Sparse Matrix and its representations | Set 1 (Using Arrays and Linked Lists)

A matrix is a two-dimensional data object made of m rows and n columns, therefore having total m x n values. If most of the elements of the matrix have 0 value, then it is called a sparse matrix.

Why to use Sparse Matrix instead of simple matrix ?

Storage: There are lesser non-zero elements than zeros and thus lesser memory can be used to store only those elements.
Computing time: Computing time can be saved by logically designing a data structure traversing only non-zero elements..

Example:

Representing a sparse matrix by a 2D array leads to wastage of lots of memory as zeroes in the matrix are of no use in most of the cases. So, instead of storing zeroes with non-zero elements, we only store non-zero elements. This means storing non-zero elements with triples- (Row, Column, value).

Sparse Matrix Representations can be done in many ways following are two common representations:

Array representation
Linked list representation

Method 1: Using Arrays

2D array is used to represent a sparse matrix in which there are three rows named as

Row: Index of row, where non-zero element is located
Column: Index of column, where non-zero element is located
Value: Value of the non zero element located at index – (row,column)

// C++ program for Sparse Matrix Representation 
// using Array 
#include<stdio.h> 
  
int main() 
{ 
    // Assume 4x5 sparse matrix 
    int sparseMatrix[4][5] = 
    { 
        {0 , 0 , 3 , 0 , 4 }, 
        {0 , 0 , 5 , 7 , 0 }, 
        {0 , 0 , 0 , 0 , 0 }, 
        {0 , 2 , 6 , 0 , 0 } 
    }; 
  
    int size = 0; 
    for (int i = 0; i < 4; i++) 
        for (int j = 0; j < 5; j++) 
            if (sparseMatrix[i][j] != 0) 
                size++; 
  
    // number of columns in compactMatrix (size) must be 
    // equal to number of non - zero elements in 
    // sparseMatrix 
    int compactMatrix[3][size]; 
  
    // Making of new matrix 
    int k = 0; 
    for (int i = 0; i < 4; i++) 
        for (int j = 0; j < 5; j++) 
            if (sparseMatrix[i][j] != 0) 
            { 
                compactMatrix[0][k] = i; 
                compactMatrix[1][k] = j; 
                compactMatrix[2][k] = sparseMatrix[i][j]; 
                k++; 
            } 
  
    for (int i=0; i<3; i++) 
    { 
        for (int j=0; j<size; j++) 
            printf("%d ", compactMatrix[i][j]); 
  
        printf("\n"); 
    } 
    return 0; 
} 

Output:


0 0 1 1 3 3 
2 4 2 3 1 2 
3 4 5 7 2 6

Method 2: Using Linked Lists

In linked list, each node has four fields. These four fields are defined as:

Row: Index of row, where non-zero element is located
Column: Index of column, where non-zero element is located
Value: Value of the non zero element located at index – (row,column)
Next node: Address of the next node

// C program for Sparse Matrix Representation 
// using Linked Lists 
#include<stdio.h> 
#include<stdlib.h> 
  
// Node to represent sparse matrix 
struct Node 
{ 
    int value; 
    int row_position; 
    int column_postion; 
    struct Node *next; 
}; 
  
// Function to create new node 
void create_new_node(struct Node** start, int non_zero_element, 
                     int row_index, int column_index ) 
{ 
    struct Node *temp, *r; 
    temp = *start; 
    if (temp == NULL) 
    { 
        // Create new node dynamically 
        temp = (struct Node *) malloc (sizeof(struct Node)); 
        temp->value = non_zero_element; 
        temp->row_position = row_index; 
        temp->column_postion = column_index; 
        temp->next = NULL; 
        *start = temp; 
  
    } 
    else
    { 
        while (temp->next != NULL) 
            temp = temp->next; 
  
        // Create new node dynamically 
        r = (struct Node *) malloc (sizeof(struct Node)); 
        r->value = non_zero_element; 
        r->row_position = row_index; 
        r->column_postion = column_index; 
        r->next = NULL; 
        temp->next = r; 
  
    } 
} 
  
// This function prints contents of linked list 
// starting from start 
void PrintList(struct Node* start) 
{ 
    struct Node *temp, *r, *s; 
    temp = r = s = start; 
  
    printf("row_position: "); 
    while(temp != NULL) 
    { 
  
        printf("%d ", temp->row_position); 
        temp = temp->next; 
    } 
    printf("\n"); 
  
    printf("column_postion: "); 
    while(r != NULL) 
    { 
        printf("%d ", r->column_postion); 
        r = r->next; 
    } 
    printf("\n"); 
    printf("Value: "); 
    while(s != NULL) 
    { 
        printf("%d ", s->value); 
        s = s->next; 
    } 
    printf("\n"); 
} 
  
  
// Driver of the program 
int main() 
{ 
   // Assume 4x5 sparse matrix 
    int sparseMatric[4][5] = 
    { 
        {0 , 0 , 3 , 0 , 4 }, 
        {0 , 0 , 5 , 7 , 0 }, 
        {0 , 0 , 0 , 0 , 0 }, 
        {0 , 2 , 6 , 0 , 0 } 
    }; 
  
    /* Start with the empty list */
    struct Node* start = NULL; 
  
    for (int i = 0; i < 4; i++) 
        for (int j = 0; j < 5; j++) 
  
            // Pass only those values which are non - zero 
            if (sparseMatric[i][j] != 0) 
                create_new_node(&start, sparseMatric[i][j], i, j); 
  
    PrintList(start); 
  
    return 0; 
} 

Output:

row_position: 0 0 1 1 3 3 
column_postion: 2 4 2 3 1 2 
Value: 3 4 5 7 2 6

Other representations:
As a Dictionary where row and column numbers are used as keys and values are matrix entries. This method saves space but sequential access of items is costly.

As a list of list. The idea is to make a list of rows and every item of list contains values. We can keep list items sorted by column numbers.

Sparse Matrix and its representations | Set 2 (Using List of Lists and Dictionary of keys)

This article is contributed by Akash Gupta.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.

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