Q15_graph_coloring.cpp

// 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 contain u).
// There are no parallel edges (graph[u] does not contain duplicate values).
// If v is in graph[u], then u is in graph[v] (the graph is undirected).
// The graph may not be connected, meaning there may be two nodes u and v 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 bipartite.

// 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}.

#include<bits/stdc++.h>
using namespace std;

class Solution {
private:
    bool dfs(vector<vector<int>>& graph, int src, vector<int>& colors) {
        if(colors[src] == -1) colors[src] = 1;
        
        for(int &v : graph[src]) {
            if(colors[v] == -1) {
                colors[v] = !colors[src];
                if(!dfs(graph, v, colors)) return false;
            }
            else if(colors[v] == colors[src]) return false;
        }
        return true;
    }
public:
    bool isBipartite(vector<vector<int>>& graph) {
        int n = graph.size();
        vector<int> colors(n, -1);
        
        for(int i=0; i<n; i++) {
            if(colors[i] == -1) 
                if(!dfs(graph, i, colors)) return false;
        }
        return true;
    }
};

// Time Complexity  : O(n)
// Space Complexity : O(n)