for (int neighbor : graph.getneighbors(vertex)) { if (colors[neighbor] == -1) { colors[neighbor] = numcolors; } } numcolors++; } // print assigned colors for (int i = 0; i < numvertices; i++) { cout << vertex << i << - color: << colors[i] << endl; }}int main() { graph graph(7); graph.addedge(0, 1); graph.addedge(0, 2); graph.addedge(0, 3); graph.addedge(1, 4); graph.addedge(1, 5); graph.addedge(2, 6); graph.addedge(3, 6); welshpowelllargestfirst(graph); return 0;}
输出vertex 0 - color: 0vertex 1 - color: 0vertex 2 - color: 1vertex 3 - color: 1vertex 4 - color: 0vertex 5 - color: 0vertex 6 - color: 1
结论这篇博文分析了使用 c++ 算法构建威尔士鲍威尔图着色技术的两种不同方法。每种方法在对顶点进行排序和分配颜色时都采取了不同的策略，从而产生了有效且优化的图形着色方法。通过使用这些技术，我们可以有效地减少所需的颜色数量，同时保证附近的顶点包含不同的颜色。凭借其适应性和简单性，威尔士鲍威尔算法仍然是各种图形着色应用中的有用工具。
以上就是威尔士-鲍威尔图着色算法的详细内容。