Efficient method to store a Lower Triangular Matrix using row-major mapping - GeeksforGeeks

08 Mar, 2021
Given a lower triangular matrix Mat[][], the task is to store the matrix using row-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 us see how to represent lower triangular matrices in our 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, row by row, as shown below:
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 Row-Major Mapping for storing lower triangular matrix, the element at index Mat[i][j] can be represented as:
Index of Mat[i][j] matrix in the array A[] = [i*(i – 1)/2 + j – 1]
Below is the implementation of the above approach:
C
Java
C#
// C program for the above approach#include <stdio.h>#include <stdlib.h> // Dimensions of a matrixconst int N = 5; // Structure of the efficient matrixstruct Matrix {    int* A;    int size;}; // Function to set the// values in the Matrixvoid Set(struct Matrix* mat,         int i, int j, int x){    if (i >= j)        mat->A[i * (i - 1) / 2 + j - 1] = x;} // Function to store the// values in the Matrixint Get(struct Matrix mat, int i, int j){    if (i >= j) {        return mat.A[i * (i - 1) / 2 + j - 1];    }    else {        return 0;    }} // Function to display the// elements of the matrixvoid Display(struct Matrix mat){    int i, j;     // Traverse the matrix    for (i = 1; i <= mat.size; i++) {        for (j = 1; j <= mat.size; j++) {             if (i >= j) {                printf("%d ",                       mat.A[i * (i - 1) / 2 + j - 1]);            }            else {                printf("0 ");            }        }        printf("\n");    }} // Function to generate an efficient matrixstruct 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));     int i, j;     // Set the values in matrix    for (i = 1; i <= mat.size; i++) {         for (j = 1; j <= mat.size; j++) {             Set(&mat, i, j, Mat[i - 1][j - 1]);        }    }     // Return the matrix    return mat;} // Driver Codeint main(){    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 } };     // Stores the efficient matrix    struct Matrix mat = createMat(Mat);     // Print the Matrix    Display(mat);     return 0;}
// Java program for the above approachclass GFG{   // Dimensions of a matrixstatic int N = 5; // Structure of the efficient matrixstatic class Matrix {    int[] A;    int size;}; // Function to set the// values in the Matrixstatic void Set(Matrix mat,         int i, int j, int x){    if (i >= j)        mat.A[i * (i - 1) / 2 + j - 1] = x;} // Function to store the// values in the Matrixstatic int Get(Matrix mat, int i, int j){    if (i >= j) {        return mat.A[i * (i - 1) / 2 + j - 1];    }    else {        return 0;    }} // Function to display the// elements of the matrixstatic void Display(Matrix mat){    int i, j;     // Traverse the matrix    for (i = 1; i <= mat.size; i++) {        for (j = 1; j <= mat.size; j++) {             if (i >= j) {                System.out.printf("%d ",                       mat.A[i * (i - 1) / 2 + j - 1]);            }            else {                System.out.printf("0 ");            }        }        System.out.printf("\n");    }} // Function to generate an efficient matrixstatic Matrix createMat(int Mat[][]){       // Declare efficient Matrix    Matrix mat = new Matrix();     // Initialize the Matrix    mat.size = N;    mat.A = new int[(mat.size*(mat.size + 1)) / 2];    int i, j;     // Set the values in matrix    for (i = 1; i <= mat.size; i++)    {        for (j = 1; j <= mat.size; j++)        {            Set(mat, i, j, Mat[i - 1][j - 1]);        }    }     // Return the matrix    return mat;} // Driver Codepublic static void main(String[] args){    int Mat[][] = { { 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 } };     // Stores the efficient matrix    Matrix mat = createMat(Mat);     // Print the Matrix    Display(mat);}} // This code is contributed by 29AjayKumar
// C# program for the above approachusing System;public class GFG{   // Dimensions of a matrix  static int N = 5;   // Structure of the efficient matrix  class Matrix {    public int[] A;    public int size;  };   // Function to set the  // values in the Matrix  static void Set(Matrix mat,                  int i, int j, int x)  {    if (i >= j)      mat.A[i * (i - 1) / 2 + j - 1] = x;  }   // Function to store the  // values in the Matrix  static int Get(Matrix mat, int i, int j)  {    if (i >= j) {      return mat.A[i * (i - 1) / 2 + j - 1];    }    else {      return 0;    }  }   // Function to display the  // elements of the matrix  static void Display(Matrix mat)  {    int i, j;     // Traverse the matrix    for (i = 1; i <= mat.size; i++) {      for (j = 1; j <= mat.size; j++) {         if (i >= j) {          Console.Write("{0} ",                        mat.A[i * (i - 1) / 2 + j - 1]);        }        else {          Console.Write("0 ");        }      }      Console.Write("\n");    }  }   // Function to generate an efficient matrix  static Matrix createMat(int [,]Mat)  {     // Declare efficient Matrix    Matrix mat = new Matrix();     // Initialize the Matrix    mat.size = N;    mat.A = new int[(mat.size*(mat.size + 1)) / 2];    int i, j;     // Set the values in matrix    for (i = 1; i <= mat.size; i++)    {      for (j = 1; j <= mat.size; j++)      {        Set(mat, i, j, Mat[i - 1,j - 1]);      }    }     // Return the matrix    return mat;  }   // Driver Code  public static void Main(String[] args)  {    int [,]Mat = { { 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 } };     // Stores the efficient matrix    Matrix mat = createMat(Mat);     // Print the Matrix    Display(mat);  }} // This code is contributed by 29AjayKumar
  
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) 
 
29AjayKumar
Mathematical
Matrix
Mathematical
Matrix
Writing code in comment?
Please use ide.geeksforgeeks.org,
generate link and share the link here.
Merge two sorted arrays
Modulo Operator (%) in C/C++ with Examples
Prime Numbers
Program to find GCD or HCF of two numbers
Print all possible combinations of r elements in a given array of size n
Program to find largest element in an array
Print a given matrix in spiral form
Rat in a Maze | Backtracking-2
Sudoku | Backtracking-7
Find the number of islands | Set 1 (Using DFS)