Sum of the minimum elements in all connected components of an undirected graph

Given an array A of N numbers where A

_i

represent the value of the (i+1)

^th

node. Also given are M pair of edges where u and v represent the nodes that are connected by an edge. The task is to find the sum of the minimum element in all the connected components of the given undirected graph. If a node has no connectivity to any other node, count it as a component with one node.

Examples:

Input: a[] = {1, 6, 2, 7, 3, 8, 4, 9, 5, 10} m = 5 1 2 3 4 5 6 7 8 9 10 Output: 15 Connected components are: 1–2, 3–4, 5–6, 7–8 and 9–10 Sum of Minimum of all them : 1 + 2 + 3 + 4 + 5 = 15 Input: a[] = {2, 5, 3, 4, 8} m = 2 1 4 4 5 Output: 10

Approach:

Finding connected components for an undirected graph is an easier task. Doing either a

BFS

or

DFS

starting from every unvisited vertex will give us our connected components. Create a

visited[]

array which has initially all nodes marked as False. Iterate all the nodes, if the node is not visited, call

DFS()

function so that all the nodes connected directly or indirectly to the node are marked as visited. While visiting all the directly or indirectly connected nodes, store the minimum value of all nodes. Create a variable

sum

which stores the summation of the minimum of all these connected components. Once all the nodes are visited,

sum

will have the answer to the problem. Below is the implementation of the above approach:

CPP

// C++ program to find the sum
// of the minimum elements in all
// connected components of an undirected graph
#include <bits/stdc++.h>
using namespace std;
const int N = 10000;
vector<int> graph[N];

// Initially all nodes
// marked as unvisited
bool visited[N];

// DFS function that visits all
// connected nodes from a given node
void dfs(int node, int a[], int mini)
{
    // Stores the minimum
    mini = min(mini, a[node]);

    // Marks node as visited
    visited[node] = true;

    // Traversed in all the connected nodes
    for (int i : graph[node]) {
        if (!visited[i])
            dfs(i, a, mini);
    }
}

// Function to add the edges
void addedge(int u, int v)
{
    graph[u - 1].push_back(v - 1);
    graph[v - 1].push_back(u - 1);
}

// Function that returns the sum of all minimums
// of connected componenets of graph
int minimumSumConnectedComponents(int a[], int n)
{
    // Initially sum is 0
    int sum = 0;

    // Traverse for all nodes
    for (int i = 0; i < n; i++) {
        if (!visited[i]) {
            int mini = a[i];
            dfs(i, a, mini);
            sum += mini;
        }
    }

    // Returns the answer
    return sum;
}

// Driver Code
int main()
{
    int a[] = {1, 6, 2, 7, 3, 8, 4, 9, 5, 10};

    // Add edges
    addedge(1, 2);
    addedge(3, 4);
    addedge(5, 6);
    addedge(7, 8);
    addedge(9, 10);

    int n = sizeof(a) / sizeof(a[0]);

    // Calling Function
    cout << minimumSumConnectedComponents(a, n);
    return 0;
}

C#

using System;
using System.Collections.Generic;

class Program {
    const int N = 10000;
    static List<int>[] graph = new List<int>[ N ];
    static bool[] visited = new bool[N];

    // DFS function that visits all
    // connected nodes from a given node
    static void DFS(int node, int[] a, ref int mini)
    {
        // Stores the minimum
        mini = Math.Min(mini, a[node]);

        // Marks node as visited
        visited[node] = true;

        // Traversed in all the connected nodes
        foreach(int i in graph[node])
        {
            if (!visited[i])
                DFS(i, a, ref mini);
        }
    }

    // Function to add the edges
    static void AddEdge(int u, int v)
    {
        graph[u - 1].Add(v - 1);
        graph[v - 1].Add(u - 1);
    }

    // Function that returns the sum of all minimums
    // of connected components of the graph
    static int MinimumSumConnectedComponents(int[] a, int n)
    {
        // Initially sum is 0
        int sum = 0;

        // Traverse for all nodes
        for (int i = 0; i < n; i++) {
            if (!visited[i]) {
                int mini = a[i];
                DFS(i, a, ref mini);
                sum += mini;
            }
        }

        // Returns the answer
        return sum;
    }

    // Driver Code
    static void Main()
    {
        int[] a = { 1, 6, 2, 7, 3, 8, 4, 9, 5, 10 };

        // Initialize graph
        for (int i = 0; i < N; i++) {
            graph[i] = new List<int>();
        }

        // Add edges
        AddEdge(1, 2);
        AddEdge(3, 4);
        AddEdge(5, 6);
        AddEdge(7, 8);
        AddEdge(9, 10);

        int n = a.Length;

        // Calling Function
        Console.WriteLine(
            MinimumSumConnectedComponents(a, n));
    }
}

Output

15