图论14-最短路径-Dijkstra算法+Bellman-Ford算法+Floyed算法

0 代码仓库

https://github.com/Chufeng-Jiang/Graph-Theory/tree/main/src/Chapter11_Min_Path

1 Dijkstra算法

2 Dijkstra算法的实现

2.1 设置距离数组

//用于存储从源点到当前节点的距离，并初始化
dis = new int[G.V()];
Arrays.fill(dis, Integer.MAX_VALUE);
dis[s] = 0;

2.2 找到当前路径的最小值 curdis，及对应的该顶点cur

int cur = -1, curdis = Integer.MAX_VALUE;

for(int v = 0; v < G.V(); v ++)
    if(!visited[v] && dis[v] < curdis){
        curdis = dis[v];
        cur = v;
    }

2.3 更新权重

visited[cur] = true;
for(int w: G.adj(cur))
   if(!visited[w]){
       if(dis[cur] + G.getWeight(cur, w) < dis[w])
           dis[w] = dis[cur] + G.getWeight(cur, w);
   }

2.4 其他接口

2.4.1 判断某个顶点的连通性

public boolean isConnectedTo(int v){
    G.validateVertex(v);
    return visited[v];
}

2.4.2 求源点s到某个顶点的最短路径

public int distTo(int v){
    G.validateVertex(v);
    return dis[v];
}

3使用优先队列优化-Dijkstra算法

3.1 设计内部类node

存放节点编号和距离

    private class Node implements Comparable<Node>{

        public int v, dis;

        public Node(int v, int dis){
            this.v = v;
            this.dis = dis;
        }

        @Override
        public int compareTo(Node another){
            return dis - another.dis;
        }
    }

3.2 入队

PriorityQueue<Node> pq = new PriorityQueue<Node>();

pq.add(new Node(s, 0));

这里的缺点就是，更新node时候，会重复添加节点相同的node，但是路径值不一样。不影响最后结果。

while(!pq.isEmpty()){

    int cur = pq.remove().v;
    if(visited[cur]) continue;

    visited[cur] = true;
    for(int w: G.adj(cur))
        if(!visited[w]){
            if(dis[cur] + G.getWeight(cur, w) < dis[w]){
                dis[w] = dis[cur] + G.getWeight(cur, w);
                pq.add(new Node(w, dis[w]));
                pre[w] = cur;
            }
        }
}

3.3 记录路径

private int[] pre;

• 更新pre数组

for(int w: G.adj(cur))
    if(!visited[w]){
        if(dis[cur] + G.getWeight(cur, w) < dis[w]){
            dis[w] = dis[cur] + G.getWeight(cur, w);
            pq.add(new Node(w, dis[w]));
            pre[w] = cur;
        }
    }

• 输出路径

    public Iterable<Integer> path(int t){

        ArrayList<Integer> res = new ArrayList<>();
        if(!isConnectedTo(t)) return res;

        int cur = t;
        while(cur != s){
            res.add(cur);
            cur = pre[cur];
        }
        res.add(s);

        Collections.reverse(res);
        return res;
    }

3.4 整体

package Chapter11_Min_Path.Dijkstra_pq;

import java.util.ArrayList;
import java.util.Arrays;
import java.util.Collections;
import java.util.PriorityQueue;


public class Dijkstra {

    private WeightedGraph G;
    private int s;
    private int[] dis;
    private boolean[] visited;
    private int[] pre;

    private class Node implements Comparable<Node>{

        public int v, dis;

        public Node(int v, int dis){
            this.v = v;
            this.dis = dis;
        }

        @Override
        public int compareTo(Node another){
            return dis - another.dis;
        }
    }

    public Dijkstra(WeightedGraph G, int s){

        this.G = G;

        G.validateVertex(s);
        this.s = s;

        dis = new int[G.V()];
        Arrays.fill(dis, Integer.MAX_VALUE);

        pre = new int[G.V()];
        Arrays.fill(pre, -1);

        dis[s] = 0;
        pre[s] = s;
        visited = new boolean[G.V()];

        PriorityQueue<Node> pq = new PriorityQueue<Node>();
        pq.add(new Node(s, 0));

        while(!pq.isEmpty()){

            int cur = pq.remove().v;
            if(visited[cur]) continue;

            visited[cur] = true;
            for(int w: G.adj(cur))
                if(!visited[w]){
                    if(dis[cur] + G.getWeight(cur, w) < dis[w]){
                        dis[w] = dis[cur] + G.getWeight(cur, w);
                        pq.add(new Node(w, dis[w]));
                        pre[w] = cur;
                    }
                }
        }
    }

    public boolean isConnectedTo(int v){

        G.validateVertex(v);
        return visited[v];
    }

    public int distTo(int v){

        G.validateVertex(v);
        return dis[v];
    }

    public Iterable<Integer> path(int t){

        ArrayList<Integer> res = new ArrayList<>();
        if(!isConnectedTo(t)) return res;

        int cur = t;
        while(cur != s){
            res.add(cur);
            cur = pre[cur];
        }
        res.add(s);

        Collections.reverse(res);
        return res;
    }

    static public void main(String[] args){

        WeightedGraph g = new WeightedGraph("g.txt");
        Dijkstra dij = new Dijkstra(g, 0);
        for(int v = 0; v < g.V(); v ++)
            System.out.print(dij.distTo(v) + " ");
        System.out.println();

        System.out.println(dij.path(3));
    }
}

4 Bellman-Ford算法

4.1 松弛操作

4.2 负权环

4.3 算法思想

4.4 进行V-1次松弛操作

// 进行V-1次松弛操作
for(int pass = 1; pass < G.V(); pass ++){
    for(int v = 0; v < G.V(); v ++)
        for(int w: G.adj(v))
            if(dis[v] != Integer.MAX_VALUE && // 避免对无穷值的点进行松弛操作
               dis[v] + G.getWeight(v, w) < dis[w]){
                dis[w] = dis[v] + G.getWeight(v, w);
                pre[w] = v;
            }
}

4.5 判断负权环

// 多进行一次操作，如果还有更新，那么存在负权换
for(int v = 0; v < G.V(); v ++)
    for(int w : G.adj(v))
        if(dis[v] != Integer.MAX_VALUE &&
           dis[v] + G.getWeight(v, w) < dis[w])
            hasNegCycle = true;

4.6 整体

package Chapter11_Min_Path.BellmanFord;

import java.util.ArrayList;
import java.util.Arrays;
import java.util.Collections;

public class BellmanFord {

    private WeightedGraph G;
    private int s;
    private int[] dis;
    private int[] pre;
    private boolean hasNegCycle = false;

    public BellmanFord(WeightedGraph G, int s){

        this.G = G;

        G.validateVertex(s);
        this.s = s;

        dis = new int[G.V()];
        Arrays.fill(dis, Integer.MAX_VALUE);
        dis[s] = 0;

        pre = new int[G.V()];
        Arrays.fill(pre, -1);

        // 进行V-1次松弛操作
        for(int pass = 1; pass < G.V(); pass ++){
            for(int v = 0; v < G.V(); v ++)
                for(int w: G.adj(v))
                    if(dis[v] != Integer.MAX_VALUE && // 避免对无穷值的点进行松弛操作
                       dis[v] + G.getWeight(v, w) < dis[w]){
                        dis[w] = dis[v] + G.getWeight(v, w);
                        pre[w] = v;
                    }
        }

        // 多进行一次操作，如果还有更新，那么存在负权换
        for(int v = 0; v < G.V(); v ++)
            for(int w : G.adj(v))
                if(dis[v] != Integer.MAX_VALUE &&
                   dis[v] + G.getWeight(v, w) < dis[w])
                    hasNegCycle = true;
    }

    public boolean hasNegativeCycle(){
        return hasNegCycle;
    }

    public boolean isConnectedTo(int v){
        G.validateVertex(v);
        return dis[v] != Integer.MAX_VALUE;
    }

    public int distTo(int v){
        G.validateVertex(v);
        if(hasNegCycle) throw new RuntimeException("exist negative cycle.");
        return dis[v];
    }

    public Iterable<Integer> path(int t){

        ArrayList<Integer> res = new ArrayList<Integer>();
        if(!isConnectedTo(t)) return res;

        int cur = t;
        while(cur != s){
            res.add(cur);
            cur = pre[cur];
        }
        res.add(s);

        Collections.reverse(res);
        return res;
    }

    static public void main(String[] args){

        WeightedGraph g = new WeightedGraph("gw2.txt");
        BellmanFord bf = new BellmanFord(g, 0);
        if(!bf.hasNegativeCycle()){
            for(int v = 0; v < g.V(); v ++)
                System.out.print(bf.distTo(v) + " ");
            System.out.println();

            System.out.println(bf.path(3));
        }
        else
            System.out.println("exist negative cycle.");

        WeightedGraph g2 = new WeightedGraph("g2.txt");
        BellmanFord bf2 = new BellmanFord(g2, 0);
        if(!bf2.hasNegativeCycle()){
            for(int v = 0; v < g2.V(); v ++)
                System.out.print(bf2.distTo(v) + " ");
            System.out.println();
        }
        else
            System.out.println("exist negative cycle.");
    }
}

5 Floyed算法

5.1 设置记录两点最短距离的数组，并初始化两点之间的距离

private int[][] dis;

• 初始化两点之间的距离

for(int v = 0; v < G.V(); v ++){
    dis[v][v] = 0;
    for(int w: G.adj(v))
        dis[v][w] = G.getWeight(v, w);
}

5.2 更新两点之间的距离

第一重循环：测试两点之间经过点t是否存在更短的路径。

        for(int t = 0; t < G.V(); t ++)
        
            for(int v = 0; v < G.V(); v ++)
                for(int w = 0; w < G.V(); w ++)
                    if(dis[v][t] != Integer.MAX_VALUE && dis[t][w] != Integer.MAX_VALUE
                       && dis[v][t] + dis[t][w] < dis[v][w])
                        dis[v][w] = dis[v][t] + dis[t][w];