Preface
This article mainly explains the graph structure in the data structure, including depth first search (DFS), breadth first search (BFS), minimum spanning tree algorithm, etc. I hope you will like it.
table of Contents
1 Introduction
- Data structure belonging to the circular structure of the logical structure
- The details are as follows
2. Basic concepts
- In the graph data structure, there are many basic concepts, such as edge types, graph vertices & relationships between edges, etc.
- See the figure below for details
3. Type
There are many types of graphs, as follows:
3.1 Directed Graph & Undirected Graph
3.2 Connected graph
-
Definition
Any vertex in the graph is a connected graph -
Specific related concepts
For directed graphs & undirected graphs, the specific concepts of connected graphs are different, as follows
For undirected graphs:
For directed graphs:
3.3 Other types of diagrams
4. Storage structure
There are 5 types of storage structure of the picture, please see the figure below for details
5. Graph traversal
Data structure: Graphical detailed binary tree (traversal, type, operation)
5.1 Definition
Starting from a certain vertex in the graph, traverse all the other vertices in the graph & make each vertex be visited only once
5.2 Traversal method
Depth-first traversal ( DFS), breadth-first traversal ( BFS)
5.3 Specific introduction
5.3.1 Depth First Traversal (DFS)
- Introduction
- Algorithm diagram
- Specific implementation: recursive & non-recursive
The storage structure of the graph here = adjacency matrix
import java.util.Stack;
public class MyGraph {
/**
* 准备工作1：设置变量
*/
private int vexnum; // 存放图中顶点数量
private char[] vertices; // 存放结点数据
private int [][] arcs; // 存放图的所有边
private boolean[] visited;// 记录节点是否已被遍历
/**
* 准备工作2：初始化图的顶点数量、数据 & 边
*/
public MyGraph(int n){
vexnum = n;
vertices = new char[n];
visited = new boolean[n];
arcs = new int[n][n];
}
/**
* 图的深度优先遍历算法实现：递归
* 类似 前序遍历
*/
public void DFSTraverse(){
// 1. 初始化所有顶点的访问标记
// 即，都是未访问状态
for (int i = 0; i < vexnum; i++) {
visited[i] = false;
}
// 2. 深度优先遍历顶点（从未被访问的顶点开始）
for(int i=0;i < vexnum;i++){
if(visited[i]==false){
// 若是连通图，只会执行一次
traverse(i); // ->>看关注1
}
}
}
/**
* 关注1：邻接矩阵的深度优先搜索递归算法
* 即，从第i个顶点开始深度优先遍历
*/
private void traverse(int i){
// 1. 标记第i个顶点已遍历
visited[i] = true;
// 2. （输出）访问当前遍历的顶点
visit(i);
// 3. 遍历邻接矩阵中第i个顶点的所有邻接顶点
for(int j=0;j<vexnum;j++){
// a. 若当前顶点的邻接顶点存在 & 未被访问，则递归 深度优先搜索 算法
if(arcs[i][j]==1 && visited[j]==false){
// b. 将当前顶点的邻接顶点作为当前顶点，递归 深度优先搜索 算法
traverse(j);
}
}
}
/**
* 辅助算法1：访问顶点值
*/
public void visit(int i){
System.out.print(vertices[i] + " ");
}
/**
* 图的深度优先遍历算法实现：非递归
* 原理：采用 栈实现
*/
public void DFSTraverse2(){
// 1. 初始化顶点访问标记
// 全部标记为：未访问
for (int i = 0; i < vexnum; i++) {
visited[i] = false;
}
// 2. 创建栈
Stack<Integer> s = new Stack<Integer>();
for(int i=0 ; i<vexnum; i++){
// 3. 若该顶点未被访问
if(!visited[i]){
// 4. 入栈该顶点
s.add(i);
do{
// 出栈
int curr = s.pop();
// 如果该节点还没有被遍历，则遍历该节点并将子节点入栈
if(visited[curr]==false){
// 遍历并打印
visit(curr);
visited[curr] = true;
// 没遍历的子节点入栈
for(int j=vexnum-1; j>=0 ; j-- ){
if(arcs[curr][j]==1 && visited[j]==false){
s.add(j);
}
}
}
}while(!s.isEmpty());
}
}
}
/**
* 测试（递归 & 非递归）
*/
public static void main(String[] args) {
// 1. 初始化图的结构（顶点数量 = 9）
MyGraph g = new MyGraph(9);
// 2. 设置顶点数据
char[] vertices = {'A','B','C','D','E','F','G','H','I'};
g.setVertices(vertices);
// 3. 设置边
g.addEdge(0, 1);
g.addEdge(0, 5);
g.addEdge(1, 0);
g.addEdge(1, 2);
g.addEdge(1, 6);
g.addEdge(1, 8);
g.addEdge(2, 1);
g.addEdge(2, 3);
g.addEdge(2, 8);
g.addEdge(3, 2);
g.addEdge(3, 4);
g.addEdge(3, 6);
g.addEdge(3, 7);
g.addEdge(3, 8);
g.addEdge(4, 3);
g.addEdge(4, 5);
g.addEdge(4, 7);
g.addEdge(5, 0);
g.addEdge(5, 4);
g.addEdge(5, 6);
g.addEdge(6, 1);
g.addEdge(6, 3);
g.addEdge(6, 5);
g.addEdge(6, 7);
g.addEdge(7, 3);
g.addEdge(7, 4);
g.addEdge(7, 6);
g.addEdge(8, 1);
g.addEdge(8, 2);
g.addEdge(8, 3);
// 4. 执行 图的深度优先遍历：（递归 & 非递归）
System.out.println("深度优先遍历（递归）");
g.DFSTraverse();
System.out.println("深度优先遍历（非递归）");
g.DFSTraverse2();
}
/**
* 辅助算法1：添加边(无向图)
*/
public void addEdge(int i, int j) {
// 边的头尾不能为同一节点
if (i == j) return;
// 将邻接矩阵的第i行第j列的元素值置为1；
arcs[i][j] = 1;
// 将邻接矩阵的第j行第i列的元素值置为1；
arcs[j][i] = 1;
// 设置为1代表2顶点之间存在 边 （设置相等原因 = 邻接矩阵 是对称的）
}
/**
* 辅助算法2：设置顶点数据
*/
public void setVertices(char[] vertices) {
this.vertices = vertices;
}
}
- Test Results
深度优先遍历（递归）
A B C D E F G H I
深度优先遍历（非递归）
A B C D E F G H I
-
Special attention
to the storage structure of FIG = adjacency table implementation, only the two-dimensional array can be stored side chain into. -
Graph storage structure = performance comparison of adjacency matrix / adjacency list
5.3.2 Breadth First Traversal (BFS)
- Introduction
- Algorithm diagram
- specific process
Note:
Gratio ofIthe reasons to use an array of memory access = vertices, theGscale ratioIsubscript smaller lower (inABCDEFGHIorder of storage)
- Implementation
Non-recursive: use queue
import java.util.LinkedList;
import java.util.Queue;
import java.util.Stack;
public class MyGraph {
/**
* 准备工作1：设置变量
*/
private int vexnum; // 存放图中顶点数量
private char[] vertices; // 存放结点数据
private int [][] arcs; // 存放图的所有边
private boolean[] visited;// 记录节点是否已被遍历
/**
* 准备工作2：初始化图的顶点数量、数据 & 边
*/
public MyGraph(int n){
vexnum = n;
vertices = new char[n];
visited = new boolean[n];
arcs = new int[n][n];
}
/**
* 广度优先搜索 算法实现
* 原理：非递归（采用队列）
*/
public void BFS(){
// 1. 初始化所有顶点的访问标记
// 即，设置为未访问状态
for (int i = 0; i < vexnum; i++) {
visited[i] = false;
}
// 2. 创建队列
Queue<Integer> q=new LinkedList<Integer>();
// 3. 对所有顶点做遍历循环（从第1个顶点开始）
// 若遍历完毕，则结束整个层序遍历
for(int i=0;i < vexnum;i++){
// 4. 若当前顶点未被访问，就进行处理
// 若当前顶点已被访问，则回到3进行判断
if( visited[i]==false ) {
// 5. （输出）访问当前顶点
visit(i);
// 6. 标记当前顶点已被访问
visited[i] = true;
// 7. 入队当前顶点
q.add(i);
// 8.判断当前队列是否为空
// 若为空则跳出循环，回到3进行判断
while(!q.isEmpty()) {
// 9. 出队队首元素 & 将出队的元素 赋值为 当前顶点
i = q.poll();
// 10. 遍历当前顶点的所有邻接点
// 若遍历完毕，则回到8判断
for(int j=0; j< vexnum ; j++){
// 11. 若当前顶点的邻接顶点存在 & 未被访问，则执行处理
// 否则回到10判断
if(arcs[i][j]==1 && visited[j]==false){
// 12. （输出）访问当前顶点的邻接顶点
visit(j);
// 13. 标记当前顶点的邻接顶点已被访问
visited[j] = true;
// 14. 入队当前顶点的邻接顶点
q.add(j);
}
}
}
}
}
}
/**
* 辅助算法1：访问该顶点
*/
public void visit(int i){
System.out.print(vertices[i] + " ");
}
/**
* 测试算法
*/
public static void main(String[] args) {
// 1. 初始化图的结构（顶点数量 = 9
MyGraph g = new MyGraph(9);
// 2. 设置顶点数据
char[] vertices = {'A','B','C','D','E','F','G','H','I'};
g.setVertices(vertices);
// 3. 设置边
g.addEdge(0, 1);
g.addEdge(0, 5);
g.addEdge(1, 0);
g.addEdge(1, 2);
g.addEdge(1, 6);
g.addEdge(1, 8);
g.addEdge(2, 1);
g.addEdge(2, 3);
g.addEdge(2, 8);
g.addEdge(3, 2);
g.addEdge(3, 4);
g.addEdge(3, 6);
g.addEdge(3, 7);
g.addEdge(3, 8);
g.addEdge(4, 3);
g.addEdge(4, 5);
g.addEdge(4, 7);
g.addEdge(5, 0);
g.addEdge(5, 4);
g.addEdge(5, 6);
g.addEdge(6, 1);
g.addEdge(6, 3);
g.addEdge(6, 5);
g.addEdge(6, 7);
g.addEdge(7, 3);
g.addEdge(7, 4);
g.addEdge(7, 6);
g.addEdge(8, 1);
g.addEdge(8, 2);
g.addEdge(8, 3);
// 4. 执行 图的广度优先遍历（非递归）
System.out.print("广度优先遍历（非递归）：");
g.BFS();
}
/**
* 辅助算法1：添加边(无向图)
*/
public void addEdge(int i, int j) {
// 边的头尾不能为同一节点
if (i == j) return;
// 将邻接矩阵的第i行第j列的元素值置为1；
arcs[i][j] = 1;
// 将邻接矩阵的第j行第i列的元素值置为1；
arcs[j][i] = 1;
// 设置为1代表2顶点之间存在 边 （设置相等原因 = 邻接矩阵 是对称的）
}
/**
* 辅助算法2：设置顶点数据
*/
public void setVertices(char[] vertices) {
this.vertices = vertices;
}
}
- Results of the
广度优先遍历（非递归）：A B F C G I E D H
5.4 Comparison of traversal methods
6. Minimum spanning tree
This section explains the drawing minimum spanning tree
6.1 Definition
Constructing minimum cost spanning tree of connected network graph
- Net graph: graph with weights
- Minimal cost: Use (n-1) edges to connect a connected graph with n vertices & minimize the sum of weights
6.2 Algorithm for finding the minimum spanning tree
- Mainly include:
Prim(Kruskal) Primm algorithm & ( ) Kruskal algorithm - The details are as follows
6.2.1 (Prim) Prim's algorithm
- Algorithm overview
- Schematic diagram of algorithm principle flow
- for example
6.2.2 (Kruskal) Kruskal Algorithm
- Algorithm overview
6.3 Algorithm comparison
7. The shortest path
7.1 Definition
- For non-net graphs (without weight), the shortest path = the path with the least number of edges between two vertices
- For the network graph (weights): the shortest path = the weight of the edge passed between two vertices and the least path
1st vertex = source point, 2nd vertex = end point
7.2 Problems to be solved
The shortest path from the source point -> the remaining vertices
7.3 Find the shortest path algorithm
-
Mainly include: Dijkstra's algorithm
（Dijkstra）, Freud's algorithm（Floyd） -
The details are as follows
8. Summary
- This article mainly explains the graphs in the data structure
- Now I will continue to data structures , are interested can continue to focus on Carson_Ho Android Development Notes