02. Santa Banta

The problem can be found at the following link: Question Link

My Approach

To solve the problem, I have used a depth-first search (DFS) algorithm to find the largest connected component in an undirected graph. Here are the steps I followed:

1. The dfs function is implemented to perform the depth-first search. It takes the current node as input and recursively visits all its unvisited neighbors.

2. Within the dfs function, I mark the current node as visited and increment the count of the component.

3. Then, for each unvisited neighbor of the current node, I call the dfs function recursively.

4. Finally, the helpSanta function is implemented to iterate through all the nodes in the graph and find the largest connected component.

5. It initializes a boolean array vis to keep track of visited nodes and an integer variable out to store the size of the largest connected component.

6. It calls the dfs function for each unvisited node, updating out with the maximum component size encountered.

7. If out is greater than 1, I retrieve the out-1th prime number from a precomputed list of prime numbers (kPrime) and assign it to out. Otherwise, I assign -1 to out.

8. Finally, the function returns the value of out.

Time and Auxiliary Space Complexity

• Time Complexity: The DFS algorithm has a time complexity of O(V + E), where V is the number of vertices and E is the number of edges in the graph. Therefore, the overall time complexity of the algorithm is O(n + m), where n is the number of nodes and m is the number of edges in the graph.

• Auxiliary Space Complexity: The auxiliary space complexity of this approach is O(n), where n is the number of nodes in the graph. This is due to the space required for the vis array and the recursive call stack.

Code (C++)

// Constant defining the maximum size of the array
const int maxn=1000001;

// Array for marking prime numbers
int a[maxn+1];

// Vector for storing prime numbers
vector<int> pl={2};

class Solution{
public:
    // Function to precompute the prime numbers
    void precompute(){
        // Marking all numbers as prime initially
        for(int i=1;i<=maxn;i++)
            a[i]=1;
        
        // Marking 0 and 1 as not prime
        a[0]=a[1]=0;
        
        // Sieve of Eratosthenes algorithm to mark non-prime numbers
        for(int i=2;i*i<=maxn;i++){
            if(a[i]==1){
                for(int j=i*i;j<=maxn;j+=i){
                    a[j]=0;
                }
            }
        }
        
        // Storing all the prime numbers in the vector
        for(int i=3;i<=maxn;i++)
            if(a[i])
                pl.push_back(i);
    }
    
    // Depth-first search function to find the number of reachable nodes in a graph
    int dfs(int i, vector<int> g[], vector<int> &vis){
        // Marking the current node as visited
        vis[i]=1;
        
        // Counter variable to keep track of the number of reachable nodes
        int cnt=1;
        
        // Recursively traversing all the adjacent nodes of the current node
        for(auto x:g[i]){
            if(!vis[x]){
                cnt+=dfs(x, g, vis);
            }
        }
        
        // Returning the total number of reachable nodes
        return cnt;
    }
    
    // Function to help Santa navigate the given graph
    int helpSanta(int n, int m, vector<vector<int>> &g)
    {
        // Initializing the visited array
        vector<int> vis(n+1, 0);
        
        // Creating an adjacency list for the given graph
        vector<int> adj[n + 1];
        for(auto i : g){
            adj[i[0]].push_back(i[1]);
            adj[i[1]].push_back(i[0]);
        }
        
        // Variable to store the largest component size
        int lc=0;
        
        // Traversing all the nodes from 0 to n and finding the largest component
        for(int i = 0; i <= n; i++){
            if(!vis[i]){
                lc=max(lc,dfs(i, adj, vis));
            }
        }
        
        // Checking if there is only one component in the graph
        // If yes, returning -1
        if(lc==1)
            return -1;
        
        // Returning the prime number at the (largest component size - 1) index
        return pl[lc-1];
    }
};

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