Given a matrix of order NxN, Find the minimum number of steps to convert given matrix into Upper Hessenberg matrix. In each step, the only operation allowed is to decrease or increase any element value by 1.
Examples:
Input : N=3
1 2 8
1 3 4
2 3 4
Output :2
Decrease the element a[2][0] 2 times.
Now the matrix is upper hessenbergInput : N=4
1 2 2 3
1 3 4 2
3 3 4 2
-1 0 1 4
Output :4
Approach:
- For a matrix to be Upper Hessenberg matrix all of its elements below sub-diagonal must be equal zero, i.e Aij = 0 for all i > j+1..
- The minimum number of steps required to convert a given matrix in the upper Hessenberg matrix is equal to the sum of the absolute values of all Aij for all i > j + 1.
- The modulus value of the element is taken into account because both the increase and decrease of the element count as a single step.
Below is the implementation of the above approach:
C++
// C++ implementation of above approach #include <bits/stdc++.h> #define N 4 using namespace std; // Function to count steps in // conversion of matrix into upper // Hessenberg matrix int stepsRequired(int arr[][N]) { int result = 0; for (int i = 0; i < N; i++) { for (int j = 0; j < N; j++) { // if element is below sub-diagonal // add abs(element) into result if (i > j + 1) result += abs(arr[i][j]); } } return result; } // Driver code int main() { int arr[N][N] = { 1, 2, 3, 4, 3, 1, 0, 3, 3, 2, 1, 3, -3, 4, 2, 1 }; // Function call cout << stepsRequired(arr); return 0; } |
Java
// Java implementation of above approach class GFG { static int N = 4; // Function to count steps in // conversion of matrix into upper // Hessenberg matrix static int stepsRequired(int arr[][]) { int result = 0; for (int i = 0; i < N; i++) { for (int j = 0; j < N; j++) { // if element is below sub-diagonal // add abs(element) into result if (i > j + 1) result += Math.abs(arr[i][j]); } } return result; } // Driver code public static void main (String[] args) { int arr [][] = new int [][] {{1, 2, 3, 4}, {3, 1, 0, 3}, {3, 2, 1, 3}, {-3, 4, 2, 1 }}; // Function call System.out.println(stepsRequired(arr)); } } // This code is contributed by ihritik |
Python3
# Python3 implementation of above approach N = 4; # Function to count steps in # conversion of matrix into upper # Hessenberg matrix def stepsRequired(arr): result = 0; for i in range(N): for j in range(N): # if element is below sub-diagonal # add abs(element) into result if (i > j + 1): result += abs(arr[i][j]); return result; # Driver code arr = [[1, 2, 3, 4], [3, 1, 0, 3], [3, 2, 1, 3], [-3, 4, 2, 1]]; # Function call print(stepsRequired(arr)); # This code is contributed by Rajput-Ji |
C#
// C# implementation of above approach using System; class GFG { static int N = 4; // Function to count steps in // conversion of matrix into upper // Hessenberg matrix static int stepsRequired(int [, ] arr) { int result = 0; for (int i = 0; i < N; i++) { for (int j = 0; j < N; j++) { // if element is below sub-diagonal // add abs(element) into result if (i > j + 1) result += Math.Abs(arr[i, j]); } } return result; } // Driver code public static void Main () { int [ , ] arr = new int [, ] { {1, 2, 3, 4}, {3, 1, 0, 3}, {3, 2, 1, 3}, {-3, 4, 2, 1}}; // Function call Console.WriteLine(stepsRequired(arr)); } } // This code is contributed by ihritik |
10
Time complexity : O(N*N)
Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.
Recommended Posts:
- Minimum steps required to convert the matrix into lower hessenberg matrix
- Print Upper Hessenberg matrix of order N
- Minimum number of steps to convert a given matrix into Diagonally Dominant Matrix
- Print Lower Hessenberg matrix of order N
- Maximize sum of N X N upper left sub-matrix from given 2N X 2N matrix
- Minimum steps required to convert X to Y where a binary matrix represents the possible conversions
- Minimum steps to convert all paths in matrix from top left to bottom right as palindromic paths
- Minimum steps to convert all paths in matrix from top left to bottom right as palindromic paths | Set 2
- Minimum steps to convert all top left to bottom right paths in Matrix as palindrome | Set 2
- Program to print Lower triangular and Upper triangular matrix of an array
- Program to swap upper diagonal elements with lower diagonal elements of matrix.
- Program to check if matrix is upper triangular
- Convert given Matrix into sorted Spiral Matrix
- Minimum cost to convert 3 X 3 matrix into magic square
- Case conversion (Lower to Upper and Vice Versa) of a string using BitWise operators in C/C++
- Sum of upper triangle and lower triangle
- Minimum steps to reach any of the boundary edges of a matrix | Set-2
- Find minimum steps required to reach the end of a matrix | Set - 1
- Minimum steps to reach any of the boundary edges of a matrix | Set 1
- Find minimum steps required to reach the end of a matrix | Set 2
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.
