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++ program for the above approach #include <bits/stdc++.h> using namespace std;
// Dimensions of a matrix static int N = 5;
// Structure of the efficient matrix class Matrix {
public :
int * A;
int size;
}; // Function to set the // values in the Matrix 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 int Get(Matrix mat, int i, int j)
{ if (i >= j)
return mat.A[i * (i - 1) / 2 + j - 1];
return 0;
} // Function to display the // elements of the matrix 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)
cout << mat.A[i * (i - 1) / 2 + j - 1]
<< " " ;
else
cout << 0 << " " ;
}
cout << endl;
}
} // Function to generate an efficient matrix Matrix createMat(vector<vector< int > >& Mat)
{ // Declare efficient Matrix
Matrix mat;
// 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 int main()
{ vector<vector< 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);
return 0;
} // This code is contributed by Tapesh (tapeshdua420) |
// C program for the above approach #include <stdio.h> #include <stdlib.h> // Dimensions of a matrix const int N = 5;
// Structure of the efficient matrix struct Matrix {
int * A;
int size;
}; // Function to set the // values in the Matrix void 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 Matrix int 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 matrix void 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 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 ));
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 int 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 approach class GFG
{ // Dimensions of a matrix static int N = 5 ;
// Structure of the efficient matrix static class Matrix {
int [] A;
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) {
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 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 |
# Python program for the above approach # Dimensions of a matrix N = 5
# Structure of the efficient matrix class Matrix:
def __init__( self , size):
self .size = size
self .A = [ None ] * ( self .size)
# Function to set the # values in the Matrix def Set (mat, i, j, x):
if i > = j:
mat.A[i * (i - 1 ) / / 2 + j - 1 ] = x
# Function to store the # values in the Matrix def get(mat, i, j):
if i > = j:
return mat.A[i * (i - 1 ) / / 2 + j - 1 ]
return 0
# Function to display the # elements of the matrix def display(mat):
# Traverse the matrix
for i in range ( 1 , mat.size + 1 ):
for j in range ( 1 , mat.size + 1 ):
if i > = j:
print (mat.A[i * (i - 1 ) / / 2 + j - 1 ], end = " " )
else :
print ( 0 , end = " " )
print ()
# Function to generate an efficient matrix def create_matrix(Mat):
# Declare efficient Matrix
mat = Matrix(N)
# Initialize the Matrix
mat.A = [ None ] * ((mat.size * (mat.size + 1 )) / / 2 )
# Set the values in matrix
for i in range ( 1 , mat.size + 1 ):
for j in range ( 1 , mat.size + 1 ):
Set (mat, i, j, Mat[i - 1 ][j - 1 ])
# Return the matrix
return mat
if __name__ = = '__main__' :
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 ]]
mat = create_matrix(Mat)
display(mat)
# This code is contributed by Tapesh (tapeshdua420) |
// C# program for the above approach using 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 |
// JavaScript program for the above approach let N = 5; class Matrix{ A = new Array();
size;
constructor(){ }
} function Set(mat, i, j, x){
if (i >= j){
mat.A[i * (i - 1) / 2 + j - 1] = x;
}
} function Get(mat, i, j){
if (i >= j) {
return mat.A[i * (i - 1) / 2 + j - 1];
}
else {
return 0;
}
} function Display(mat){
let i, j;
// Traverse the matrix
for (i = 1; i <= mat.size; i++) {
for (j = 1; j <= mat.size; j++) {
if (i >= j) {
console.log(mat.A[i * (i - 1) / 2 + j - 1] + " " );
}
else {
console.log( "0 " );
}
}
console.log( "<br>" );
}
} function createMat(Mat){
var mat = new Matrix();
mat.size = N;
mat.A = new Array((mat.size*(mat.size + 1))/2);
let 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;
} let 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] ];
var mat = createMat(Mat);
Display(mat); // This code is contributed by lokesh. |
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)