Maximum sum of values of nodes among all connected components of an undirected graph

Given an undirected graph with V vertices and E edges. Every node has been assigned a given value. The task is to find the connected chain with the maximum sum of of values among all the connected components in the graph.

Examples:

Input: V = 7, E = 4
Values = {10, 25, 5, 15, 5, 20, 0}

Output : Max Sum value = 35
Explanation:
Component {1, 2} – Value {10, 25}: sumValue = 10 + 25 = 35
Component {3, 4, 5} – Value {5, 15, 5}: sumValue = 5 + 15 + 5 = 25
Component {6, 7} – Value {20, 0}: sumValue = 20 + 0 = 20
Max Sum value chain is {1, 2} with values {10, 25}, hence 35 is answer.

Input: V = 10, E = 6
Values = {5, 10, 15, 20, 25, 30, 35, 40, 45, 50}

Output : Max Sum value = 105



Approach: The idea is to use the Depth First Search traversal method to keep a track of all the connected components. A temporary variable is used to sum up all the values of the individual values of the connected chains. At each traversal of a connected component, the heaviest value till now is compared with the current value and updated accordingly. After all connected components have been traversed, the maximum among all will the answer.

Below is the implementation of the above approach:

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// C++ program to find Maximum sum of values
// of nodes among all connected
// components of an undirected graph
#include <bits/stdc++.h>
using namespace std;
  
// Function to implement DFS
void depthFirst(int v, vector<int> graph[],
                vector<bool>& visited,
                int& sum,
                vector<int> values)
{
    // Marking the visited vertex as true
    visited[v] = true;
  
    // Updating the value of connection
    sum += values[v - 1];
  
    // Traverse for all adjacent nodes
    for (auto i : graph[v]) {
  
        if (visited[i] == false) {
  
            // Recursive call to the DFS algorithm
            depthFirst(i, graph, visited,
                       sum, values);
        }
    }
}
  
void maximumSumOfValues(vector<int> graph[],
                        int vertices, vector<int> values)
{
    // Initializing boolean array to mark visited vertices
    vector<bool> visited(values.size() + 1, false);
  
    // maxChain stores the maximum chain size
    int maxValueSum = INT_MIN;
  
    // Following loop invokes DFS algorithm
    for (int i = 1; i <= vertices; i++) {
        if (visited[i] == false) {
  
            // Variable to hold temporary values
            int sum = 0;
  
            // DFS algorithm
            depthFirst(i, graph, visited,
                       sum, values);
  
            // Conditional to update max value
            if (sum > maxValueSum) {
                maxValueSum = sum;
            }
        }
    }
  
    // Printing the heaviest chain value
    cout << "Max Sum value = ";
    cout << maxValueSum << "\n";
}
  
// Driver function to test above function
int main()
{
    // Initializing graph in the form of adjacency list
    vector<int> graph[1001];
  
    // Defining the number of edges and vertices
    int E = 4, V = 7;
  
    // Assigning the values for each
    // vertex of the undirected graph
    vector<int> values;
    values.push_back(10);
    values.push_back(25);
    values.push_back(5);
    values.push_back(15);
    values.push_back(5);
    values.push_back(20);
    values.push_back(0);
  
    // Constructing the undirected graph
    graph[1].push_back(2);
    graph[2].push_back(1);
    graph[3].push_back(4);
    graph[4].push_back(3);
    graph[3].push_back(5);
    graph[5].push_back(3);
    graph[6].push_back(7);
    graph[7].push_back(6);
  
    maximumSumOfValues(graph, V, values);
    return 0;
}

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Output:

Max Sum value = 35

Time Complexity: O(E + V)

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