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Maximum product of a pair of nodes from largest connected component in a Graph

  • Last Updated : 05 Jul, 2021

Given an undirected weighted graph G consisting of N vertices and M edges, and two arrays Edges[][2] and Weight[] consisting of M edges of the graph and weights of each edge respectively, the task is to find the maximum product of any two vertices of the largest connected component of the graph, formed by connecting all edges with the same weight.

Examples:

Input: N = 4, Edges[][] = {{1, 2}, {1, 2}, {2, 3}, {2, 3}, {2, 4}}, Weight[] = {1, 2, 1, 3, 3}
Output: 12
Explanation:

  • Components of edges of weight 1, 1 ↔ 2 ↔ 3. The maximum product of any two vertices of this component is 6.
  • Components of edges of weight 2, 1 ↔ 2. The maximum product of any two vertices of this component is 2.
  • Components of edges of weight 3, 4 ↔ 2 ↔ 3. The maximum product of any two vertices of this component is 12.

Therefore, the maximum product among all the connected components of size 3 (which is maximum) is 12.



Input: N = 5, Edges[][] = {{1, 5}, {2, 5}, {3, 5}, {4, 5}, {1, 2}, {2, 3}, {3, 4}}, Weight[] = {1, 1, 1, 1, 2, 2, 2}
Output: 20

Approach: The given problem can be solved by performing the DFS traversal on the given graph and maximize the product of the first and second maximum node for all the connected components of the same weight. Follow the steps below to solve the problem:

  • Store all the edges corresponding to all the unique weight in a map M.
  • Initialize a variable, say res as 0 to store the maximum product of any two nodes of the connected components of the same weights.
  • Traverse the map and for each key as weight create a graph by connecting all the edges mapped with the particular weight and perform the following operations:
    • Find the value of the maximum(say M1) and the second maximum(say M2) node’s value and the size of all the connected components of the graph by performing the DFS Traversal on the created graph.
    • Update the value of res to the maximum of res, M1, and M2 if the size of the currently connected components is at least the largest size connected component found previously.
  • After completing the above steps, print the value of res as the maximum product.

Below is the implementation of the above approach:

C++




// C++ program for the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Stores the first and second largest
// element in a connected component
int Max, sMax;
 
// Stores the count of nodes
// in the connected components
int cnt = 0;
 
// Function to perform DFS Traversal
// on a given graph and find the first
// and the second largest elements
void dfs(int u, int N, vector<bool>& vis,
         vector<vector<int> >& adj)
{
    // Update the maximum value
    if (u > Max) {
        sMax = Max;
        Max = u;
    }
 
    // Update the second max value
    else if (u > sMax) {
        sMax = u;
    }
 
    // Increment size of component
    cnt++;
 
    // Mark current node visited
    vis[u] = true;
 
    // Traverse the adjacent nodes
    for (auto to : adj[u]) {
 
        // If to is not already visited
        if (!vis[to]) {
            dfs(to, N, vis, adj);
        }
    }
 
    return;
}
 
// Function to find the maximum
// product of a connected component
int MaximumProduct(
    int N, vector<pair<int, int> > Edge,
    vector<int> wt)
{
    // Stores the count of edges
    int M = wt.size();
 
    // Stores all the edges mapped
    // with a particular weight
    unordered_map<int,
                  vector<pair<int, int> > >
        mp;
 
    // Update the map mp
    for (int i = 0; i < M; i++)
        mp[wt[i]].push_back(Edge[i]);
 
    // Stores the result
    int res = 0;
 
    // Traverse the map mp
    for (auto i : mp) {
 
        // Stores the adjacency list
        vector<vector<int> > adj(N + 1);
 
        // Stores the edges of
        // a particular weight
        vector<pair<int, int> > v = i.second;
 
        // Traverse the vector v
        for (int j = 0; j < v.size(); j++) {
 
            int U = v[j].first;
            int V = v[j].second;
 
            // Add an edge
            adj[U].push_back(V);
            adj[V].push_back(U);
        }
 
        // Stores if a vertex
        // is visited or not
        vector<bool> vis(N + 1, 0);
 
        // Stores the maximum
        // size of a component
        int cntMax = 0;
 
        // Iterate over the range [1, N]
        for (int u = 1; u <= N; u++) {
 
            // Assign Max, sMax, count = 0
            Max = sMax = cnt = 0;
 
            // If vertex u is not visited
            if (!vis[u]) {
 
                dfs(u, N, vis, adj);
 
                // If cnt is greater
                // than cntMax
                if (cnt > cntMax) {
 
                    // Update the res
                    res = Max * sMax;
                    cntMax = cnt;
                }
 
                // If already largest
                // connected component
                else if (cnt == cntMax) {
 
                    // Update res
                    res = max(res, Max * sMax);
                }
            }
        }
    }
 
    // Return res
    return res;
}
 
// Driver Code
int main()
{
    int N = 5;
    vector<pair<int, int> > Edges
        = { { 1, 2 }, { 2, 5 }, { 3, 5 }, { 4, 5 }, { 1, 2 }, { 2, 3 }, { 3, 4 } };
 
    vector<int> Weight = { 1, 1, 1, 1,
                           2, 2, 2 };
    cout << MaximumProduct(N, Edges, Weight);
 
    return 0;
}
Output: 
20

 

Time Complexity: O(N2 * log N + M)
Auxiliary Space: O(N2)

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