# Shortest path in an unweighted graph

• Difficulty Level : Medium
• Last Updated : 15 Oct, 2022

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

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 Bellmanâ€“Ford. If there are no negative weight cycles, then we can solve in O(E + VLogV) time using Dijkstraâ€™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.

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 the algorithm is spent in doing the Breadth-first search from a given source which we know takes O(V+E) time. Thus the 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:

Implementation:

## C++

 `// CPP code for printing shortest path between``// two vertices of unweighted graph``#include ``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;``}`

## Java

 `// Java program to find shortest path in an undirected``// graph``import` `java.util.ArrayList;``import` `java.util.Iterator;``import` `java.util.LinkedList;` `public` `class` `pathUnweighted {` `    ``// Driver Program``    ``public` `static` `void` `main(String args[])``    ``{``        ``// No of vertices``        ``int` `v = ``8``;` `        ``// Adjacency list for storing which vertices are connected``        ``ArrayList> adj =``                            ``new` `ArrayList>(v);``        ``for` `(``int` `i = ``0``; i < v; i++) {``            ``adj.add(``new` `ArrayList());``        ``}` `        ``// 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.``        ``addEdge(adj, ``0``, ``1``);``        ``addEdge(adj, ``0``, ``3``);``        ``addEdge(adj, ``1``, ``2``);``        ``addEdge(adj, ``3``, ``4``);``        ``addEdge(adj, ``3``, ``7``);``        ``addEdge(adj, ``4``, ``5``);``        ``addEdge(adj, ``4``, ``6``);``        ``addEdge(adj, ``4``, ``7``);``        ``addEdge(adj, ``5``, ``6``);``        ``addEdge(adj, ``6``, ``7``);``        ``int` `source = ``0``, dest = ``7``;``        ``printShortestDistance(adj, source, dest, v);``    ``}` `    ``// function to form edge between two vertices``    ``// source and dest``    ``private` `static` `void` `addEdge(ArrayList> adj, ``int` `i, ``int` `j)``    ``{``        ``adj.get(i).add(j);``        ``adj.get(j).add(i);``    ``}` `    ``// function to print the shortest distance and path``    ``// between source vertex and destination vertex``    ``private` `static` `void` `printShortestDistance(``                     ``ArrayList> 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[] = ``new` `int``[v];``        ``int` `dist[] = ``new` `int``[v];` `        ``if` `(BFS(adj, s, dest, v, pred, dist) == ``false``) {``            ``System.out.println(``"Given source and destination"` `+``                                         ``"are not connected"``);``            ``return``;``        ``}` `        ``// LinkedList to store path``        ``LinkedList path = ``new` `LinkedList();``        ``int` `crawl = dest;``        ``path.add(crawl);``        ``while` `(pred[crawl] != -``1``) {``            ``path.add(pred[crawl]);``            ``crawl = pred[crawl];``        ``}` `        ``// Print distance``        ``System.out.println(``"Shortest path length is: "` `+ dist[dest]);` `        ``// Print path``        ``System.out.println(``"Path is ::"``);``        ``for` `(``int` `i = path.size() - ``1``; i >= ``0``; i--) {``            ``System.out.print(path.get(i) + ``" "``);``        ``}``    ``}` `    ``// a modified version of BFS that stores predecessor``    ``// of each vertex in array pred``    ``// and its distance from source in array dist``    ``private` `static` `boolean` `BFS(ArrayList> 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``        ``// BFS algorithm using LinkedList of Integer type``        ``LinkedList queue = ``new` `LinkedList();` `        ``// boolean array visited[] which stores the``        ``// information whether ith vertex is reached``        ``// at least once in the Breadth first search``        ``boolean` `visited[] = ``new` `boolean``[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] = Integer.MAX_VALUE;``            ``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.add(src);` `        ``// bfs Algorithm``        ``while` `(!queue.isEmpty()) {``            ``int` `u = queue.remove();``            ``for` `(``int` `i = ``0``; i < adj.get(u).size(); i++) {``                ``if` `(visited[adj.get(u).get(i)] == ``false``) {``                    ``visited[adj.get(u).get(i)] = ``true``;``                    ``dist[adj.get(u).get(i)] = dist[u] + ``1``;``                    ``pred[adj.get(u).get(i)] = u;``                    ``queue.add(adj.get(u).get(i));` `                    ``// stopping condition (when we find``                    ``// our destination)``                    ``if` `(adj.get(u).get(i) == dest)``                        ``return` `true``;``                ``}``            ``}``        ``}``        ``return` `false``;``    ``}``}``// This code is contributed by Sahil Vaid`

## Python3

 `# Python3 code for printing shortest path between``# two vertices of unweighted graph`` ` `# utility function to form edge between two vertices``# source and dest``def` `add_edge(adj, src, dest):` `    ``adj[src].append(dest);``    ``adj[dest].append(src);`` ` `# a modified version of BFS that stores predecessor``# of each vertex in array p``# and its distance from source in array d``def` `BFS(adj, src, dest, v, pred, dist):` `    ``# a queue to maintain queue of vertices whose``    ``# adjacency list is to be scanned as per normal``    ``# DFS algorithm``    ``queue ``=` `[]`` ` `    ``# boolean array visited[] which stores the``    ``# information whether ith vertex is reached``    ``# at least once in the Breadth first search``    ``visited ``=` `[``False` `for` `i ``in` `range``(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` `i ``in` `range``(v):` `        ``dist[i] ``=` `1000000``        ``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.append(src);`` ` `    ``# standard BFS algorithm``    ``while` `(``len``(queue) !``=` `0``):``        ``u ``=` `queue[``0``];``        ``queue.pop(``0``);``        ``for` `i ``in` `range``(``len``(adj[u])):``        ` `            ``if` `(visited[adj[u][i]] ``=``=` `False``):``                ``visited[adj[u][i]] ``=` `True``;``                ``dist[adj[u][i]] ``=` `dist[u] ``+` `1``;``                ``pred[adj[u][i]] ``=` `u;``                ``queue.append(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``def` `printShortestDistance(adj, s, dest, v):``    ` `    ``# predecessor[i] array stores predecessor of``    ``# i and distance array stores distance of i``    ``# from s``    ``pred``=``[``0` `for` `i ``in` `range``(v)]``    ``dist``=``[``0` `for` `i ``in` `range``(v)];`` ` `    ``if` `(BFS(adj, s, dest, v, pred, dist) ``=``=` `False``):``        ``print``(``"Given source and destination are not connected"``)`` ` `    ``# vector path stores the shortest path``    ``path ``=` `[]``    ``crawl ``=` `dest;``    ``path.append(crawl);``    ` `    ``while` `(pred[crawl] !``=` `-``1``):``        ``path.append(pred[crawl]);``        ``crawl ``=` `pred[crawl];``    ` ` ` `    ``# distance from source is in distance array``    ``print``(``"Shortest path length is : "` `+` `str``(dist[dest]), end ``=` `'')`` ` `    ``# printing path from source to destination``    ``print``(``"\nPath is : : "``)``    ` `    ``for` `i ``in` `range``(``len``(path)``-``1``, ``-``1``, ``-``1``):``        ``print``(path[i], end``=``' '``)``        ` `# Driver program to test above functions``if` `__name__``=``=``'__main__'``:``    ` `    ``# no. of vertices``    ``v ``=` `8``;`` ` `    ``# array of vectors is used to store the graph``    ``# in the form of an adjacency list``    ``adj ``=` `[[] ``for` `i ``in` `range``(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``);``    ``source ``=` `0``    ``dest ``=` `7``;``    ``printShortestDistance(adj, source, dest, v);` `    ``# This code is contributed by rutvik_56`

## C#

 `// C# program to find shortest``// path in an undirected graph``using` `System;``using` `System.Collections.Generic;``class` `pathUnweighted{` `// Driver code``public` `static` `void` `Main(String []args)``{``  ``// No of vertices``  ``int` `v = 8;` `  ``// Adjacency list for storing``  ``// which vertices are connected``  ``List> adj =``            ``new` `List>(v);``  ` `  ``for` `(``int` `i = 0; i < v; i++)``  ``{``    ``adj.Add(``new` `List<``int``>());``  ``}` `  ``// 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.``  ``addEdge(adj, 0, 1);``  ``addEdge(adj, 0, 3);``  ``addEdge(adj, 1, 2);``  ``addEdge(adj, 3, 4);``  ``addEdge(adj, 3, 7);``  ``addEdge(adj, 4, 5);``  ``addEdge(adj, 4, 6);``  ``addEdge(adj, 4, 7);``  ``addEdge(adj, 5, 6);``  ``addEdge(adj, 6, 7);``  ``int` `source = 0, dest = 7;``  ``printShortestDistance(adj, source,``                        ``dest, v);``}` `// function to form edge between``// two vertices source and dest``private` `static` `void` `addEdge(List> adj,``                            ``int` `i, ``int` `j)``{``  ``adj[i].Add(j);``  ``adj[j].Add(i);``}` `// function to print the shortest``// distance and path between source``// vertex and destination vertex``private` `static` `void` `printShortestDistance(List> 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 = ``new` `int``[v];``  ``int` `[]dist = ``new` `int``[v];` `  ``if` `(BFS(adj, s, dest,``          ``v, pred, dist) == ``false``)``  ``{``    ``Console.WriteLine(``"Given source and destination"` `+``                      ``"are not connected"``);``    ``return``;``  ``}` `  ``// List to store path``  ``List<``int``> path = ``new` `List<``int``>();``  ``int` `crawl = dest;``  ``path.Add(crawl);``  ` `  ``while` `(pred[crawl] != -1)``  ``{``    ``path.Add(pred[crawl]);``    ``crawl = pred[crawl];``  ``}` `  ``// Print distance``  ``Console.WriteLine(``"Shortest path length is: "` `+``                     ``dist[dest]);` `  ``// Print path``  ``Console.WriteLine(``"Path is ::"``);``  ` `  ``for` `(``int` `i = path.Count - 1;``           ``i >= 0; i--)``  ``{``    ``Console.Write(path[i] + ``" "``);``  ``}``}` `// a modified version of BFS that``// stores predecessor of each vertex``// in array pred and its distance``// from source in array dist``private` `static` `bool` `BFS(List> 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``  ``// BFS algorithm using List of int type``  ``List<``int``> queue = ``new` `List<``int``>();` `  ``// bool array visited[] which``  ``// stores the information whether``  ``// ith vertex is reached at least``  ``// once in the Breadth first search``  ``bool` `[]visited = ``new` `bool``[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``.MaxValue;``    ``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.Add(src);` `  ``// bfs Algorithm``  ``while` `(queue.Count != 0)``  ``{``    ``int` `u = queue[0];``    ``queue.RemoveAt(0);``    ` `    ``for` `(``int` `i = 0;``             ``i < adj[u].Count; 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.Add(adj[u][i]);` `        ``// stopping condition (when we``        ``// find our destination)``        ``if` `(adj[u][i] == dest)``          ``return` `true``;``      ``}``    ``}``  ``}``  ``return` `false``;``}``}` `// This code is contributed by Rajput-Ji`

Output

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

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

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