Given a directed graph where every edge has weight as either 1 or 2, find the shortest path from a given source vertex ‘s’ to a given destination vertex ‘t’. Expected time complexity is O(V+E).

A **Simple Solution** is to use Dijkstra’s shortest path algorithm, we can get a shortest path in O(E + VLogV) time.

**How to do it in O(V+E) time?** The idea is to use BFS. One important observation about BFS is, the path used in BFS always has least number of edges between any two vertices. So if all edges are of same weight, we can use BFS to find the shortest path. For this problem, we can modify the graph and split all edges of weight 2 into two edges of weight 1 each. In the modified graph, we can use BFS to find the shortest path.

**How many new intermediate vertices are needed?** We need to add a new intermediate vertex for every source vertex. The reason is simple, if we add a intermediate vertex x between u and v and if we add same vertex between y and z, then new paths u to z and y to v are added to graph which might have note been there in original graph. Therefore in a graph with V vertices, we need V extra vertices.

Below is C++ implementation of above idea. In the below implementation 2*V vertices are created in a graph and for every edge (u, v), we split it into two edges (u, u+V) and (u+V, w). This way we make sure that a different intermediate vertex is added for every source vertex.

## C/C++

`// Program to shortest path from a given source vertex ‘s’ to ` `// a given destination vertex ‘t’. Expected time complexity ` `// is O(V+E). ` `#include<bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// This class represents a directed graph using adjacency ` `// list representation ` `class` `Graph ` `{ ` ` ` `int` `V; ` `// No. of vertices ` ` ` `list<` `int` `> *adj; ` `// adjacency lists ` `public` `: ` ` ` `Graph(` `int` `V); ` `// Constructor ` ` ` `void` `addEdge(` `int` `v, ` `int` `w, ` `int` `weight); ` `// adds an edge ` ` ` ` ` `// finds shortest path from source vertex ‘s’ to ` ` ` `// destination vertex ‘d’. ` ` ` `int` `findShortestPath(` `int` `s, ` `int` `d); ` ` ` ` ` `// print shortest path from a source vertex ‘s’ to ` ` ` `// destination vertex ‘d’. ` ` ` `int` `printShortestPath(` `int` `parent[], ` `int` `s, ` `int` `d); ` `}; ` ` ` `Graph::Graph(` `int` `V) ` `{ ` ` ` `this` `->V = V; ` ` ` `adj = ` `new` `list<` `int` `>[2*V]; ` `} ` ` ` `void` `Graph::addEdge(` `int` `v, ` `int` `w, ` `int` `weight) ` `{ ` ` ` `// split all edges of weight 2 into two ` ` ` `// edges of weight 1 each. The intermediate ` ` ` `// vertex number is maximum vertex number + 1, ` ` ` `// that is V. ` ` ` `if` `(weight==2) ` ` ` `{ ` ` ` `adj[v].push_back(v+V); ` ` ` `adj[v+V].push_back(w); ` ` ` `} ` ` ` `else` `// Weight is 1 ` ` ` `adj[v].push_back(w); ` `// Add w to v’s list. ` `} ` ` ` `// To print the shortest path stored in parent[] ` `int` `Graph::printShortestPath(` `int` `parent[], ` `int` `s, ` `int` `d) ` `{ ` ` ` `static` `int` `level = 0; ` ` ` ` ` `// If we reached root of shortest path tree ` ` ` `if` `(parent[s] == -1) ` ` ` `{ ` ` ` `cout << ` `"Shortest Path between "` `<< s << ` `" and "` ` ` `<< d << ` `" is "` `<< s << ` `" "` `; ` ` ` `return` `level; ` ` ` `} ` ` ` ` ` `printShortestPath(parent, parent[s], d); ` ` ` ` ` `level++; ` ` ` `if` `(s < V) ` ` ` `cout << s << ` `" "` `; ` ` ` ` ` `return` `level; ` `} ` ` ` `// This function mainly does BFS and prints the ` `// shortest path from src to dest. It is assumed ` `// that weight of every edge is 1 ` `int` `Graph::findShortestPath(` `int` `src, ` `int` `dest) ` `{ ` ` ` `// Mark all the vertices as not visited ` ` ` `bool` `*visited = ` `new` `bool` `[2*V]; ` ` ` `int` `*parent = ` `new` `int` `[2*V]; ` ` ` ` ` `// Initialize parent[] and visited[] ` ` ` `for` `(` `int` `i = 0; i < 2*V; i++) ` ` ` `{ ` ` ` `visited[i] = ` `false` `; ` ` ` `parent[i] = -1; ` ` ` `} ` ` ` ` ` `// Create a queue for BFS ` ` ` `list<` `int` `> queue; ` ` ` ` ` `// Mark the current node as visited and enqueue it ` ` ` `visited[src] = ` `true` `; ` ` ` `queue.push_back(src); ` ` ` ` ` `// 'i' will be used to get all adjacent vertices of a vertex ` ` ` `list<` `int` `>::iterator i; ` ` ` ` ` `while` `(!queue.empty()) ` ` ` `{ ` ` ` `// Dequeue a vertex from queue and print it ` ` ` `int` `s = queue.front(); ` ` ` ` ` `if` `(s == dest) ` ` ` `return` `printShortestPath(parent, s, dest); ` ` ` ` ` `queue.pop_front(); ` ` ` ` ` `// Get all adjacent vertices of the dequeued vertex s ` ` ` `// If a adjacent has not been visited, then mark it ` ` ` `// visited and enqueue it ` ` ` `for` `(i = adj[s].begin(); i != adj[s].end(); ++i) ` ` ` `{ ` ` ` `if` `(!visited[*i]) ` ` ` `{ ` ` ` `visited[*i] = ` `true` `; ` ` ` `queue.push_back(*i); ` ` ` `parent[*i] = s; ` ` ` `} ` ` ` `} ` ` ` `} ` `} ` ` ` `// Driver program to test methods of graph class ` `int` `main() ` `{ ` ` ` `// Create a graph given in the above diagram ` ` ` `int` `V = 4; ` ` ` `Graph g(V); ` ` ` `g.addEdge(0, 1, 2); ` ` ` `g.addEdge(0, 2, 2); ` ` ` `g.addEdge(1, 2, 1); ` ` ` `g.addEdge(1, 3, 1); ` ` ` `g.addEdge(2, 0, 1); ` ` ` `g.addEdge(2, 3, 2); ` ` ` `g.addEdge(3, 3, 2); ` ` ` ` ` `int` `src = 0, dest = 3; ` ` ` `cout << ` `"\nShortest Distance between "` `<< src ` ` ` `<< ` `" and "` `<< dest << ` `" is "` ` ` `<< g.findShortestPath(src, dest); ` ` ` ` ` `return` `0; ` `} ` |

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## Python

`''' Program to shortest path from a given source vertex s to ` ` ` `a given destination vertex t. Expected time complexity ` ` ` `is O(V+E)'''` `from` `collections ` `import` `defaultdict ` ` ` `#This class represents a directed graph using adjacency list representation ` `class` `Graph: ` ` ` ` ` `def` `__init__(` `self` `,vertices): ` ` ` `self` `.V ` `=` `vertices ` `#No. of vertices ` ` ` `self` `.V_org ` `=` `vertices ` ` ` `self` `.graph ` `=` `defaultdict(` `list` `) ` `# default dictionary to store graph ` ` ` ` ` `# function to add an edge to graph ` ` ` `def` `addEdge(` `self` `,u,v,w): ` ` ` `if` `w ` `=` `=` `1` `: ` ` ` `self` `.graph[u].append(v) ` ` ` `else` `: ` ` ` `'''split all edges of weight 2 into two ` ` ` `edges of weight 1 each. The intermediate ` ` ` `vertex number is maximum vertex number + 1, ` ` ` `that is V.'''` ` ` `self` `.graph[u].append(` `self` `.V) ` ` ` `self` `.graph[` `self` `.V].append(v) ` ` ` `self` `.V ` `=` `self` `.V ` `+` `1` ` ` ` ` `# To print the shortest path stored in parent[] ` ` ` `def` `printPath(` `self` `, parent, j): ` ` ` `Path_len ` `=` `1` ` ` `if` `parent[j] ` `=` `=` `-` `1` `and` `j < ` `self` `.V_org : ` `#Base Case : If j is source ` ` ` `print` `j, ` ` ` `return` `0` `# when parent[-1] then path length = 0 ` ` ` `l ` `=` `self` `.printPath(parent , parent[j]) ` ` ` ` ` `#incerement path length ` ` ` `Path_len ` `=` `l ` `+` `Path_len ` ` ` ` ` `# print node only if its less than original node length. ` ` ` `# i.e do not print any new node that has been added later ` ` ` `if` `j < ` `self` `.V_org : ` ` ` `print` `j, ` ` ` ` ` `return` `Path_len ` ` ` ` ` `''' This function mainly does BFS and prints the ` ` ` `shortest path from src to dest. It is assumed ` ` ` `that weight of every edge is 1'''` ` ` `def` `findShortestPath(` `self` `,src, dest): ` ` ` ` ` `# Mark all the vertices as not visited ` ` ` `# Initialize parent[] and visited[] ` ` ` `visited ` `=` `[` `False` `]` `*` `(` `self` `.V) ` ` ` `parent ` `=` `[` `-` `1` `]` `*` `(` `self` `.V) ` ` ` ` ` `# Create a queue for BFS ` ` ` `queue` `=` `[] ` ` ` ` ` `# Mark the source node as visited and enqueue it ` ` ` `queue.append(src) ` ` ` `visited[src] ` `=` `True` ` ` ` ` `while` `queue : ` ` ` ` ` `# Dequeue a vertex from queue ` ` ` `s ` `=` `queue.pop(` `0` `) ` ` ` ` ` `# if s = dest then print the path and return ` ` ` `if` `s ` `=` `=` `dest: ` ` ` `return` `self` `.printPath(parent, s) ` ` ` ` ` ` ` `# Get all adjacent vertices of the dequeued vertex s ` ` ` `# If a adjacent has not been visited, then mark it ` ` ` `# visited and enqueue it ` ` ` `for` `i ` `in` `self` `.graph[s]: ` ` ` `if` `visited[i] ` `=` `=` `False` `: ` ` ` `queue.append(i) ` ` ` `visited[i] ` `=` `True` ` ` `parent[i] ` `=` `s ` ` ` ` ` `# Create a graph given in the above diagram ` `g ` `=` `Graph(` `4` `) ` `g.addEdge(` `0` `, ` `1` `, ` `2` `) ` `g.addEdge(` `0` `, ` `2` `, ` `2` `) ` `g.addEdge(` `1` `, ` `2` `, ` `1` `) ` `g.addEdge(` `1` `, ` `3` `, ` `1` `) ` `g.addEdge(` `2` `, ` `0` `, ` `1` `) ` `g.addEdge(` `2` `, ` `3` `, ` `2` `) ` `g.addEdge(` `3` `, ` `3` `, ` `2` `) ` ` ` `src ` `=` `0` `; dest ` `=` `3` `print` `(` `"Shortest Path between %d and %d is "` `%` `(src, dest)), ` `l ` `=` `g.findShortestPath(src, dest) ` `print` `(` `"\nShortest Distance between %d and %d is %d "` `%` `(src, dest, l)), ` ` ` `#This code is contributed by Neelam Yadav ` |

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Output :

Shortest Path between 0 and 3 is 0 1 3 Shortest Distance between 0 and 3 is 3

How is this approach O(V+E)? In worst case, all edges are of weight 2 and we need to do O(E) operations to split all edges and 2V vertices, so the time complexity becomes O(E) + O(V+E) which is O(V+E).

This article is contributed by **Aditya Goel**. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

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