Shortest path in an unweighted graph

Given a unweighted graph, a source and a destination, we need to find shortest path from source to destination in the graph in most optimal way.

unweighted graph

unweighted graph of 8 vertices

Input: source vertex = 0 and destination vertex is = 7.
Output: Shortest path length is:2
        Path is::
        0 3 7

Input: source vertex is = 2 and destination vertex is = 6.
Output: Shortest path length is:5
        Path is::
        2 1 0 3 4 6
 

One solution is to solve in O(VE) time using Bellman–Ford. If there are no negative weight cycles, then we can solve in O(E + VLogV) time using Dijkstra’s algorithm.

Since the graph is unweighted, we can solve this problem in O(V + E) time. The idea is to use a modified version of Breadth-first search in which we keep storing the predecessor of a given vertex while doing the breadth first search. This algorithm will work even when negative weight cycles are present in the graph.
We first initialize an array dist[0, 1, …., v-1] such that dist[i] stores the distance of vertex i from the source vertex and array pred[0, 1, ….., v-1] such that pred[i] represents the immediate predecessor of the vertex i in the breadth first search starting from the source.
Now we get the length of the path from source to any other vertex in O(1) time from array d, and for printing the path from source to any vertex we can use array p and that will take O(V) time in worst case as V is the size of array P. So most of the time of algorithm is spent in doing the Breadth first search from given source which we know takes O(V+E) time. Thus time complexity of our algorithm is O(V+E).

Take the following unweighted graph as an example:

Following is the complete algorithm for finding the shortest path:

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// CPP code for printing shortest path between
// two vertices of unweighted graph
#include <bits/stdc++.h>
using namespace std;
  
// utility function to form edge between two vertices
// source and dest
void add_edge(vector<int> adj[], int src, int dest)
{
    adj[src].push_back(dest);
    adj[dest].push_back(src);
}
  
// a modified version of BFS that stores predecessor
// of each vertex in array p
// and its distance from source in array d
bool BFS(vector<int> adj[], int src, int dest, int v,
                            int pred[], int dist[])
{
    // a queue to maintain queue of vertices whose
    // adjacency list is to be scanned as per normal
    // DFS algorithm
    list<int> queue;
  
    // boolean array visited[] which stores the
    // information whether ith vertex is reached
    // at least once in the Breadth first search
    bool visited[v];
  
    // initially all vertices are unvisited
    // so v[i] for all i is false
    // and as no path is yet constructed
    // dist[i] for all i set to infinity
    for (int i = 0; i < v; i++) {
        visited[i] = false;
        dist[i] = INT_MAX;
        pred[i] = -1;
    }
  
    // now source is first to be visited and
    // distance from source to itself should be 0
    visited[src] = true;
    dist[src] = 0;
    queue.push_back(src);
  
    // standard BFS algorithm
    while (!queue.empty()) {
        int u = queue.front();
        queue.pop_front();
        for (int i = 0; i < adj[u].size(); i++) {
            if (visited[adj[u][i]] == false) {
                visited[adj[u][i]] = true;
                dist[adj[u][i]] = dist[u] + 1;
                pred[adj[u][i]] = u;
                queue.push_back(adj[u][i]);
  
                // We stop BFS when we find
                // destination.
                if (adj[u][i] == dest)
                   return true;
            }
        }
    }
  
    return false;
}
  
// utility function to print the shortest distance 
// between source vertex and destination vertex
void printShortestDistance(vector<int> adj[], int s, 
                                    int dest, int v)
{
    // predecessor[i] array stores predecessor of
    // i and distance array stores distance of i
    // from s
    int pred[v], dist[v];
  
    if (BFS(adj, s, dest, v, pred, dist) == false)
    {
        cout << "Given source and destination"
             << " are not connected";
        return;
    }
  
    // vector path stores the shortest path
    vector<int> path;
    int crawl = dest;
    path.push_back(crawl);
    while (pred[crawl] != -1) {
        path.push_back(pred[crawl]);
        crawl = pred[crawl];
    }
  
    // distance from source is in distance array
    cout << "Shortest path length is : "
        << dist[dest];
  
    // printing path from source to destination
    cout << "\nPath is::\n";
    for (int i = path.size() - 1; i >= 0; i--)
        cout << path[i] << " ";
}
  
// Driver program to test above functions
int main()
{
    // no. of vertices
    int v = 8; 
  
    // array of vectors is used to store the graph
    // in the form of an adjacency list
    vector<int> adj[v];
  
    // Creating graph given in the above diagram.
    // add_edge function takes adjacency list, source 
    // and destination vertex as argument and forms
    // an edge between them.
    add_edge(adj, 0, 1);
    add_edge(adj, 0, 3);
    add_edge(adj, 1, 2);
    add_edge(adj, 3, 4);
    add_edge(adj, 3, 7);
    add_edge(adj, 4, 5);
    add_edge(adj, 4, 6);
    add_edge(adj, 4, 7);
    add_edge(adj, 5, 6);
    add_edge(adj, 6, 7);
    int source = 0, dest = 7;
    printShortestDistance(adj, source, dest, v);
    return 0;
}

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Output:

Shortest path length is : 2
Path is::
0 3 7

Time Complexity : O(V + E)
Auxiliary Space : O(V)



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