单源/多源最短路径

单源最短路径

• 单源即一个起点，指从图中某个结点开始到其他所有结点

单源无权最短路径

关于本例中图的邻接矩阵创建参见 图的深搜和广搜

int dist[MAX_VERTEX];
int path[MAX_VERTEX];

void unWeightedSP(int v)
{
	for (int i = 0; i < nv; ++i) {
		dist[i] = -1;
	}
	dist[v] = 0;
	queue<int> q;
	q.push(v);
	while (!q.empty()) {
		int u = q.front();
		q.pop();
		for (int j = 0; j < nv; ++j) {
			if (MGraph[u][j] == 1&& dist[j] == -1) {
				dist[j] = dist[u] + 1;
				path[j] = u;
				q.push(j);
			}
		}
	}
}

有权图的单源最短路算法 Dijstra

Dijkstra算法

void Dijsktra(int v)
{
	int i;
	//初始化
	for (i = 0; i < nv; ++i) {
		dist[i] = 9999;
		path[i] = -1;
	}
	dist[v] = 0;

	while (1) {
		//获取未收录结点中距离源点距离最小者
		int u = getMinDistV();
		if (u == -1)break;//循环退出条件
		collected[u] = 1;
		//依次遍历u的邻接点
		for (int j = 0; j < nv; ++j) {
			if ((MGraph[u][j] > 0) && (collected[j] == 0)) {
				if (dist[u] + MGraph[u][j] < dist[j]) {
					dist[j] = dist[u] + MGraph[u][j];
					path[j] = u;
				}
			}
		}
	}
}
int getMinDistV()
{
	int i;
	int minDist = 9999;
	int minV = -1;
	for (i = 0; i < nv; ++i) {
		if ((minDist > dist[i])&&(collected[i]==0)) {
			minDist = dist[i];
			minV = i;
		}
	}
	return minV;
}

• Full Code

  #include
  #include
  
  using namespace std;
  
  #define MAX_VERTEX 100
  
  int MGraph[MAX_VERTEX][MAX_VERTEX];
  int visit[MAX_VERTEX];
  
  int nv, ne;
  void creatMGraph();
  void printMGraph();
  void Dijsktra(int v);
  int getMinDistV();
  int dist[MAX_VERTEX];
  int collected[MAX_VERTEX];
  int path[MAX_VERTEX];
  
  int main()
  {	
  	creatMGraph();
  	printMGraph();
  	Dijsktra(0);
  	for (int i = 0; i < nv; ++i)cout << dist[i] << " ";
  	system("pause");
  	return 0;
  }
  void creatMGraph()
  {
  	cout << "请输入结点数nv: ";
  	cin >> nv;
  	//初始化
  	int i, j;
  	for (i = 0; i < nv; ++i) {
  		visit[i] = 0;	
  	}
  	for (i = 0; i < nv; ++i) {
  		for (j = 0; j < nv; ++j) {
  			MGraph[i][j] = 0;
  		}
  	}
  	//添加边
  	cout << "请输入边数ne：";
  	cin >> ne;
  	cout << "请依次输入边及权重（例如：3 1 2）: " << endl;
  	int w;
  	for (int k = 0; k < ne; ++k) {
  		cin >> i >> j >> w;
  		MGraph[i][j] = w;
  	}
  }
  void printMGraph()
  {
  	int i, j;
  	for (i = 0; i < nv; ++i) {
  		for (j = 0; j < nv; ++j) {
  			cout << MGraph[i][j] << " ";
  		}
  		cout << endl;
  	}
  }
  void Dijsktra(int v)
  {
  	int i;
  	//初始化
  	for (i = 0; i < nv; ++i) {
  		dist[i] = 9999;
  		path[i] = -1;
  	}
  	dist[v] = 0;
  
  	while (1) {
  		//获取未收录结点中距离源点距离最小者
  		int u = getMinDistV();
  		if (u == -1)break;//循环退出条件
  		collected[u] = 1;
  		//依次遍历u的邻接点
  		for (int j = 0; j < nv; ++j) {
  			if ((MGraph[u][j] > 0) && (collected[j] == 0)) {
  				if (dist[u] + MGraph[u][j] < dist[j]) {
  					dist[j] = dist[u] + MGraph[u][j];
  					path[j] = u;
  				}
  			}
  		}
  	}
  }
  int getMinDistV()
  {
  	int i;
  	int minDist = 9999;
  	int minV = -1;
  	for (i = 0; i < nv; ++i) {
  		if ((minDist > dist[i])&&(collected[i]==0)) {
  			minDist = dist[i];
  			minV = i;
  		}
  	}
  	return minV;
  }
  

多源最短路径 Floyd

• 多源即图中任意两个结点
  

void Floyd()
{
	int i, j, k;
	for (i = 0; i < nv; ++i) {
		for (j = 0; j < nv; ++j) {
			Dist[i][j] = MGraph[i][j] > 0 ? MGraph[i][j] : 9999;
			Path[i][j] = -1;
		}
	}
	for (k = 0; k < nv; ++k) {
		for (i = 0; i < nv; ++i) {
			for (j = 0; j < nv; ++j) {
				if (Dist[i][k] + Dist[k][j] < Dist[i][j]) {
					Dist[i][j] = Dist[i][k] + Dist[k][j];
					Path[i][j] = k;
				}
			}
		}
	}
}