Shortest Path Faster Algorithm

Prerequisites: Bellman-Ford Algorithm
Given a directed weighted graph with V vertices, E edges and a source vertex S. The task is to find the shortest path from the source vertex to all other vertices in the given graph.
Example:

Input: V = 5, S = 1, arr = {{1, 2, 1}, {2, 3, 7}, {2, 4, -2}, {1, 3, 8}, {1, 4, 9}, {3, 4, 3}, {2, 5, 3}, {4, 5, -3}} 
Output:
1, 0 
2, 1 
3, 8 
4, -1 
5, -4 
Explanation: For the given input, the shortest path from 1 to 1 is 0, 1 to 2 is 1 and so on.
Input: V = 5, S = 1, arr = {{1, 2, -1}, {1, 3, 4}, {2, 3, 3}, {2, 4, 2}, {2, 5, 2}, {4, 3, 5}, {4, 2, 1}, {5, 4, 3}} 
Output:
1, 0 
2, -1 
3, 2 
4, 1 
5, 1 

Approach: The shortest path faster algorithm is based on Bellman-Ford algorithm where every vertex is used to relax its adjacent vertices but in SPF algorithm, a queue of vertices is maintained and a vertex is added to the queue only if that vertex is relaxed. This process repeats until no more vertex can be relaxed. 
The following steps can be performed to compute the result: 

1. Create an array d[] to store the shortest distance of all vertex from the source vertex. Initialize this array by infinity except for d[S] = 0 where S is the source vertex.
2. Create a queue Q and push starting source vertex in it. 
   • while Queue is not empty, do the following for each edge(u, v) in the graph 
     • If d[v] > d[u] + weight of edge(u, v)
     • d[v] = d[u] + weight of edge(u, v)
     • If vertex v is not present in Queue, then push the vertex v into the Queue.

Note: The term relaxation means updating the cost of all vertices connected to a vertex v if those costs would be improved by including the path via vertex v. This can be understood better from an analogy between the estimate of the shortest path and the length of a helical tension spring, which is not designed for compression. Initially, the cost of the shortest path is an overestimate, likened to a stretched-out spring. As shorter paths are found, the estimated cost is lowered, and the spring is relaxed. Eventually, the shortest path, if one exists, is found and the spring has been relaxed to its resting length.
Below is the implementation of the above approach: 

C++

// C++ implementation of SPFA

#include <bits/stdc++.h>
using namespace std;

// Graph is stored as vector of vector of pairs
// first element of pair store vertex
// second element of pair store weight
vector<vector<pair<int, int> >> graph;

// Function to add edges in the graph
// connecting a pair of vertex(frm) and weight
// to another vertex(to) in graph
void addEdge(int frm, int to, int weight)
{

    graph[frm].push_back({ to, weight });
}

// Function to print shortest distance from source
void print_distance(int d[], int V)
{
    cout << "Vertex \t\t Distance"
         << " from source" << endl;

    for (int i = 1; i <= V; i++) {
        cout << i << "             " << d[i] << '\n';
    }
}

// Function to compute the SPF algorithm
void shortestPathFaster(int S, int V)
{
    // Create array d to store shortest distance
    int d[V + 1];

    // Boolean array to check if vertex
    // is present in queue or not
    bool inQueue[V + 1] = { false };

    // Initialize the distance from source to
    // other vertex as INT_MAX(infinite)
    for (int i = 0; i <= V; i++) {
        d[i] = INT_MAX;
    }
    d[S] = 0;

    queue<int> q;
    q.push(S);
    inQueue[S] = true;

    while (!q.empty()) {

        // Take the front vertex from Queue
        int u = q.front();
        q.pop();
        inQueue[u] = false;

        // Relaxing all the adjacent edges of
        // vertex taken from the Queue
        for (int i = 0; i < graph[u].size(); i++) {

            int v = graph[u][i].first;
            int weight = graph[u][i].second;

            if (d[v] > d[u] + weight) {
                d[v] = d[u] + weight;

                // Check if vertex v is in Queue or not
                // if not then push it into the Queue
                if (!inQueue[v]) {
                    q.push(v);
                    inQueue[v] = true;
                }
            }
        }
    }

    // Print the result
    print_distance(d, V);
}

// Driver code
int main()
{
    int V = 5;
    int S = 1;

      graph = vector<vector<pair<int,int>>> (V+1);
    // Connect vertex a to b with weight w
    // addEdge(a, b, w)

    addEdge(1, 2, 1);
    addEdge(2, 3, 7);
    addEdge(2, 4, -2);
    addEdge(1, 3, 8);
    addEdge(1, 4, 9);
    addEdge(3, 4, 3);
    addEdge(2, 5, 3);
    addEdge(4, 5, -3);

    // Calling shortestPathFaster function
    shortestPathFaster(S, V);

    return 0;
}

Java

// Java implementation of SPFA
import java.util.*;

class GFG
{
    static class pair
    {
        int first, second;
        public pair(int first, int second)
        {
            this.first = first;
            this.second = second;
        }
    }

// Graph is stored as vector of vector of pairs
// first element of pair store vertex
// second element of pair store weight
static Vector<pair > []graph = new Vector[100000];

// Function to add edges in the graph
// connecting a pair of vertex(frm) and weight
// to another vertex(to) in graph
static void addEdge(int frm, int to, int weight)
{

    graph[frm].add(new pair( to, weight ));
}

// Function to print shortest distance from source
static void print_distance(int d[], int V)
{
    System.out.print("Vertex \t\t Distance"
        + " from source" +"\n");

    for (int i = 1; i <= V; i++)
    {
        System.out.printf("%d \t\t %d\n", i, d[i]);
    }
}

// Function to compute the SPF algorithm
static void shortestPathFaster(int S, int V)
{
    // Create array d to store shortest distance
    int []d = new int[V + 1];

    // Boolean array to check if vertex
    // is present in queue or not
    boolean []inQueue = new boolean[V + 1];

    // Initialize the distance from source to
    // other vertex as Integer.MAX_VALUE(infinite)
    for (int i = 0; i <= V; i++)
    {
        d[i] = Integer.MAX_VALUE;
    }
    d[S] = 0;

    Queue<Integer> q = new LinkedList<>();
    q.add(S);
    inQueue[S] = true;

    while (!q.isEmpty())
    {

        // Take the front vertex from Queue
        int u = q.peek();
        q.remove();
        inQueue[u] = false;

        // Relaxing all the adjacent edges of
        // vertex taken from the Queue
        for (int i = 0; i < graph[u].size(); i++)
        {

            int v = graph[u].get(i).first;
            int weight = graph[u].get(i).second;

            if (d[v] > d[u] + weight)
            {
                d[v] = d[u] + weight;

                // Check if vertex v is in Queue or not
                // if not then push it into the Queue
                if (!inQueue[v])
                {
                    q.add(v);
                    inQueue[v] = true;
                }
            }
        }
    }

    // Print the result
    print_distance(d, V);
}

// Driver code
public static void main(String[] args)
{
    int V = 5;
    int S = 1;
    for (int i = 0; i < graph.length; i++)
    {
        graph[i] = new Vector<pair>();
    }

    // Connect vertex a to b with weight w
    // addEdge(a, b, w)
    addEdge(1, 2, 1);
    addEdge(2, 3, 7);
    addEdge(2, 4, -2);
    addEdge(1, 3, 8);
    addEdge(1, 4, 9);
    addEdge(3, 4, 3);
    addEdge(2, 5, 3);
    addEdge(4, 5, -3);

    // Calling shortestPathFaster function
    shortestPathFaster(S, V);
}
}

// This code is contributed by 29AjayKumar

C#

// C# implementation of SPFA
using System;
using System.Collections.Generic;

class GFG
{
    class pair
    {
        public int first, second;
        public pair(int first, int second)
        {
            this.first = first;
            this.second = second;
        }
    }

// Graph is stored as vector of vector of pairs
// first element of pair store vertex
// second element of pair store weight
static List<pair> []graph = new List<pair>[100000];

// Function to add edges in the graph
// connecting a pair of vertex(frm) and weight
// to another vertex(to) in graph
static void addEdge(int frm, int to, int weight)
{

    graph[frm].Add(new pair( to, weight ));
}

// Function to print shortest distance from source
static void print_distance(int []d, int V)
{
    Console.Write("Vertex \t\t Distance"
        + " from source" +"\n");

    for (int i = 1; i <= V; i++)
    {
        Console.Write("{0} \t\t {1}\n", i, d[i]);
    }
}

// Function to compute the SPF algorithm
static void shortestPathFaster(int S, int V)
{
    // Create array d to store shortest distance
    int []d = new int[V + 1];

    // Boolean array to check if vertex
    // is present in queue or not
    bool []inQueue = new bool[V + 1];

    // Initialize the distance from source to
    // other vertex as int.MaxValue(infinite)
    for (int i = 0; i <= V; i++)
    {
        d[i] = int.MaxValue;
    }
    d[S] = 0;

    Queue<int> q = new Queue<int>();
    q.Enqueue(S);
    inQueue[S] = true;

    while (q.Count!=0)
    {

        // Take the front vertex from Queue
        int u = q.Peek();
        q.Dequeue();
        inQueue[u] = false;

        // Relaxing all the adjacent edges of
        // vertex taken from the Queue
        for (int i = 0; i < graph[u].Count; i++)
        {

            int v = graph[u][i].first;
            int weight = graph[u][i].second;

            if (d[v] > d[u] + weight)
            {
                d[v] = d[u] + weight;

                // Check if vertex v is in Queue or not
                // if not then push it into the Queue
                if (!inQueue[v])
                {
                    q.Enqueue(v);
                    inQueue[v] = true;
                }
            }
        }
    }

    // Print the result
    print_distance(d, V);
}

// Driver code
public static void Main(String[] args)
{
    int V = 5;
    int S = 1;
    for (int i = 0; i < graph.Length; i++)
    {
        graph[i] = new List<pair>();
    }

    // Connect vertex a to b with weight w
    // addEdge(a, b, w)
    addEdge(1, 2, 1);
    addEdge(2, 3, 7);
    addEdge(2, 4, -2);
    addEdge(1, 3, 8);
    addEdge(1, 4, 9);
    addEdge(3, 4, 3);
    addEdge(2, 5, 3);
    addEdge(4, 5, -3);

    // Calling shortestPathFaster function
    shortestPathFaster(S, V);
}
}

// This code is contributed by PrinciRaj1992

Python3

# Python3 implementation of SPFA
from collections import deque

# Graph is stored as vector of vector of pairs
# first element of pair store vertex
# second element of pair store weight
graph = [[] for _ in range(100000)]

# Function to add edges in the graph
# connecting a pair of vertex(frm) and weight
# to another vertex(to) in graph
def addEdge(frm, to, weight):

    graph[frm].append([to, weight])

# Function to print shortest distance from source
def print_distance(d, V):
    print("Vertex","\t","Distance from source")

    for i in range(1, V + 1):
        print(i,"\t",d[i])

# Function to compute the SPF algorithm
def shortestPathFaster(S, V):

    # Create array d to store shortest distance
    d = [10**9]*(V + 1)

    # Boolean array to check if vertex
    # is present in queue or not
    inQueue = [False]*(V + 1)

    d[S] = 0

    q = deque()
    q.append(S)
    inQueue[S] = True

    while (len(q) > 0):

        # Take the front vertex from Queue
        u = q.popleft()
        inQueue[u] = False

        # Relaxing all the adjacent edges of
        # vertex taken from the Queue
        for i in range(len(graph[u])):

            v = graph[u][i][0]
            weight = graph[u][i][1]

            if (d[v] > d[u] + weight):
                d[v] = d[u] + weight

                # Check if vertex v is in Queue or not
                # if not then append it into the Queue
                if (inQueue[v] == False):
                    q.append(v)
                    inQueue[v] = True

    # Print the result
    print_distance(d, V)

# Driver code
if __name__ == '__main__':
    V = 5
    S = 1

    # Connect vertex a to b with weight w
    # addEdge(a, b, w)

    addEdge(1, 2, 1)
    addEdge(2, 3, 7)
    addEdge(2, 4, -2)
    addEdge(1, 3, 8)
    addEdge(1, 4, 9)
    addEdge(3, 4, 3)
    addEdge(2, 5, 3)
    addEdge(4, 5, -3)

    # Calling shortestPathFaster function
    shortestPathFaster(S, V)

# This code is contributed by mohit kumar 29

Javascript

<script>

// JavaScript implementation of SPFA

 // Graph is stored as vector of vector of pairs
// first element of pair store vertex
// second element of pair store weight
    let graph=new Array(100000);

    // Function to add edges in the graph
// connecting a pair of vertex(frm) and weight
// to another vertex(to) in graph
    function addEdge(frm,to,weight)
    {
        graph[frm].push([to, weight ]);
    }

    // Function to print shortest distance from source
    function print_distance(d,V)
    {
        document.write(
   "Vertex", " ", "Distance" + " from source" +"<br>"
        );

    for (let i = 1; i <= V; i++)
    {
        document.write( i+"     "+
        d[i]+"<br>");
    }
    }

    // Function to compute the SPF algorithm
    function shortestPathFaster(S,V)
    {
        // Create array d to store shortest distance
    let d = new Array(V + 1);

    // Boolean array to check if vertex
    // is present in queue or not
    let inQueue = new Array(V + 1);

    // Initialize the distance from source to
    // other vertex as Integer.MAX_VALUE(infinite)
    for (let i = 0; i <= V; i++)
    {
        d[i] = Number.MAX_VALUE;
    }
    d[S] = 0;

    let q = [];
    q.push(S);
    inQueue[S] = true;

    while (q.length!=0)
    {

        // Take the front vertex from Queue
        let u = q[0];
        q.shift();
        inQueue[u] = false;

        // Relaxing all the adjacent edges of
        // vertex taken from the Queue
        for (let i = 0; i < graph[u].length; i++)
        {

            let v = graph[u][i][0];
            let weight = graph[u][i][1];

            if (d[v] > d[u] + weight)
            {
                d[v] = d[u] + weight;

                // Check if vertex v is in Queue or not
                // if not then push it into the Queue
                if (!inQueue[v])
                {
                    q.push(v);
                    inQueue[v] = true;
                }
            }
        }
    }

    // Print the result
    print_distance(d, V);
    }

    // Driver code
    let V = 5;
    let S = 1;
    for (let i = 0; i < graph.length; i++)
    {
        graph[i] = [];
    }

    // Connect vertex a to b with weight w
    // addEdge(a, b, w)
    addEdge(1, 2, 1);
    addEdge(2, 3, 7);
    addEdge(2, 4, -2);
    addEdge(1, 3, 8);
    addEdge(1, 4, 9);
    addEdge(3, 4, 3);
    addEdge(2, 5, 3);
    addEdge(4, 5, -3);

    // Calling shortestPathFaster function
    shortestPathFaster(S, V);

// This code is contributed by unknown2108

</script>

Output:

Vertex    Distance from source
1          0
2          1
3          8
4          -1
5          -4

Time Complexity:
Average Time Complexity: O(|E|) 
Worstcase Time Complexity: O(|V|.|E|) 

Space complexity: O(V),The space complexity is O(V) since we need to store the distances for each vertex in an array.

Note: Bound on average runtime has not been proved yet.
References: Shortest Path Faster Algorithm