圖論系列（一）——圖的基本概念和基本表示

1. 圖的分類

2. 圖的基本概念（聯通分量、聯通圖的生成樹）

3. 圖的基本表示（無向無權圖爲例）

實現代碼：

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;

    AdjMatrix(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");
            E = scanner.nextInt();
            if (E < 0) throw new IllegalArgumentException("E must be non-negative");
            adj = new int[V][V];

            for (int i = 0; i < E; i++) {
                int a = scanner.nextInt();
                validateVertex(a);
                int b = scanner.nextInt();
                validateVertex(b);

                if(a == b)  throw new IllegalArgumentException("Self Loop is Detected");
                if(adj[a][b] == 1)  throw new IllegalArgumentException("Parallel Edges is Detected");

                adj[a][b] = 1;
                adj[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;
    }

    //返回頂點a的所有鄰接點
    public ArrayList<Integer> adj(int v){
        validateVertex(v);
        ArrayList<Integer> ret = new ArrayList<Integer>();
        for (int i = 0; i < V; i++) {
            if(adj[v][i] == 1){
                ret.add(i);
            }
        }

        return ret;
    }

    //返回頂點a的度
    public int degree(int v){
        return adj(v).size();
    }

    @Override
    public String toString() {
        StringBuilder sb = new StringBuilder();
        sb.append(String.format("V = %d, E = %d\n", V, E));
        for (int i = 0; i < V; i++) {
            for (int j = 0; j < V; j++) {
                sb.append(String.format("%d ", adj[i][j]));
            }
            sb.append("\n");
        }

        return sb.toString();
    }

    public static void main(String[] args) {
        AdjMatrix adjMatrix = new AdjMatrix("g.txt");

        System.out.println(adjMatrix);
    }
}

實現方式一：鏈表存儲

import javax.sound.sampled.Line;
import java.io.File;
import java.io.IOException;
import java.util.LinkedList;
import java.util.Scanner;

public class AdjList {
    private int V;
    private int E;
    private LinkedList<Integer>[] adj;

    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");
            E = scanner.nextInt();
            if (E < 0) throw new IllegalArgumentException("E must be non-negative");
            adj = new LinkedList[V];
            for (int i = 0; i < V; i++) {
                adj[i] = new LinkedList<Integer>();
            }

            for (int i = 0; i < E; i++) {
                int a = scanner.nextInt();
                validateVertex(a);
                int b = scanner.nextInt();
                validateVertex(b);

                if (a == b) throw new IllegalArgumentException("Self Loop is Detected");
                //此處查找時間複雜度爲 O(V)
                if (adj[a].contains(b)) throw new IllegalArgumentException("Parallel Edges is Detected");

                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);
    }

    //返回頂點a的所有鄰接點
    public LinkedList<Integer> adj(int v) {
        validateVertex(v);
        return adj[v];
    }

    //返回頂點a的度
    public int degree(int v) {
        return adj(v).size();
    }

    @Override
    public String toString() {
        StringBuilder sb = new StringBuilder();
        sb.append(String.format("V = %d, E = %d\n", V, E));
        for (int v = 0; v < V; v++) {
            sb.append(String.format("%d: ", v));
            for (int w : adj[v]) {
                sb.append(String.format("%d ", w));
            }
            sb.append("\n");
        }

        return sb.toString();
    }

    public static void main(String[] args) {
        AdjList adjList = new AdjList("g.txt");

        System.out.println(adjList);
    }
}

實現方式二：紅黑樹存儲

import com.sun.source.tree.Tree;

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;

    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");
            E = scanner.nextInt();
            if (E < 0) throw new IllegalArgumentException("E must be non-negative");
            adj = new TreeSet[V];
            for (int i = 0; i < V; i++) {
                adj[i] = new TreeSet<Integer>();
            }

            for (int i = 0; i < E; i++) {
                int a = scanner.nextInt();
                validateVertex(a);
                int b = scanner.nextInt();
                validateVertex(b);

                if (a == b) throw new IllegalArgumentException("Self Loop is Detected");
                //此處查找時間複雜度爲 O(logV)
                if (adj[a].contains(b)) throw new IllegalArgumentException("Parallel Edges is Detected");

                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);
    }

    //返回頂點a的所有鄰接點
    public Iterable<Integer> adj(int v) {
        validateVertex(v);
        return adj[v];
    }

    //返回頂點a的度
    public int degree(int v) {
        validateVertex(v);
        return adj[v].size();
    }

    @Override
    public String toString() {
        StringBuilder sb = new StringBuilder();
        sb.append(String.format("V = %d, E = %d\n", V, E));
        for (int v = 0; v < V; v++) {
            sb.append(String.format("%d: ", v));
            for (int w : adj[v]) {
                sb.append(String.format("%d ", w));
            }
            sb.append("\n");
        }

        return sb.toString();
    }

    public static void main(String[] args) {
        AdjSet adjSet = new AdjSet("g.txt");

        System.out.println(adjSet);
    }
}

綜合比較

• 注：本系列參考玩轉算法系列–圖論精講