# 0-1 BFS (Shortest Path in a Binary Weight Graph)

Given a graph where every edge has weight as either 0 or 1. A source vertex is also given in the graph. Find the shortest path from source vertex to every other vertex.

Input : Source Vertex = 0 and below graph Output : Shortest distances from given source 0 0 1 1 2 1 2 1 2 Explanation : Shortest distance from 0 to 0 is 0 Shortest distance from 0 to 1 is 0 Shortest distance from 0 to 2 is 1 ..................

In normal BFS of a graph all edges have equal weight but in 0-1 BFS some edges may have 0 weight and some may have 1 weight. In this we will not use bool array to mark visited nodes but at each step we will check for the optimal distance condition. We use double ended queue to store the node. While performing BFS if a edge having weight = 0 is found node is pushed at front of double ended queue and if a edge having weight = 1 is found, it is pushed at back of double ended queue.

The approach is similar to Dijkstra that the if the shortest distance to node is relaxed by the previous node then only it will be pushed in the queue.

The above idea works in all cases, when pop a vertex (like Dijkstra), it is the minimum weight vertex among remaining vertices. If there is a 0 weight vertex adjacent to it, then this adjacent has same distance. If there is a 1 weight adjacent, then this adjacent has maximum distance among all vertices in dequeue (because all other vertices are either adjacent of currently popped vertex or adjacent of previously popped vertices).

Below is C++ implementation of the above idea.

## C++

`// C++ program to implement single source ` `// shortest path for a Binary Graph ` `#include<bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `/* no.of vertices */` `#define V 9 ` ` ` `// a structure to represent edges ` `struct` `node ` `{ ` ` ` `// two variable one denote the node ` ` ` `// and other the weight ` ` ` `int` `to, weight; ` `}; ` ` ` `// vector to store edges ` `vector <node> edges[V]; ` ` ` `// Prints shortest distance from given source to ` `// every other vertex ` `void` `zeroOneBFS(` `int` `src) ` `{ ` ` ` `// Initialize distances from given source ` ` ` `int` `dist[V]; ` ` ` `for` `(` `int` `i=0; i<V; i++) ` ` ` `dist[i] = INT_MAX; ` ` ` ` ` `// double ende queue to do BFS. ` ` ` `deque <` `int` `> Q; ` ` ` `dist[src] = 0; ` ` ` `Q.push_back(src); ` ` ` ` ` `while` `(!Q.empty()) ` ` ` `{ ` ` ` `int` `v = Q.front(); ` ` ` `Q.pop_front(); ` ` ` ` ` `for` `(` `int` `i=0; i<edges[v].size(); i++) ` ` ` `{ ` ` ` `// checking for the optimal distance ` ` ` `if` `(dist[edges[v][i].to] > dist[v] + edges[v][i].weight) ` ` ` `{ ` ` ` `dist[edges[v][i].to] = dist[v] + edges[v][i].weight; ` ` ` ` ` `// Put 0 weight edges to front and 1 weight ` ` ` `// edges to back so that vertices are processed ` ` ` `// in increasing order of weights. ` ` ` `if` `(edges[v][i].weight == 0) ` ` ` `Q.push_front(edges[v][i].to); ` ` ` `else` ` ` `Q.push_back(edges[v][i].to); ` ` ` `} ` ` ` `} ` ` ` `} ` ` ` ` ` `// printing the shortest distances ` ` ` `for` `(` `int` `i=0; i<V; i++) ` ` ` `cout << dist[i] << ` `" "` `; ` `} ` ` ` `void` `addEdge(` `int` `u, ` `int` `v, ` `int` `wt) ` `{ ` ` ` `edges[u].push_back({v, wt}); ` ` ` `edges[v].push_back({u, wt}); ` `} ` ` ` `// Driver function ` `int` `main() ` `{ ` ` ` `addEdge(0, 1, 0); ` ` ` `addEdge(0, 7, 1); ` ` ` `addEdge(1, 7, 1); ` ` ` `addEdge(1, 2, 1); ` ` ` `addEdge(2, 3, 0); ` ` ` `addEdge(2, 5, 0); ` ` ` `addEdge(2, 8, 1); ` ` ` `addEdge(3, 4, 1); ` ` ` `addEdge(3, 5, 1); ` ` ` `addEdge(4, 5, 1); ` ` ` `addEdge(5, 6, 1); ` ` ` `addEdge(6, 7, 1); ` ` ` `addEdge(7, 8, 1); ` ` ` `int` `src = 0;` `//source node ` ` ` `zeroOneBFS(src); ` ` ` `return` `0; ` `} ` |

*chevron_right*

*filter_none*

## Java

`// Java Program to implement 0-1 BFS ` `import` `java.util.ArrayDeque; ` `import` `java.util.ArrayList; ` `import` `java.util.Deque; ` ` ` `public` `class` `ZeroOneBFS { ` ` ` `private` `static` `class` `Node { ` ` ` `int` `to; ` `// the ending vertex ` ` ` `int` `weight; ` `// the weight of the edge ` ` ` ` ` `public` `Node(` `int` `to, ` `int` `wt) { ` ` ` `this` `.to = to; ` ` ` `this` `.weight = wt; ` ` ` `} ` ` ` `} ` ` ` ` ` `private` `static` `final` `int` `numVertex = ` `9` `; ` ` ` `private` `ArrayList<Node>[] edges = ` `new` `ArrayList[numVertex]; ` ` ` ` ` `public` `ZeroOneBFS() { ` ` ` `for` `(` `int` `i = ` `0` `; i < edges.length; i++) { ` ` ` `edges[i] = ` `new` `ArrayList<Node>(); ` ` ` `} ` ` ` `} ` ` ` ` ` `public` `void` `addEdge(` `int` `u, ` `int` `v, ` `int` `wt) { ` ` ` `edges[u].add(edges[u].size(), ` `new` `Node(v, wt)); ` ` ` `edges[v].add(edges[v].size(), ` `new` `Node(u, wt)); ` ` ` `} ` ` ` ` ` `public` `void` `zeroOneBFS(` `int` `src) { ` ` ` ` ` `// initialize distances from given source ` ` ` `int` `[] dist = ` `new` `int` `[numVertex]; ` ` ` `for` `(` `int` `i = ` `0` `; i < numVertex; i++) { ` ` ` `dist[i] = Integer.MAX_VALUE; ` ` ` `} ` ` ` ` ` `// double ended queue to do BFS ` ` ` `Deque<Integer> queue = ` `new` `ArrayDeque<Integer>(); ` ` ` `dist[src] = ` `0` `; ` ` ` `queue.addLast(src); ` ` ` ` ` `while` `(!queue.isEmpty()) { ` ` ` `int` `v = queue.removeFirst(); ` ` ` `for` `(` `int` `i = ` `0` `; i < edges[v].size(); i++) { ` ` ` ` ` `// checking for optimal distance ` ` ` `if` `(dist[edges[v].get(i).to] > ` ` ` `dist[v] + edges[v].get(i).weight) { ` ` ` ` ` `// update the distance ` ` ` `dist[edges[v].get(i).to] = ` ` ` `dist[v] + edges[v].get(i).weight; ` ` ` ` ` `// put 0 weight edges to front and 1 ` ` ` `// weight edges to back so that vertices ` ` ` `// are processed in increasing order of weight ` ` ` `if` `(edges[v].get(i).weight == ` `0` `) { ` ` ` `queue.addFirst(edges[v].get(i).to); ` ` ` `} ` `else` `{ ` ` ` `queue.addLast(edges[v].get(i).to); ` ` ` `} ` ` ` `} ` ` ` `} ` ` ` `} ` ` ` ` ` `for` `(` `int` `i = ` `0` `; i < dist.length; i++) { ` ` ` `System.out.print(dist[i] + ` `" "` `); ` ` ` `} ` ` ` `} ` ` ` ` ` `public` `static` `void` `main(String[] args) { ` ` ` `ZeroOneBFS graph = ` `new` `ZeroOneBFS(); ` ` ` `graph.addEdge(` `0` `, ` `1` `, ` `0` `); ` ` ` `graph.addEdge(` `0` `, ` `7` `, ` `1` `); ` ` ` `graph.addEdge(` `1` `, ` `7` `, ` `1` `); ` ` ` `graph.addEdge(` `1` `, ` `2` `, ` `1` `); ` ` ` `graph.addEdge(` `2` `, ` `3` `, ` `0` `); ` ` ` `graph.addEdge(` `2` `, ` `5` `, ` `0` `); ` ` ` `graph.addEdge(` `2` `, ` `8` `, ` `1` `); ` ` ` `graph.addEdge(` `3` `, ` `4` `, ` `1` `); ` ` ` `graph.addEdge(` `3` `, ` `5` `, ` `1` `); ` ` ` `graph.addEdge(` `4` `, ` `5` `, ` `1` `); ` ` ` `graph.addEdge(` `5` `, ` `6` `, ` `1` `); ` ` ` `graph.addEdge(` `6` `, ` `7` `, ` `1` `); ` ` ` `graph.addEdge(` `7` `, ` `8` `, ` `1` `); ` ` ` `int` `src = ` `0` `;` `//source node ` ` ` `graph.zeroOneBFS(src); ` ` ` `return` `; ` ` ` `} ` `} ` |

*chevron_right*

*filter_none*

Output:

0 0 1 1 2 1 2 1 2

This problem can also be solved by Dijkstra but the time complexity will be O(E + V Log V) whereas by BFS it will be O(V+E).

**Reference :**

http://codeforces.com/blog/entry/22276

This article is contributed by **Ayush Jha**. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

## Recommended Posts:

- Shortest Path in a weighted Graph where weight of an edge is 1 or 2
- Multistage Graph (Shortest Path)
- Shortest path in an unweighted graph
- Shortest Path in Directed Acyclic Graph
- Shortest path with exactly k edges in a directed and weighted graph
- Multi Source Shortest Path in Unweighted Graph
- Shortest path in a Binary Maze
- Shortest path from source to destination such that edge weights along path are alternatively increasing and decreasing
- Convert the undirected graph into directed graph such that there is no path of length greater than 1
- Check if there is a cycle with odd weight sum in an undirected graph
- Find minimum weight cycle in an undirected graph
- k'th heaviest adjacent node in a graph where each vertex has weight
- Dijkstra’s shortest path algorithm using set in STL
- Some interesting shortest path questions | Set 1
- Shortest Path using Meet In The Middle