Is Graph Bipartite?#
Problem statement#
[1]There is an undirected graph with n
nodes, where each node is numbered between 0
and n - 1
. You are given a 2D array graph
, where graph[u]
is an array of nodes that node u
is adjacent to. More formally, for each v
in graph[u]
, there is an undirected edge between node u
and node v
. The graph has the following properties:
There are no self-edges (
graph[u]
does not containu
).There are no parallel edges (
graph[u]
does not contain duplicate values).If
v
is ingraph[u]
, thenu
is ingraph[v]
(the graph is undirected).The graph may not be connected, meaning there may be two nodes
u
andv
such that there is no path between them.
A graph is bipartite if the nodes can be partitioned into two independent sets A
and B
such that every edge in the graph connects a node in set A
and a node in set B
.
Return true
if and only if it is a bipartite graph.
Example 1#
Input: graph = [[1,2,3],[0,2],[0,1,3],[0,2]]
Output: false
Explanation: There is no way to partition the nodes into two independent sets such that every edge connects a node in one and a node in the other.
Example 2#
Input: graph = [[1,3],[0,2],[1,3],[0,2]]
Output: true
Explanation: We can partition the nodes into two sets: {0, 2} and {1, 3}.
Constraints#
graph.length == n
.1 <= n <= 100
.0 <= graph[u].length < n
.0 <= graph[u][i] <= n - 1
.graph[u]
does not containu
.All the values of
graph[u]
are unique.If
graph[u]
containsv
, thengraph[v]
containsu
.
Solution: Coloring the nodes by Depth First Search#
You could color the nodes in set A with one color and those in B with another color. Then two ends of every edge have different colors.
Now you can use the DFS algorithm[2] to perform the coloring on each connected component of the graph.
During the traversal, if there is an edge having the same color at two ends then return false
.
Code#
#include <vector>
#include <iostream>
using namespace std;
bool isBipartite(vector<vector<int>>& graph) {
vector<int> color(graph.size(), 0);
for (int i = 0; i < graph.size(); i++) {
if (color[i] != 0) continue;
vector<int> s;
s.push_back(i);
color[i] = 1;
while (!s.empty()) {
int u = s.back();
s.pop_back();
for (int v : graph[u]) {
if (color[v] == 0) {
color[v] = -color[u];
s.push_back(v);
} else if (color[v] == color[u]) {
return false;
}
}
}
}
return true;
}
int main() {
vector<vector<int>> graph{{1,2,3},{0,2},{0,1,3},{0,2}};
cout << isBipartite(graph) << endl;
graph = {{1,3},{0,2},{1,3},{0,2}};
cout << isBipartite(graph) << endl;
}
Output:
0
1
Complexity#
Runtime:
O(n)
, wheren = graph.length
.Extra space:
O(n)
.
Implementation tips#
This is the non-recursive implementation of DFS algorithm where you could use the stack data structure to avoid the recursion.
The stack’s methods needed in the DFS algorithm are only
push
andpop
. There are similar ones instd::vector
. They arepush_back
[3] andpop_back
[4] which you could use well in this case.