Leetcode学习之二叉树与图（3）

开宗明义：本系列基于小象学院林沐老师课程《面试算法 LeetCode 刷题班》，刷题小白，旨在理解和交流，重在记录，望各位大牛指点！

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Leetcode学习之二叉树与图（3）

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文章目录

1、图的基础知识1.1 图的构造与表示（邻接矩阵）1.2 图的构造与表示（临接表） 2、图的深度优先遍历3、图的宽度优先遍历4、课程安排（有向图判断环）Leetcode 207.4.1 深度优先搜索4.2 拓扑排序（宽度优先搜索 ）

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1、图的基础知识

图：由顶点的有穷非空集合和顶点之间边的集合组成，通常表示为G(V,E)。其中，G表示一个图，V表示图G中顶点的集合，E表示图G中边的集合。图分为有向图与无向图，根据图的边长又分带权图与不带权图。

1.1 图的构造与表示（邻接矩阵）

测试代码：

#include <stdio.h> #include <iostream> int main() { const int MAX_N = 5;//一共5个顶点 int Graph[MAX_N][MAX_N] = { 0 };//使用邻接矩阵表示 Graph[0][2] = 1; Graph[0][4] = 1; Graph[1][0] = 1;//将图连通，且不带权，一般用邻接矩阵表示稠密图 Graph[1][2] = 1; Graph[2][3] = 1; Graph[3][4] = 1; Graph[4][3] = 1; printf("Graph:\n"); for (int i = 0; i < MAX_N; i++) { for (int j = 0; j < MAX_N; j++) { printf("%d ", Graph[i][j]); } printf("\n"); } system("pause"); return 0; }

效果图：

1.2 图的构造与表示（临接表）

测试代码：

#include <stdio.h> #include <vector> using namespace std; struct GraphNode { //图的邻接表数据结构 int label; //图的顶点的值 vector<GraphNode *> neighbors; //相邻节点指针数组 GraphNode(int x) :label(x) {}; }; int main() { const int MAX_N = 5; GraphNode *Graph[MAX_N];//5个顶点 for (int i = 0; i < MAX_N; i++) { Graph[i] = new GraphNode(i); }//添加边 Graph[0]->neighbors.push_back(Graph[2]); Graph[0]->neighbors.push_back(Graph[4]); Graph[1]->neighbors.push_back(Graph[0]); Graph[1]->neighbors.push_back(Graph[2]); Graph[2]->neighbors.push_back(Graph[3]); Graph[3]->neighbors.push_back(Graph[4]); Graph[4]->neighbors.push_back(Graph[3]); printf("Graph:\n"); for (int i = 0; i < MAX_N; i++) { printf("Label(%d): ", i); for (int j = 0; j < Graph[i]->neighbors.size(); j++) { printf("%d ", Graph[i]->neighbors[j]->label); } printf("\n"); } for (int i = 0; i < MAX_N; i++) { delete Graph[i]; } system("pause"); return 0; }

效果图：

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2、图的深度优先遍历

描述：从图中某个顶点v出发，首先访问该顶点，然后依次从它的各个未被访问的邻接点出发深度优先搜索遍历图，直至图中所以和v有路径想通且未被访问的顶点都被访问到。若此时尚有其他顶点未被访问到，则另选一个未被访问的顶点做起始点，重复上述过程，直至图中所有顶点都被访问到为止。 分析：

测试代码：

#include <stdio.h> #include <vector> using namespace std; struct GraphNode { //图的邻接表数据结构 int label; //图的顶点的值 vector<GraphNode *> neighbors; //相邻节点指针数组 GraphNode(int x) :label(x) {}; }; void DFS_graph(GraphNode *node, int visit[]) { visit[node->label] = 1;//标记已访问的顶点 printf("%d ", node->label); //访问相邻且没有被访问的顶点 for (int i = 0; i < node->neighbors.size(); i++) { if (visit[node->neighbors[i]->label] == 0) { //没有标记的 DFS_graph(node->neighbors[i], visit); } } } int main() { const int MAX_N = 5; GraphNode *Graph[MAX_N];//5个顶点 for (int i = 0; i < MAX_N; i++) { Graph[i] = new GraphNode(i); }//添加边 Graph[0]->neighbors.push_back(Graph[4]); Graph[0]->neighbors.push_back(Graph[2]); Graph[1]->neighbors.push_back(Graph[0]); Graph[1]->neighbors.push_back(Graph[2]); Graph[2]->neighbors.push_back(Graph[3]); Graph[3]->neighbors.push_back(Graph[4]); Graph[4]->neighbors.push_back(Graph[3]); int visit[MAX_N] = { 0 };//标记已访问的顶点 for (int i = 0; i < MAX_N; i++) { if (visit[i] == 0) { printf("From label(%d): ", Graph[i]->label); DFS_graph(Graph[i], visit); printf("\n"); } } for (int i = 0; i < MAX_N; i++) { delete Graph[i]; } system("pause"); return 0; }

效果图：

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3、图的宽度优先遍历

描述：从图中某个顶点v出发，在访问了v之后依次访问v的各个未曾访问过的邻接点，然后分别从这些邻接点出发依次访问它们的邻接点，并使得“先被访问的顶点的邻接点先于后被访问的顶点的邻接点被访问”，直至图中所有已被访问的邻接点都被访问到。如果还有顶点没有被访问到，需要另选一个顶点作为起始点，重复上述过程，直至所有的顶点都被访问到为止。 分析： 测试代码：

#include <stdio.h> #include <vector> #include <queue> using namespace std; struct GraphNode { //图的邻接表数据结构 int label; //图的顶点的值 vector<GraphNode *> neighbors; //相邻节点指针数组 GraphNode(int x) :label(x) {}; }; void DFS_graph(GraphNode *node, int visit[]) { queue<GraphNode *> Q;//宽度优先搜索使用队列，队列不空即一直循环 Q.push(node); visit[node->label] = 1;//先给标签 while (!Q.empty()) { GraphNode *node = Q.front(); Q.pop(); printf("%d ", node->label); for (int i = 0; i < node->neighbors.size(); i++) { //相邻 if (visit[node->neighbors[i]->label] == 0) { Q.push(node->neighbors[i]); visit[node->neighbors[i]->label] = 1; } } } } int main() { const int MAX_N = 5; GraphNode *Graph[MAX_N];//5个顶点 for (int i = 0; i < MAX_N; i++) { Graph[i] = new GraphNode(i); }//添加边 Graph[0]->neighbors.push_back(Graph[4]); Graph[0]->neighbors.push_back(Graph[2]); Graph[1]->neighbors.push_back(Graph[0]); Graph[1]->neighbors.push_back(Graph[2]); Graph[2]->neighbors.push_back(Graph[3]); Graph[3]->neighbors.push_back(Graph[4]); Graph[4]->neighbors.push_back(Graph[3]); int visit[MAX_N] = { 0 };//标记已访问的顶点 for (int i = 0; i < MAX_N; i++) { //5个顶点 if (visit[i] == 0) { printf("From label(%d): ", Graph[i]->label); DFS_graph(Graph[i], visit); printf("\n"); } } for (int i = 0; i < MAX_N; i++) { delete Graph[i]; } system("pause"); return 0; }

效果图：

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4、课程安排（有向图判断环）Leetcode 207.

题目来源：$Leetcode207.Course/Schedule$ 描述：已知有n个课程，标记从0至n-1，课程之间是有依赖关系的，例如希望完成A课程，可能需要先完成B课程。已知n个课程的依赖关系，求是否可以将n个课程全部完成。 分析：

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4.1 深度优先搜索

描述：二叉树层次遍历，又称宽度优先搜索，按树的层次依次访问树的结点。层次遍历使用队列对遍历节点进行存储，先进入队列的结点，优先遍历拓展其左孩子与右孩子。 分析：

测试代码：

#include <stdio.h> #include <vector> #include <queue> using namespace std; struct GraphNode { //图的邻接表数据结构 int label; //图的顶点的值 vector<GraphNode *> neighbors; //相邻节点指针数组 GraphNode(int x) :label(x) {}; }; bool DFS_graph(GraphNode *node,vector<int> &visit) { visit[node->label] = 0; for (int i = 0; i < node->neighbors.size(); i++) { if (visit[node->neighbors[i]->label] == -1) { if (DFS_graph(node->neighbors[i], visit) == 0) { return false; } } else if (visit[node->neighbors[i]->label] == 0) { return false; } } visit[node->label] = 1; return true; } class Solution { public: bool canFinish(int numCourses, vector<pair<int, int>>& prerequisites) { vector<GraphNode*> graph;//邻接表 vector<int> visit;//节点访问状态，-1没有访问过，0正在访问，1代表已经完成访问 for (int i = 0; i < numCourses; i++) { graph.push_back(new GraphNode(i));//创建图的节点，并赋访问状态为空 visit.push_back(-1); } //创建图，连接图的顶点 for (int i = 0; i < prerequisites.size(); i++) { GraphNode *begin = graph[prerequisites[i].second]; GraphNode *end = graph[prerequisites[i].first]; begin->neighbors.push_back(end);//课程2指向课程1 } for (int i = 0; i < graph.size(); i++) { if (visit[i] == -1 && !DFS_graph(graph[i], visit)) { return false;//如果节点没有访问到，进行DFS，如果DFS遇到环，返回无法完成 } } for (int i = 0; i < numCourses; i++) { delete graph[i]; } return true;//返回可以完成 } }; int main() { vector<pair<int, int>> prerequisites; prerequisites.push_back(make_pair(1, 0)); prerequisites.push_back(make_pair(2, 0)); prerequisites.push_back(make_pair(3, 1)); prerequisites.push_back(make_pair(3, 2)); Solution solve; printf("%d\n", solve.canFinish(4, prerequisites)); system("pause"); return 0; }

效果图：

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4.2 拓扑排序（宽度优先搜索 ）

分析：在宽度优先搜索时，只将入度为0的点添加至队列。当完成一个顶点的搜索（从队列取出），它指向的所有顶点入度都减一，若此时某顶点入度为0则添加至队列，若完成宽度搜索后，所有的入度都为0，则图无环，否则有环。

测试代码：

#include <stdio.h> #include <vector> #include <queue> using namespace std; struct GraphNode { //图的邻接表数据结构 int label; //图的顶点的值 vector<GraphNode *> neighbors; //相邻节点指针数组 GraphNode(int x) :label(x) {}; }; class Solution { public: bool canFinish(int numCourses,vector<pair<int, int>>& prequisites) { vector<GraphNode*> graph; vector<int> degree;//入度数组 for (int i = 0; i < numCourses; i++) { degree.push_back(0); graph.push_back(new GraphNode(i)); } for (int i = 0; i < prequisites.size(); i++) { GraphNode *begin = graph[prequisites[i].second]; GraphNode *end = graph[prequisites[i].first]; begin->neighbors.push_back(end); degree[prequisites[i].first]++; //入度++，即pair<课程1，课程2>课程1的入度++ } queue<GraphNode *> Q; for (int i = 0; i < numCourses; i++) { if (degree[i] == 0) { Q.push(graph[i]); } } while (!Q.empty()) { GraphNode *node = Q.front(); Q.pop(); for (int i = 0; i < node->neighbors.size(); i++) { degree[node->neighbors[i]->label]--; if (degree[node->neighbors[i]->label] == 0) { Q.push(node->neighbors[i]); } } } for (int i = 0; i < graph.size(); i++) { delete graph[i]; } for (int i = 0; i < degree.size(); i++) { if (degree[i]) { return false; } } return true; } }; int main() { vector<pair<int, int>> prerequisites; prerequisites.push_back(make_pair(1, 0)); prerequisites.push_back(make_pair(2, 0)); prerequisites.push_back(make_pair(3, 1)); prerequisites.push_back(make_pair(3, 2)); Solution solve; printf("%d\n", solve.canFinish(4, prerequisites)); system("pause"); return 0; }

效果图：