**Prim算法**是求图(无向图)中最小生成树的一种算法，另外一种是**Kruskal算法**。
**Prim算法思想**：Prim算法的每一步都会为一棵生长中的树添加一条边。一开始这棵树只有一个顶点，然后会向它添加V-1条边，每次总是将下一条连接树中的顶点与不在树中的顶点且权重最小的边加入树中。
**源代码**示例：
~~~
#include <iostream>
#include <vector>
#include <set>
#include <algorithm>
#include <functional>
using namespace std;
class Edge
{
public:
Edge(size_t _v, size_t _w, double _weight) : v(_v), w(_w), weight(_weight) {}
double getWeight() const { return weight; }
size_t either() const { return v; }
size_t other(size_t v) const
{
if (v == this->v)
return this->w;
else if (v == this->w)
return this->v;
else
{
cout << "error in Edge::other()" << endl;
exit(1);
}
}
void print() const{ cout << "(" << v << ", " << w << ", " << weight << ") "; }
private:
size_t v, w; // 两个顶点
double weight; // 权重
};
class EdgeWeithtedGraph
{
public:
EdgeWeithtedGraph(size_t vertax_nums) : vertaxs(vertax_nums), edges(0), arr(vertax_nums) {}
void addEdge(const Edge e);
vector<Edge> adj(size_t v) const { return v < arrSize() ? arr[v] : vector<Edge>(); }
vector<Edge> allEdges() const; // 返回加权无向图的所有边
size_t arrSize() const { return arr.size(); }
size_t vertax() const { return vertaxs; }
size_t edge() const { return edges; }
void printVertax(size_t v) const;
void printGraph() const;
private:
size_t vertaxs; // 顶点个数
size_t edges; // 边的个数
vector<vector<Edge>> arr; // 邻接表
};
/// 为了存放Prim算法过程中的边，这些边按照权值从小到大排列
template <typename T>
struct greator
{
bool operator()(const T &x, const T &y)
{
return x.getWeight() < y.getWeight();
}
};
/// 最小生成树的Prim算法的延时实现
class LazyPrimMST
{
public:
LazyPrimMST(const EdgeWeithtedGraph &graph) : marked(graph.arrSize())
{
if (graph.arrSize() == 0) // 防止为空图
return;
for (size_t i = 0; i < marked.size(); i++)
marked[i] = false;
visit(graph, 0);
set<Edge>::iterator miniter;
while (!pq.empty())
{
miniter = pq.begin();
size_t v = miniter->either();
size_t w = miniter->other(v);
if (marked[v] && marked[w])
{
pq.erase(miniter);
continue;
}
mst.push_back(*miniter);
pq.erase(miniter);
if (!marked[v])
visit(graph, v);
if (!marked[w])
visit(graph, w);
}
}
vector<Edge> edges() const { return mst; }
void printEdges() const
{
for (size_t i = 0; i < mst.size(); i++)
{
mst[i].print();
}
cout << endl;
}
private:
void visit(const EdgeWeithtedGraph &graph, size_t v)
{
marked[v] = true;
vector<Edge> vec = graph.adj(v);
for (size_t i = 0; i < vec.size(); i++)
{
if (!marked[vec[i].other(v)])
pq.insert(vec[i]);
}
}
vector<bool> marked;
vector<Edge> mst; // 最小生成树的边
set<Edge, greator<Edge>> pq; // 计算过程保存边的set
};
int main(void)
{
EdgeWeithtedGraph graph(8);
graph.addEdge(Edge(0, 7, 0.16));
graph.addEdge(Edge(0, 2, 0.26));
graph.addEdge(Edge(0, 4, 0.38));
graph.addEdge(Edge(0, 6, 0.58));
graph.addEdge(Edge(1, 7, 0.19));
graph.addEdge(Edge(5, 7, 0.28));
graph.addEdge(Edge(2, 7, 0.34));
graph.addEdge(Edge(4, 7, 0.37));
graph.addEdge(Edge(1, 3, 0.29));
graph.addEdge(Edge(1, 5, 0.32));
graph.addEdge(Edge(1, 2, 0.36));
graph.addEdge(Edge(2, 3, 0.17));
graph.addEdge(Edge(6, 2, 0.40));
graph.addEdge(Edge(3, 6, 0.52));
graph.addEdge(Edge(4, 5, 0.35));
graph.addEdge(Edge(6, 4, 0.93));
cout << "arrSize: " << graph.arrSize() << endl;
cout << "vertax: " << graph.vertax() << endl;
cout << "edge: " << graph.edge() << endl;
cout << "----------------" << endl;
graph.printGraph();
cout << endl;
// 输出无向图的所有边
vector<Edge> vec = graph.allEdges();
for (size_t i = 0; i < vec.size(); i++)
vec[i].print();
cout << endl << endl;
// prim算法
LazyPrimMST prim(graph);
prim.printEdges();
system("pause");
return 0;
}
void EdgeWeithtedGraph::addEdge(const Edge e)
{
size_t v = e.either();
size_t w = e.other(v);
if (!(v < arrSize() && w < arrSize()))
return;
arr[v].push_back(e);
arr[w].push_back(e);
this->edges++;
}
vector<Edge> EdgeWeithtedGraph::allEdges() const
{
vector<Edge> vec;
for (size_t i = 0; i < arrSize(); i++)
{
for (size_t j = 0; j < arr[i].size(); j++)
{
if (arr[i][j].other(i) > i) // 所有边的权重各不不同，可以这样判断，每个边只保留一个
vec.push_back(arr[i][j]);
}
}
return vec;
}
void EdgeWeithtedGraph::printVertax(size_t v) const
{
if (v >= arrSize())
return;
for (size_t i = 0; i < arr[v].size(); i++)
arr[v][i].print();
cout << endl;
}
void EdgeWeithtedGraph::printGraph() const
{
for (size_t i = 0; i < arrSize(); i++)
{
cout << i << ": ";
printVertax(i);
}
}
~~~
**参考资料**：
1、[https://github.com/luoxn28/algorithm_data_structure](https://github.com/luoxn28/algorithm_data_structure) （里面有常用的数据结构源代码，图相关算法在graph文件夹中）
2、《算法 第4版》 图的Prim算法章节
