Sum of shortest distance on source to destination and back having at least a common vertex

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Given a directed weighted graph and the source and destination vertex. The task is to find the sum of shortest distance on the path going from source to destination and then from destination to source such that both the paths have at least a common vertex other than the source and the destination.
Note: On going from destination to source, all the directions of the edges are reversed.
Examples:

Input: src = 0, des = 1

Output: 17
Explanation:
Common vertex is 4 and path is 0 -> 4 -> 3 -> 1 -> 4 -> 0

Approach: The idea is to use Dijkstra’s algorithm. On finding the shortest path from source to destination and shortest path from destination to the source using Dijkstra’s algorithm, it may not result in a path where there is at least one node in common except the source and destination vertex.

Let s be the source vertex and d be destination vertex and v be the intermediate node common in both the paths from source to destination and destination to source. The shortest pair of paths, so that v is in intersection of this two paths is a path: s -> v -> d -> v -> s and it’s length is

dis[s][v] + dis[v][d] + dis[d][v] + dis[v][s]

Since s and d are fixed, just find v such that it gives shortest path.
In order to find such v, follow the below steps:
1. Find shortest distance from all vertices to s and d which gives us the values of dis[v][s] and dis[v][d]. For finding the shortest path from all the vertices to a given node refer Shortest paths from all vertices to a destination.
2. Find shortest distance of all vertex from s and d which gives us d[s][v] and d[d][v].
3. Iterate for all v and find minimum of d[s][v] + d[v][d] + d[d][v] + d[v][s].

Below is the implementation of the above approach:

CPP

// CPP implementation of the approach
 
#include <bits/stdc++.h>
using namespace std;
#define INF 0x3f3f3f3f
 
// iPair represents the Integer Pair
typedef pair<int, int> iPair;
 
// This class represents
// a directed graph using
// adjacency list representation
class Graph {
 
    // Number of vertices
    int V;
 
    // In a weighted graph, store vertex
    // and weight pair for every edge
    list<pair<int, int> >* adj;
 
public:
    // Constructor
    Graph(int V);
 
    // Function to add an edge to graph
    void addEdge(int u, int v, int w);
 
    // Find shortest path from
    // source vertex to all vertex
    void shortestPath(int src,
                      vector<int>& dist);
};
 
// Allocates memory for adjacency list
Graph::Graph(int V)
{
    this->V = V;
    adj = new list<iPair>[V];
}
 
// Function to add an edge to the graph
void Graph::addEdge(int u, int v, int w)
{
    adj[v].push_back(make_pair(u, w));
}
 
// Function to find the shortest paths
// from source to all other vertices
void Graph::shortestPath(int src,
                         vector<int>& dist)
{
 
    // Create a priority queue to
    // store vertices that
    // are being preprocessed
    priority_queue<iPair,
                   vector<iPair>,
                   greater<iPair> >
        pq;
 
    // Insert source itself in priority
    // queue and initialize
    // its distance as 0
    pq.push(make_pair(0, src));
    dist[src] = 0;
 
    // Loop till priority queue
    // becomes empty (or all
    // distances are not finalized)
    while (!pq.empty()) {
 
        // The first vertex in pair
        // is the minimum distance
        // vertex, extract it from
        // priority queue
        int u = pq.top().second;
        pq.pop();
 
        // 'i' is used to get all
        // adjacent vertices of a vertex
        list<pair<int, int> >::iterator i;
        for (i = adj[u].begin(); i != adj[u].end(); ++i) {
 
            // Get vertex label and
            // weight of current
            // adjacent of u
            int v = (*i).first;
            int weight = (*i).second;
 
            // If there is shorted
            // path to v through u
            if (dist[v] > dist[u] + weight) {
 
                // Updating distance of v
                dist[v] = dist[u] + weight;
                pq.push(make_pair(dist[v], v));
            }
        }
    }
}
 
// Function to return the
// required minimum path
int minPath(int V, int src, int des,
            Graph g, Graph r)
{
    // Create a vector for
    // distances and
    // initialize all distances
    // as infinite (INF)
 
    // To store distance of all
    // vertex from source
    vector<int> dist(V, INF);
 
    // To store distance of all
    // vertex from destination
    vector<int> dist2(V, INF);
 
    // To store distance of source
    // from all vertex
    vector<int> dist3(V, INF);
 
    // To store distance of
    // destination from all vertex
    vector<int> dist4(V, INF);
 
    // Computing shortest path from
    // source vertex to all vertices
    g.shortestPath(src, dist);
 
    // Computing shortest path from
    // destination vertex to all vertices
    g.shortestPath(des, dist2);
 
    // Computing shortest path from
    // all the vertices to source
    r.shortestPath(src, dist3);
 
    // Computing shortest path from
    // all the vertices to destination
    r.shortestPath(des, dist4);
 
    // Finding the intermediate node (IN)
    // such that the distance of path
    // src -> IN -> des -> IN -> src is minimum
 
    // To store the shortest distance
    int ans = INT_MAX;
 
    for (int i = 0; i < V; i++) {
 
        // Intermediate node should not be
        // the source and destination
        if (i != des && i != src)
            ans = min(
                ans,
                dist[i] + dist2[i]
                    + dist3[i] + dist4[i]);
    }
 
    // Return the minimum path required
    return ans;
}
 
// Driver code
int main()
{
 
    // Create the graph
    int V = 5;
    int src = 0, des = 1;
 
    // To store the original graph
    Graph g(V);
 
    // To store the reverse graph
    // and compute distance from all
    // vertex to a particular vertex
    Graph r(V);
 
    // Adding edges
    g.addEdge(0, 2, 1);
    g.addEdge(0, 4, 5);
    g.addEdge(1, 4, 1);
    g.addEdge(2, 0, 10);
    g.addEdge(2, 3, 5);
    g.addEdge(3, 1, 1);
    g.addEdge(4, 0, 5);
    g.addEdge(4, 2, 100);
    g.addEdge(4, 3, 5);
 
    // Adding edges in reverse direction
    r.addEdge(2, 0, 1);
    r.addEdge(4, 0, 5);
    r.addEdge(4, 1, 1);
    r.addEdge(0, 2, 10);
    r.addEdge(3, 2, 5);
    r.addEdge(1, 3, 1);
    r.addEdge(0, 4, 5);
    r.addEdge(2, 4, 100);
    r.addEdge(3, 4, 5);
 
    cout << minPath(V, src, des, g, r);
 
    return 0;
}

Java

// Java implementation of the approach
import java.util.*;
 
// This class represents
// a directed graph using
// adjacency list representation
class Graph {
     
    // Number of vertices
    private int V;
     
      // In a weighted graph, store vertex
    // and weight pair for every edge
    private List<List<iPair> > adj;
 
    public Graph(int V)
    {
        this.V = V;
        adj = new ArrayList<>();
        for (int i = 0; i < V; i++) {
            adj.add(new ArrayList<>());
        }
    }
   // Function to add an edge to graph
    public void addEdge(int u, int v, int w)
    {
        adj.get(v).add(new iPair(u, w));
    }
   // Find shortest path from
    // source vertex to all vertex
    public void shortestPath(int src, List<Integer> dist)
    {
        PriorityQueue<iPair> pq = new PriorityQueue<>(
            Comparator.comparingInt(ip -> ip.first));
        pq.add(new iPair(0, src));
        dist.set(src, 0);
 
        while (!pq.isEmpty()) {
            int u = pq.poll().second;
 
            for (iPair i : adj.get(u)) {
                int v = i.first;
                int weight = i.second;
 
                if (dist.get(v) > dist.get(u) + weight) {
                    dist.set(v, dist.get(u) + weight);
                    pq.add(new iPair(dist.get(v), v));
                }
            }
        }
    }
}
// iPair represents the Integer Pair
class iPair {
    int first, second;
 
    public iPair(int first, int second)
    {
        this.first = first;
        this.second = second;
    }
}
class Main {
    static int INF = 0x3f3f3f3f;
 // Function to return the
// required minimum path
    public static int minPath(int V, int src, int des,
                              Graph g, Graph r)
    {
          // Create a vector for
    // distances and
    // initialize all distances
    // as infinite (INF)
 
    // To store distance of all
    // vertex from source
        List<Integer> dist
            = new ArrayList<>(Collections.nCopies(V, INF));
     
      // To store distance of all
    // vertex from destination
        List<Integer> dist2
            = new ArrayList<>(Collections.nCopies(V, INF));
         
         // To store distance of source
    // from all vertex
        List<Integer> dist3
            = new ArrayList<>(Collections.nCopies(V, INF));
             
       // To store distance of
    // destination from all vertex
        List<Integer> dist4
            = new ArrayList<>(Collections.nCopies(V, INF));
 
       // Computing shortest path from
    // source vertex to all vertices
    g.shortestPath(src, dist);
 
    // Computing shortest path from
    // destination vertex to all vertices
    g.shortestPath(des, dist2);
 
    // Computing shortest path from
    // all the vertices to source
    r.shortestPath(src, dist3);
 
    // Computing shortest path from
    // all the vertices to destination
    r.shortestPath(des, dist4);
 
 // Finding the intermediate node (IN)
    // such that the distance of path
    // src -> IN -> des -> IN -> src is minimum
 
    // To store the shortest distance
        int ans = Integer.MAX_VALUE;
 
 // Intermediate node should not be
        // the source and destination
        for (int i = 0; i < V; i++) {
            if (i != des && i != src) {
                ans = Math.min(
                    ans, dist.get(i) + dist2.get(i)
                             + dist3.get(i) + dist4.get(i));
            }
        }
 
        return ans;
    }
// Driver Code
    public static void main(String[] args)
    {
         
    // Create the graph
        int V = 5, src = 0, des = 1;
         
    // To store the original graph
        Graph g = new Graph(V);
         
            // To store the reverse graph
    // and compute distance from all
    // vertex to a particular vertex
        Graph r = new Graph(V);
 
  // Adding edges
        g.addEdge(0, 2, 1);
        g.addEdge(0, 4, 5);
        g.addEdge(1, 4, 1);
        g.addEdge(2, 0, 10);
        g.addEdge(2, 3, 5);
        g.addEdge(3, 1, 1);
        g.addEdge(4, 0, 5);
        g.addEdge(4, 2, 100);
        g.addEdge(4, 3, 5);
 
    // Adding edges in reverse direction
        r.addEdge(2, 0, 1);
        r.addEdge(4, 0, 5);
        r.addEdge(4, 1, 1);
        r.addEdge(0, 2, 10);
        r.addEdge(3, 2, 5);
        r.addEdge(1, 3, 1);
        r.addEdge(0, 4, 5);
        r.addEdge(2, 4, 100);
        r.addEdge(3, 4, 5);
 
        System.out.println(minPath(V, src, des, g, r));
    }
}

Python3

# Python implementation of the approach
from typing import List
from queue import PriorityQueue
from sys import maxsize as INT_MAX
INF = 0x3f3f3f3f
 
# This class represents
# a directed graph using
# adjacency list representation
class Graph:
    def __init__(self, V: int) -> None:
 
        # Number of vertices
        self.V = V
 
        # In a weighted graph, store vertex
        # and weight pair for every edge
        self.adj = [[] for _ in range(V)]
 
    # Function to add an edge to the graph
    def addEdge(self, u: int, v: int, w: int) -> None:
        self.adj[v].append((u, w))
 
    # Function to find the shortest paths
    # from source to all other vertices
    def shortestPath(self, src: int, dist: List[int]) -> None:
 
        # Create a priority queue to
        # store vertices that
        # are being preprocessed
        pq = PriorityQueue()
 
        # Insert source itself in priority
        # queue and initialize
        # its distance as 0
        pq.put((0, src))
        dist[src] = 0
 
        # Loop till priority queue
        # becomes empty (or all
        # distances are not finalized)
        while not pq.empty():
 
            # The first vertex in pair
            # is the minimum distance
            # vertex, extract it from
            # priority queue
            u = pq.get()[1]
 
            # 'i' is used to get all
            # adjacent vertices of a vertex
            for i in self.adj[u]:
 
                # Get vertex label and
                # weight of current
                # adjacent of u
                v = i[0]
                weight = i[1]
 
                # If there is shorted
                # path to v through u
                if dist[v] > dist[u] + weight:
 
                    # Updating distance of v
                    dist[v] = dist[u] + weight
                    pq.put((dist[v], v))
 
# Function to return the
# required minimum path
def minPath(V: int, src: int, des: int, g: Graph, r: Graph) -> int:
 
    # Create a vector for
    # distances and
    # initialize all distances
    # as infinite (INF)
 
    # To store distance of all
    # vertex from source
    dist = [INF for _ in range(V)]
 
    # To store distance of all
    # vertex from destination
    dist2 = [INF for _ in range(V)]
 
    # To store distance of source
    # from all vertex
    dist3 = [INF for _ in range(V)]
 
    # To store distance of
    # destination from all vertex
    dist4 = [INF for _ in range(V)]
 
    # Computing shortest path from
    # source vertex to all vertices
    g.shortestPath(src, dist)
 
    # Computing shortest path from
    # destination vertex to all vertices
    g.shortestPath(des, dist2)
 
    # Computing shortest path from
    # all the vertices to source
    r.shortestPath(src, dist3)
 
    # Computing shortest path from
    # all the vertices to destination
    r.shortestPath(des, dist4)
 
    # Finding the intermediate node (IN)
    # such that the distance of path
    # src -> IN -> des -> IN -> src is minimum
 
    # To store the shortest distance
    ans = INT_MAX
    for i in range(V):
 
        # Intermediate node should not be
        # the source and destination
        if (i != des and i != src):
            ans = min(ans, dist[i] + dist2[i] + dist3[i] + dist4[i])
 
    # Return the minimum path required
    return ans
 
# Driver code
if __name__ == "__main__":
 
    # Create the graph
    V = 5
    src = 0
    des = 1
 
    # To store the original graph
    g = Graph(V)
 
    # To store the reverse graph
    # and compute distance from all
    # vertex to a particular vertex
    r = Graph(V)
 
    # Adding edges
    g.addEdge(0, 2, 1)
    g.addEdge(0, 4, 5)
    g.addEdge(1, 4, 1)
    g.addEdge(2, 0, 10)
    g.addEdge(2, 3, 5)
    g.addEdge(3, 1, 1)
    g.addEdge(4, 0, 5)
    g.addEdge(4, 2, 100)
    g.addEdge(4, 3, 5)
 
    # Adding edges in reverse direction
    r.addEdge(2, 0, 1)
    r.addEdge(4, 0, 5)
    r.addEdge(4, 1, 1)
    r.addEdge(0, 2, 10)
    r.addEdge(3, 2, 5)
    r.addEdge(1, 3, 1)
    r.addEdge(0, 4, 5)
    r.addEdge(2, 4, 100)
    r.addEdge(3, 4, 5)
 
    print(minPath(V, src, des, g, r))
 
# This code is contributed by sanjeev2552

C#

// C# implementation of the approach
 
using System;
using System.Collections.Generic;
 
// Priority Queue
class PriorityQueue<T> where T : IComparable<T>
{
    private List<Tuple<T, int>> queue = new List<Tuple<T, int>>();
 
    public void Enqueue(T item, int priority)
    {
        queue.Add(new Tuple<T, int>(item, priority));
        queue.Sort((a, b) => a.Item2.CompareTo(b.Item2));
    }
 
    public Tuple<T, int> Dequeue()
    {
        var item = queue[0];
        queue.RemoveAt(0);
        return item;
    }
 
    public bool IsEmpty()
    {
        return queue.Count == 0;
    }
}
 
// This class represents
// a directed graph using
// adjacency list representation
class Graph
{
    // Number of vertices
    private int V;
    private List<Tuple<int, int>>[] adj;
 
    public Graph(int V)
    {
         
        this.V = V;
    // In a weighted graph, store vertex
    // and weight pair for every edge
        adj = new List<Tuple<int, int>>[V];
        for (int i = 0; i < V; i++)
        {
            adj[i] = new List<Tuple<int, int>>();
        }
    }
// Function to add an edge to graph
    public void AddEdge(int u, int v, int w)
    {
        adj[v].Add(new Tuple<int, int>(u, w));
    }
 
// Function to find the shortest paths
// from source to all other vertices
    public void ShortestPath(int src, int[] dist)
    {
         
    // Create a priority queue to store vertices that
    // are being preprocessed
        var pq = new PriorityQueue<int>();
 
    // Insert source and initialize
    // its distance as 0
        pq.Enqueue(src, 0);
        dist[src] = 0;
 
    // Loop till priority queue
    // becomes empty
        while (!pq.IsEmpty())
        {
        // The first vertex in pair
        // is the minimum distance
        // vertex, extract it from
        // priority queue
            var u = pq.Dequeue().Item1;
 
           // 'i' is used to get all
        // adjacent vertices of a vertex
            foreach (var item in adj[u])
            {
                var v = item.Item1;
                var weight = item.Item2;
                  // If there is shorted
            // path to v through u
                if (dist[v] > dist[u] + weight)
                {
                    dist[v] = dist[u] + weight;
                    pq.Enqueue(v, dist[v]);
                }
            }
        }
    }
}
 
class Program
{
    // Function to return the
// required minimum path
    static int MinPath(int V, int src, int des, Graph g, Graph r)
    {
        const int INF = 0x3f3f3f3f;
 
    // To store distance of all
    // vertex from source
        var dist = new int[V];
         
    // To store distance of all
    // vertex from destination
        var dist2 = new int[V];
     
    // To store distance of all
    // vertex from destination
        var dist3 = new int[V];
     
    // To store distance of all
    // vertex from source
        var dist4 = new int[V];
 
        for (int i = 0; i < V; i++)
        {
            dist[i] = INF;
            dist2[i] = INF;
            dist3[i] = INF;
            dist4[i] = INF;
        }
 
// Computing shortest path from
// source vertex to all vertices
        g.ShortestPath(src, dist);
         
// Computing shortest path from
// destination vertex to all vertices
        g.ShortestPath(des, dist2);
 
// Computing shortest path from
// all the vertices to source
        r.ShortestPath(src, dist3);
 
// Computing shortest path from
// all the vertices to destination
        r.ShortestPath(des, dist4);
 
// To store the shortest distance
        int ans = INF;
        for (int i = 0; i < V; i++)
        {
            if (i != src && i != des)
            {
                ans = Math.Min(ans, dist[i] + dist2[i] + dist3[i] + dist4[i]);
            }
        }
 
        return ans;
    }
 
// Driver code
static void Main(string[] args)
    {
         
    // To store the original graph
        int V = 5;
        int src = 0;
        int des = 1;
 
       
    // To store the original graph
        var g = new Graph(V);
  
    // Adding edges
    g.AddEdge(0, 2, 1);
    g.AddEdge(0, 4, 5);
    g.AddEdge(1, 4, 1);
    g.AddEdge(2, 0, 10);
    g.AddEdge(2, 3, 5);
    g.AddEdge(3, 1, 1);
    g.AddEdge(4, 0, 5);
    g.AddEdge(4, 2, 100);
    g.AddEdge(4, 3, 5);
  
    // To store the reverse graph
    // and compute distance from all
    // vertex to a particular vertex
    var r = new Graph(V);
  
    // Adding edges in reverse direction
    r.AddEdge(2, 0, 1);
    r.AddEdge(4, 0, 5);
    r.AddEdge(4, 1, 1);
    r.AddEdge(0, 2, 10);
    r.AddEdge(3, 2, 5);
    r.AddEdge(1, 3, 1);
    r.AddEdge(0, 4, 5);
    r.AddEdge(2, 4, 100);
    r.AddEdge(3, 4, 5);
 
    Console.WriteLine(MinPath(V, src, des, g, r));
    }
}

Javascript

// JS implementation of the approach
 
class PriorityQueue {
  constructor() {
    this.queue = [];
  }
 
  enqueue(node, priority) {
    this.queue.push({ node, priority });
    this.queue.sort((a, b) => a.priority - b.priority);
  }
 
  dequeue() {
    return this.queue.shift();
  }
 
  isEmpty() {
    return this.queue.length === 0;
  }
}
// This class represents
// a directed graph using
// adjacency list representation
class Graph {
  constructor(V) {
   
       // Number of vertices
    this.V = V;
     
      // In a weighted graph, store vertex
    // and weight pair for every edge
    this.adj = Array.from({ length: V }, () => []);
  }
   
  // Function to add an edge to graph
  addEdge(u, v, w) {
    this.adj[v].push([u, w]);
  }
 
// Function to find the shortest paths
// from source to all other vertices
  shortestPath(src, dist) {
       
    // Create a priority queue to
    // store vertices that
    // are being preprocessed
    const pq = new PriorityQueue();
     
     // Insert source itself in priority
    // queue and initialize
    // its distance as 0
    pq.enqueue(src, 0);
    dist[src] = 0;
     
  // Loop till priority queue
    // becomes empty (or all
    // distances are not finalized)
    while (!pq.isEmpty()) {
     
           // The first vertex in pair
        // is the minimum distance
        // vertex, extract it from
        // priority queue
      const u = pq.dequeue().node;
       
         // 'i' is used to get all
        // adjacent vertices of a vertex
      for (const [v, weight] of this.adj[u]) {
       
           // If there is shorted
            // path to v through u
        if (dist[v] > dist[u] + weight) {
         
               // Updating distance of v
          dist[v] = dist[u] + weight;
          pq.enqueue(v, dist[v]);
        }
      }
    }
  }
}
 
// Function to return the
// required minimum path
function minPath(V, src, des, g, r) {
    
  const INF = 0x3f3f3f3f;
    // Create a vector for
    // distances and
    // initialize all distances
    // as infinite (INF)
 
    // To store distance of all
    // vertex from source
  const dist = Array.from({ length: V }, () => INF);
   
    // To store distance of all
    // vertex from destination
  const dist2 = Array.from({ length: V }, () => INF);
   
    // To store distance of all
    // vertex from destination
  const dist3 = Array.from({ length: V }, () => INF);
   
      // To store distance of
    // destination from all vertex
  const dist4 = Array.from({ length: V }, () => INF);
   
  // Computing shortest path from
    // source vertex to all vertices
    g.shortestPath(src, dist);
 
    // Computing shortest path from
    // destination vertex to all vertices
    g.shortestPath(des, dist2);
 
    // Computing shortest path from
    // all the vertices to source
    r.shortestPath(src, dist3);
 
    // Computing shortest path from
    // all the vertices to destination
    r.shortestPath(des, dist4);
 
    // Finding the intermediate node (IN)
    // such that the distance of path
    // src -> IN -> des -> IN -> src is minimum
 
    // To store the shortest distance
 
  let ans = INF;
  for (let i = 0; i < V; i++) {
    if (i !== src && i !== des) {
      ans = Math.min(ans, dist[i] + dist2[i] + dist3[i] + dist4[i]);
    }
  }
 
  return ans;
}
 
const V = 5;
const src = 0;
const des = 1;
 
// Driver code
 
 
    // To store the original graph
const g = new Graph(V);
 
    // Adding edges
g.addEdge(0, 2, 1);
g.addEdge(0, 4, 5);
g.addEdge(1, 4, 1);
g.addEdge(2, 0, 10);
g.addEdge(2, 3, 5);
g.addEdge(3, 1, 1);
g.addEdge(4, 0, 5);
g.addEdge(4, 2, 100);
g.addEdge(4, 3, 5);
 
    // To store the reverse graph
    // and compute distance from all
    // vertex to a particular vertex
const r = new Graph(V);
 
    // Adding edges in reverse direction
r.addEdge(2, 0, 1);
r.addEdge(4, 0, 5);
r.addEdge(4, 1, 1);
r.addEdge(0, 2, 10);
r.addEdge(3, 2, 5);
r.addEdge(1, 3, 1);
r.addEdge(0, 4, 5);
r.addEdge(2, 4, 100);
r.addEdge(3, 4, 5);
 
console.log(minPath(V, src, des, g, r));

Output:

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