邻接矩阵

import java.io.File;
import java.io.IOException;
import java.util.ArrayList;
import java.util.Scanner;public class AdjMatrix {private int V;private int E;private int[][] adj;// 构造函数，从文件内容初始化邻接矩阵public AdjMatrix(String filename){File file = new File(filename);try(Scanner scanner = new Scanner(file)){V = scanner.nextInt(); //读取第一行第一个数if(V < 0) throw new IllegalArgumentException("V must be non-negative");adj = new int[V][V]; //构造邻接矩阵if(E < 0) throw new IllegalArgumentException("E must be non-negative");E = scanner.nextInt(); //读取第一行第二个数// E是边的数量，在g.txt中表示为第二行开始的E行for (int i = 0; i < E; i++) {int a = scanner.nextInt(); //读取边的一个顶点validateVertex(a);int b = scanner.nextInt(); //读取边的另一个顶点validateVertex(b);if(a == b) throw new IllegalArgumentException("Self Loop is Detected"); //判断是够存在自环边if(adj[a][b] == 1) throw new IllegalArgumentException("Parallel Edges are Detected"); //判断是够存在平行l边adj[a][b] = 1; //存在边则设置为1adj[b][a] = 1;}}catch (IOException e){e.printStackTrace();}}private void validateVertex(int v){if(v < 0 || v >= V)throw new IllegalArgumentException("vertex" + v + "is invalid");}public int V(){return V;}public int E(){return E;}public boolean hasEdge(int v, int w){validateVertex(v);validateVertex(w);return adj[v][w] == 1;}public ArrayList<Integer> adj(int v){validateVertex(v);ArrayList<Integer> res = new ArrayList<>();for (int i = 0; i < V; i++) {if(adj[v][i] == 1) res.add(i);}return res;}public int degree(int v){return adj(v).size(); //adj(v)是上方的adj方法，size()是ArrayList的接口}// 用于在控制台打印该临接矩阵@Overridepublic String toString(){StringBuilder sb = new StringBuilder();sb.append(String.format("V = %d, E = %d\n", V, E)); //打印顶点数和边的数量for (int i = 0; i < V; i++) { //行for (int j = 0; j < V; j++) { //列sb.append(String.format("%d",adj[i][j])); //读取矩阵的值}sb.append('\n'); //行尾换行}return sb.toString(); //返回该邻接矩阵}public static void main(String[] args){AdjMatrix adjMatrix = new AdjMatrix("g.txt"); //新建邻接矩阵，并从文件内容初始化System.out.println(adjMatrix);}
}

空间复杂度：O（V^2）
时间复杂度分析：
建图：O(E)
查看两点是否相邻：O(1)
求一个点的相邻节点：O(V)

如果一个图有3000个节点，空间需要3000^2的空间

稀疏图和稠密图: 稀疏和稠密指的是边的多少。

邻接表

import java.io.File;
import java.io.IOException;
import java.util.ArrayList;
import java.util.LinkedList;
import java.util.Scanner;public class AdjList {private int V;private int E;private LinkedList<Integer>[] adj;public AdjList(String filename){File file = new File(filename);try(Scanner scanner = new Scanner(file)){V = scanner.nextInt();if(V < 0) throw new IllegalArgumentException("V must be non-negative");adj = new LinkedList[V]; //构造邻接表, V行，V个LinkedListfor (int i = 0; i < V; i++) {adj[i] = new LinkedList<Integer>();}E = scanner.nextInt(); //读取第一行第二个数if(E < 0) throw new IllegalArgumentException("E must be non-negative");// E是边的数量，在g.txt中表示为第二行开始的E行for (int i = 0; i < E; i++) {int a = scanner.nextInt(); //读取边的一个顶点validateVertex(a);int b = scanner.nextInt(); //读取边的另一个顶点validateVertex(b);if(a == b) throw new IllegalArgumentException("Self Loop is Detected"); //判断是够存在自环边if(adj[a].contains(b)) throw new IllegalArgumentException("Parallel Edges are Detected"); //判断是够存在平行l边adj[a].add(b);adj[b].add(a);}}catch (IOException e){e.printStackTrace();}}private void validateVertex(int v){if(v < 0 || v >= V)throw new IllegalArgumentException("vertex" + v + "is invalid");}public int V(){return V;}public int E(){return E;}public boolean hasEdge(int v, int w){validateVertex(v);validateVertex(w);return adj[v].contains(w);}public LinkedList<Integer> adj(int v){validateVertex(v);return adj[v];}public int degree(int v){return adj(v).size();}@Overridepublic String toString(){StringBuilder sb = new StringBuilder();sb.append(String.format("V = %d, E = %d\n", V, E));for (int i = 0; i < V; i++) {sb.append(String.format("%d:",i));for (int w: adj[i]) {sb.append(String.format("%d ",w));}sb.append('\n');}return sb.toString();}public static void main(String[] args){AdjList adjList = new AdjList("g.txt"); //新建邻接矩阵，并从文件内容初始化System.out.println(adjList);}
}

邻接表的问题和改进

空间复杂度：O(V+E)
时间复杂度：
建图：O(E*V)
查看两点是否相邻：O(degree(v))
求一个点的相邻节点：O(degree(v))

突破时间瓶颈：
快速查重
快速查看两点是否相邻

解决方案：不适用链表，使用哈希表Hashset ( O(1) ) 或者红黑树TreeSet ( O(logV) ).

改为使用红黑树，可以保证数据的顺序，并且时间复杂度相对小

import java.io.File;
import java.io.IOException;
import java.util.LinkedList;
import java.util.Scanner;
import java.util.TreeSet;public class AdjSet {private int V;private int E;private TreeSet<Integer>[] adj;public AdjSet(String filename){File file = new File(filename);try(Scanner scanner = new Scanner(file)){V = scanner.nextInt();if(V < 0) throw new IllegalArgumentException("V must be non-negative");adj = new TreeSet[V]; //构造邻接表, V行，V个LinkedListfor (int i = 0; i < V; i++) {adj[i] = new TreeSet<Integer>();}E = scanner.nextInt(); //读取第一行第二个数if(E < 0) throw new IllegalArgumentException("E must be non-negative");// E是边的数量，在g.txt中表示为第二行开始的E行for (int i = 0; i < E; i++) {int a = scanner.nextInt(); //读取边的一个顶点validateVertex(a);int b = scanner.nextInt(); //读取边的另一个顶点validateVertex(b);if(a == b) throw new IllegalArgumentException("Self Loop is Detected"); //判断是够存在自环边if(adj[a].contains(b)) throw new IllegalArgumentException("Parallel Edges are Detected"); //判断是够存在平行l边adj[a].add(b);adj[b].add(a);}}catch (IOException e){e.printStackTrace();}}private void validateVertex(int v){if(v < 0 || v >= V)throw new IllegalArgumentException("vertex" + v + "is invalid");}public int V(){return V;}public int E(){return E;}public boolean hasEdge(int v, int w){validateVertex(v);validateVertex(w);return adj[v].contains(w);}public Iterable<Integer> adj(int v){ // 可以是TreeSet，但是数组、链表、红黑树都是实现了Iterable的接口，因此可以统一写成这样validateVertex(v);return adj[v];}public int degree(int v){validateVertex(v);return adj[v].size(); // Iterable没有size()方法}@Overridepublic String toString(){StringBuilder sb = new StringBuilder();sb.append(String.format("V = %d, E = %d\n", V, E));for (int i = 0; i < V; i++) {sb.append(String.format("%d:",i));for (int w: adj[i]) {sb.append(String.format("%d ",w));}sb.append('\n');}return sb.toString();}public static void main(String[] args){AdjSet adjSet = new AdjSet("g.txt"); //新建邻接矩阵，并从文件内容初始化System.out.println(adjSet);}
}

总结