Efficient method to store a Lower Triangular Matrix using Column-major mapping
Given a lower triangular matrix Mat[][], the task is to store the matrix using column-major mapping.
Lower Triangular Matrix: A Lower Triangular Matrix is a square matrix in which the lower triangular part of a matrix consists of non-zero elements and the upper triangular part consists of 0s. The Lower Triangular Matrix for a 2D matrix Mat[][] is mathematically defined as:
- If i < j, set Mat[i][j] = 0.
- If i >= j, set Mat[i][j] > 0.
Illustration:
Below is a 5×5 lower triangular matrix. In general, such matrices can be stored in a 2D array, but when it comes to matrices of large size, it is not a good choice because of its high memory consumption due to the storage of unwanted 0s.
Such a matrix can be implemented in an optimized manner.
The efficient way to store the lower triangular matrix of size N:
- Count of non-zero elements = 1 + 2 + 3 + … + N = N * (N + 1) /2.
- Count of 0s = N2 – (N * (N + 1) /2 = (N * (N – 1)/2.
Now let see how to represent lower triangular matrices in the program. Notice that storing 0s must be avoided to reduce memory consumption. As calculated, for storing non-zero elements, N*(N + 1)/2 space is needed. Taking the above example, N = 5. Array of size 5 * (5 + 1)/2 = 15 is required to store the non-zero elements.
Now, elements of the 2D matrix can be stored in a 1D array, column by column, as shown below:

Array to store Lower Triangular Elements
Apart from storing the elements in an array, a procedure for extracting the element corresponding to the row and column number is also required. Using Column-Major-Mapping for storing a lower triangular matrix, the element at index Mat[i][j] can be represented as:
Index of Mat[i][j] matrix in the array A[] = [n*(j-1)-(((j-2)*(j-1))/2)+ (i-j))]
Below is the implementation of the above article:
C++
// C++ program for the above approach #include <bits/stdc++.h> #include<stdio.h> using namespace std; // Dimensions of the matrix const int N = 5; // Structure of a memory // efficient matrix struct Matrix { int * A; int size; }; // Function to set the // values in the Matrix void Set( struct Matrix* m, int i, int j, int x) { if (i >= j) m->A[((m->size)*(j-1)-(((j-2) *(j-1))/2)+(i-j))] = x; } // Function to store the // values in the Matrix int Get( struct Matrix m, int i, int j) { if (i >= j) return m.A[((m.size)*(j-1)-(((j-2) *(j-1))/2)+(i-j))]; else return 0; } // Function to display the // elements of the matrix void Display( struct Matrix m) { // Traverse the matrix for ( int i = 1; i <= m.size; i++) { for ( int j = 1; j <= m.size; j++) { if (i >= j) cout<< m.A[((m.size)*(j-1)-(((j-2) *(j-1))/2)+(i-j))] << " " ; else cout<< "0 " ; } cout<<endl; } } // Function to generate an efficient matrix struct Matrix createMat( int Mat[N][N]) { // Declare efficient Matrix struct Matrix mat; // Initialize the Matrix mat.size = N; mat.A = ( int *) malloc ( mat.size * (mat.size + 1) / 2 * sizeof ( int )); // Set the values in matrix for ( int i = 1; i <= mat.size; i++) { for ( int j = 1; j <= mat.size; j++) { Set(&mat, i, j, Mat[i - 1][j - 1]); } } // Return the matrix return mat; } // Driver Code int main() { // Given Input int Mat[5][5] = { { 1, 0, 0, 0, 0 }, { 1, 2, 0, 0, 0 }, { 1, 2, 3, 0, 0 }, { 1, 2, 3, 4, 0 }, { 1, 2, 3, 4, 5 } }; // Function call to create a memory // efficient matrix struct Matrix mat = createMat(Mat); // Function call to // print the Matrix Display(mat); return 0; } // This code is contributed by rrrtnx. |
C
// C program for the above approach #include <stdio.h> #include <stdlib.h> // Dimensions of the matrix const int N = 5; // Structure of a memory // efficient matrix struct Matrix { int * A; int size; }; // Function to set the // values in the Matrix void Set( struct Matrix* m, int i, int j, int x) { if (i >= j) m->A[((m->size)*(j-1)-(((j-2) *(j-1))/2)+(i-j))] = x; } // Function to store the // values in the Matrix int Get( struct Matrix m, int i, int j) { if (i >= j) return m.A[((m.size)*(j-1)-(((j-2) *(j-1))/2)+(i-j))]; else return 0; } // Function to display the // elements of the matrix void Display( struct Matrix m) { // Traverse the matrix for ( int i = 1; i <= m.size; i++) { for ( int j = 1; j <= m.size; j++) { if (i >= j) printf ( "%d " , m.A[((m.size)*(j-1)-(((j-2) *(j-1))/2)+(i-j))]); else printf ( "0 " ); } printf ( "\n" ); } } // Function to generate an efficient matrix struct Matrix createMat( int Mat[N][N]) { // Declare efficient Matrix struct Matrix mat; // Initialize the Matrix mat.size = N; mat.A = ( int *) malloc ( mat.size * (mat.size + 1) / 2 * sizeof ( int )); // Set the values in matrix for ( int i = 1; i <= mat.size; i++) { for ( int j = 1; j <= mat.size; j++) { Set(&mat, i, j, Mat[i - 1][j - 1]); } } // Return the matrix return mat; } // Driver Code int main() { // Given Input int Mat[5][5] = { { 1, 0, 0, 0, 0 }, { 1, 2, 0, 0, 0 }, { 1, 2, 3, 0, 0 }, { 1, 2, 3, 4, 0 }, { 1, 2, 3, 4, 5 } }; // Function call to create a memory // efficient matrix struct Matrix mat = createMat(Mat); // Function call to // print the Matrix Display(mat); return 0; } |
1 0 0 0 0 1 2 0 0 0 1 2 3 0 0 1 2 3 4 0 1 2 3 4 5
Time Complexity: O(N2)
Auxiliary Space: O(N2)
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