Bellman–Ford and SPFA Algorithm

Bellman–Ford Algorithm 是由 Richard Bellman（DP创始人） 和 Ford 共同创立的，求解单源最短路径问题的一种算法。它的原理是对图进行顶点个数减一次的松弛操作（逐渐逼近最终解的过程），得到所有顶点到源点的最短路径。该算法优于dijkstra的方面在于其边的权值可以是负数（负权环可以无限降低cost）、实现简单，缺点是时间复杂度过高，为O(V*E)。

package datastructure.graph;

// A Java program for Bellman-Ford's single source shortest path
// algorithm.

// A class to represent a connected, directed and weighted graph
class BF {
    // A class to represent a weighted edge in graph
    class Edge {
        int src, dest, weight;
        Edge()
        {
            src = dest = weight = 0;
        }
    };

    int V, E;
    Edge edge[];

    // Creates a graph with V vertices and E edges
    BF(int v, int e)
    {
        V = v;
        E = e;
        edge = new Edge[e];
        for (int i = 0; i < e; ++i)
            edge[i] = new Edge();
    }

    // The main function that finds shortest distances from src
    // to all other vertices using Bellman-Ford algorithm. The
    // function also detects negative weight cycle
    void BellmanFord(BF graph, int src)
    {
        int V = graph.V, E = graph.E;
        int dist[] = new int[V];

        // Step 1: Initialize distances from src to all other
        // vertices as INFINITE
        for (int i = 0; i < V; ++i)
            dist[i] = Integer.MAX_VALUE;
        dist[src] = 0;

        // Step 2: Relax all edges |V| - 1 times. A simple
        // shortest path from src to any other vertex can
        // have at-most |V| - 1 edges
        for (int i = 1; i < V; ++i) {
            for (int j = 0; j < E; ++j) {
                int u = graph.edge[j].src;
                int v = graph.edge[j].dest;
                int weight = graph.edge[j].weight;
                if (dist[u] != Integer.MAX_VALUE && dist[u] + weight < dist[v])
                    dist[v] = dist[u] + weight;
            }
        }

        // Step 3: check for negative-weight cycles. The above
        // step guarantees shortest distances if graph doesn't
        // contain negative weight cycle. If we get a shorter
        // path, then there is a cycle.
        for (int j = 0; j < E; ++j) {
            int u = graph.edge[j].src;
            int v = graph.edge[j].dest;
            int weight = graph.edge[j].weight;
            if (dist[u] != Integer.MAX_VALUE && dist[u] + weight < dist[v]) {
                System.out.println("Graph contains negative weight cycle");
                return;
            }
        }
        printArr(dist, V);
    }

    // A utility function used to print the solution
    void printArr(int dist[], int V)
    {
        System.out.println("Vertex Distance from Source");
        for (int i = 0; i < V; ++i)
            System.out.println(i + "		" + dist[i]);
    }

    // Driver method to test above function
    public static void main(String[] args)
    {
        int V = 5; // Number of vertices in graph
        int E = 8; // Number of edges in graph

        BF graph = new BF(V, E);

        // add edge 0-1 (or A-B in above figure)
        graph.edge[0].src = 0;
        graph.edge[0].dest = 1;
        graph.edge[0].weight = -1;

        // add edge 0-2 (or A-C in above figure)
        graph.edge[1].src = 0;
        graph.edge[1].dest = 2;
        graph.edge[1].weight = 4;

        // add edge 1-2 (or B-C in above figure)
        graph.edge[2].src = 1;
        graph.edge[2].dest = 2;
        graph.edge[2].weight = 3;

        // add edge 1-3 (or B-D in above figure)
        graph.edge[3].src = 1;
        graph.edge[3].dest = 3;
        graph.edge[3].weight = 2;

        // add edge 1-4 (or A-E in above figure)
        graph.edge[4].src = 1;
        graph.edge[4].dest = 4;
        graph.edge[4].weight = 2;

        // add edge 3-2 (or D-C in above figure)
        graph.edge[5].src = 3;
        graph.edge[5].dest = 2;
        graph.edge[5].weight = 5;

        // add edge 3-1 (or D-B in above figure)
        graph.edge[6].src = 3;
        graph.edge[6].dest = 1;
        graph.edge[6].weight = 1;

        // add edge 4-3 (or E-D in above figure)
        graph.edge[7].src = 4;
        graph.edge[7].dest = 3;
        graph.edge[7].weight = -3;

        graph.BellmanFord(graph, 0);
    }
}
// Contributed by Aakash Hasija

Shortest Path Faster Algorithm (SPFA)  - Bellman–Ford的队列改进算法

The Shortest Path Faster Algorithm (SPFA) is an improvement of the Bellman–Ford algorithm which computes single-source shortest paths in a weighted directed graph. The algorithm is believed to work well on random sparse graphs（稀疏图） and is particularly suitable for graphs that contain negative-weight edges.

• Worst case time complexity: Θ(VE)

SPFA(v):
    d[i] = infinity for each vertex i
    d[v] = 0
    queue q
    q.push(v)
    while q is not empty
        u = q.front()
        q.pop()
        for each i in adj[u]
            if d[i] > d[u] + w(u,i)
                then d[i] = d[u] + w(u,i)
                if i is not in q
                    then q.push(i)

https://www.geeksforgeeks.org/bellman-ford-algorithm-dp-23/

https://iq.opengenus.org/shortest-path-faster-algorithm/