Given a tree of N nodes, the task is to convert the given tree to its Sum Tree(including its own weight) and find the minimum difference between any two node’s weight of the sum tree.
Note: The N nodes of the given tree are given in the form of top to bottom with N-1 line where each line describes two nodes which are connected.
Examples:
Input:
Output: 1
Explanation:
total weight of node 1: 3 (own weight) + (10 + 6 + 5 + 8 + 2 + 7 + 11) (sub-tree node’s weight) = 52
total weight of node 2: 5 (own weight) + (2 + 7 + 11) (sub-tree node’s weight) = 25
total weight of node 3: 8 (own weight) + (0) (sub-tree node’s weight) = 8
total weight of node 4: 10 (own weight) + (0) (sub-tree node’s weight) = 10
total weight of node 5: 2 (own weight) + (0) (sub-tree node’s weight) = 2
total weight of node 6: 6 (own weight) + (5 + 8 + 2 + 7 + 11) (sub-tree node’s weight) = 39
total weight of node 7: 7 (own weight) + (0) (sub-tree node’s weight) = 7
total weight of node 8: 11 (own weight) + (0) (sub-tree node’s weight) = 11By observing the total weight of each node, Node 4 and 8 have a minimum difference(11-10) = 1
Input:
Output: 0
Approach:
- We will traverse the given tree from below and store the weight of that node plus its sub-tree node’s weight in one array and mark index of each node as visited. So in between if we revisit that node then we don’t have to count the weight of that node again.
- We will sort the array where we have stored the total weight of each node.
- Now find the pairwise difference in the sorted array and whichever pair gave minimum difference print that minimum difference at last.
Below is the implementation of the above approach:
C++
// C++ program for the above approach #include <bits/stdc++.h> using namespace std; // Function to find minimum // difference between any two node void MinimumDifference(int total_weight[], int N) { int min_difference = INT_MAX; for (int i = 1; i < N; i++) { // Pairwise difference if (total_weight[i] - total_weight[i - 1] < min_difference) { min_difference = total_weight[i] - total_weight[i - 1]; } } cout << min_difference << endl; } // Function to find total weight // of each individual node void SumTree(vector<pair<int, int> > v, int individual_weight[], int N) { // Array to store total weight // of each node from 1 to N int total_weight[N] = { 0 }; // Array to keep track of node // previously counted or not int visited[N] = { 0 }; // To store node no. from /// N-1 lines int first, second; // To traverse from (N-1) // line to 1 line for (int i = (N - 2); i >= 0; i--) { first = v[i].first; second = v[i].second; // Node is note visited if (visited[second - 1] == 0) { total_weight[second - 1] += individual_weight[second - 1]; // Make node visited visited[second - 1] = 1; } total_weight[first - 1] += total_weight[second - 1]; // Node is note visited if (visited[first - 1] == 0) { total_weight[first - 1] += individual_weight[first - 1]; // Make node visited visited[first - 1] = 1; } } // Sort the total weight of each node sort(total_weight, total_weight + N); // Call function to find minimum // difference MinimumDifference(total_weight, N); } // Driver code int main() { // Total node of rooted tree int N = 8; vector<pair<int, int> > v; // N-1 lines describing // rooted tree from top // to bottom v.push_back(make_pair(1, 4)); v.push_back(make_pair(1, 6)); v.push_back(make_pair(6, 2)); v.push_back(make_pair(6, 3)); v.push_back(make_pair(2, 5)); v.push_back(make_pair(2, 7)); v.push_back(make_pair(2, 8)); // Array describing weight // of each node from 1 to N int individual_weight[N] = { 3, 5, 8, 10, 2, 6, 7, 11 }; SumTree(v, individual_weight, N); return 0; } |
Python3
# Python3 program for the above approach import sys # Function to find minimum difference # between any two node def minimum_difference(total_weight, n): min_difference = sys.maxsize for i in range(1, n): # Pairwise difference if (total_weight[i] - total_weight[i - 1] < min_difference): min_difference = (total_weight[i] - total_weight[i - 1]) print(min_difference) # Function to find total weight # of each individual node def SumTree(v, individual_weight, N): # Array to store total weight of # each node from 1 to n total_weight = [0 for i in range(N)] # Array to keep track of node # previously counted or not visited = [0 for i in range(N)] # To traverse from (n-1) line to 1 line for i in range(N - 2, -1, -1): first = v[i][0] second = v[i][1] if visited[second - 1] == 0: total_weight[second - 1] += ( individual_weight[second - 1]) # Make node visited visited[second - 1] = 1 total_weight[first - 1] += ( total_weight[second - 1]) # Node is note visited if visited[first - 1] == 0: total_weight[first - 1] += ( individual_weight[first - 1]) # Make node visited visited[first - 1] = 1 # Sort the total weight of each node total_weight.sort() # Call function to find minimum difference minimum_difference(total_weight, n) # Driver Code if __name__=='__main__': # Total node of rooted tree n = 8 v = [] # n-1 lines describing rooted # tree from top to bottom v.append([1, 4]) v.append([1, 6]) v.append([6, 2]) v.append([6, 3]) v.append([2, 5]) v.append([2, 7]) v.append([2, 8]) # Array describing weight of each # node from 1 to n individual_weight = [ 3, 5, 8, 10, 2, 6, 7, 11 ] SumTree(v, individual_weight, n) # This code is contributed by rutvik_56 |
1
Time Complexity: O(N * Log(N)), where N is total nodes in rooted tree.
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Improved By : rutvik_56
