Sum of the minimum elements in all connected components of an undirected graph
Given an array A of N numbers where Ai 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 = 15Input: 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:
// 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; } |
15
Recommended Posts:
- Connected Components in an undirected graph
- Clone an undirected graph with multiple connected components
- Maximum number of edges among all connected components of an undirected graph
- Program to count Number of connected components in an undirected graph
- Cycles of length n in an undirected and connected graph
- Number of single cycle components in an undirected graph
- Kth largest node among all directly connected nodes to the given node in an undirected graph
- Find minimum weight cycle in an undirected graph
- Strongly Connected Components
- Number of connected components in a 2-D matrix of strings
- Tarjan's Algorithm to find Strongly Connected Components
- Convert the undirected graph into directed graph such that there is no path of length greater than 1
- Check if a directed graph is connected or not
- Clone an Undirected Graph
- Check if a graph is strongly connected | Set 1 (Kosaraju using DFS)
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.