Print all shortest paths between given source and destination in an undirected graph

Given an undirected and unweighted graph and two nodes as source and destination, the task is to print all the paths of the shortest length between the given source and destination.

Examples:

Input: source = 0, destination = 5

Output:
0 -> 1 -> 3 -> 5
0 -> 2 -> 3 -> 5
0 -> 1 -> 4 -> 5
Explanation:
All the above paths are of length 3, which is the shortest distance between 0 and 5.

Input: source = 0, destination = 4

Output:
0 -> 1 -> 4

Approach: The is to do a Breadth First Traversal (BFS) for a graph. Below are the steps:



  1. Start BFS traversal from source vertex.
  2. While doing BFS, store the shortest distance to each of the other nodes and also maintain a parent vector for each of the nodes.
  3. Make the parent of source node as “-1”. For each node, it will store all the parents for which it has the shortest distance from the source node.
  4. Recover all the paths using parent array. At any instant, we will push one vertex in the path array and then call for all its parents.
  5. If we encounter “-1” in the above steps, then it means a path has been found and can be stored in the paths array.

Below is the implementation of the above approach:

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// Cpp program for the above approach
#include <bits/stdc++.h>
using namespace std;
  
// Function to form edge between
// two vertices src and dest
void add_edge(vector<int> adj[],
              int src, int dest)
{
    adj[src].push_back(dest);
    adj[dest].push_back(src);
}
  
// Function which finds all the paths
// and stores it in paths array
void find_paths(vector<vector<int> >& paths,
                vector<int>& path,
                vector<int> parent[],
                int n, int u)
{
    // Base Case
    if (u == -1) {
        paths.push_back(path);
        return;
    }
  
    // Loop for all the parents
    // of the given vertex
    for (int par : parent[u]) {
  
        // Insert the current
        // vertex in path
        path.push_back(u);
  
        // Recursive call for its parent
        find_paths(paths, path, parent,
                   n, par);
  
        // Remove the current vertex
        path.pop_back();
    }
}
  
// Function which performs bfs
// from the given souce vertex
void bfs(vector<int> adj[],
         vector<int> parent[],
         int n, int start)
{
    // dist will contain shortest distance
    // from start to every other vertex
    vector<int> dist(n, INT_MAX);
  
    queue<int> q;
  
    // Insert source vertex in queue and make
    // its parent -1 and distance 0
    q.push(start);
    parent[start] = { -1 };
    dist[start] = 0;
  
    // Untill Queue is empty
    while (!q.empty()) {
        int u = q.front();
        q.pop();
        for (int v : adj[u]) {
            if (dist[v] > dist[u] + 1) {
  
                // A shorter distance is found
                // So erase all the previous parents
                // and insert new parent u in parent[v]
                dist[v] = dist[u] + 1;
                q.push(v);
                parent[v].clear();
                parent[v].push_back(u);
            }
            else if (dist[v] == dist[u] + 1) {
  
                // Another candidate parent for
                // shortes path found
                parent[v].push_back(u);
            }
        }
    }
}
  
// Function which prints all the paths
// from start to end
void print_paths(vector<int> adj[],
                 int n, int start, int end)
{
    vector<vector<int> > paths;
    vector<int> path;
    vector<int> parent[n];
  
    // Function call to bfs
    bfs(adj, parent, n, start);
  
    // Function call to find_paths
    find_paths(paths, path, parent, n, end);
  
    for (auto v : paths) {
  
        // Since paths contain each
        // path in reverse order,
        // so reverse it
        reverse(v.begin(), v.end());
  
        // Print node for the current path
        for (int u : v)
            cout << u << " ";
        cout << endl;
    }
}
  
// Driver Code
int main()
{
    // Number of vertices
    int n = 6;
  
    // array of vectors is used
    // to store the graph
    // in the form of an adjacency list
    vector<int> adj[n];
  
    // Given Graph
    add_edge(adj, 0, 1);
    add_edge(adj, 0, 2);
    add_edge(adj, 1, 3);
    add_edge(adj, 1, 4);
    add_edge(adj, 2, 3);
    add_edge(adj, 3, 5);
    add_edge(adj, 4, 5);
  
    // Given source and destination
    int src = 0;
    int dest = n - 1;
  
    // Function Call
    print_paths(adj, n, src, dest);
  
    return 0;
}

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Output:

0 1 3 5 
0 2 3 5 
0 1 4 5

Time Complexity: O(V + E) where V is the number of vertices and E is the number of edges.
Auxiliary Space: O(V) where V is the number of vertices.

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