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

Difficulty Level : Hard
Last Updated : 15 Feb, 2021

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)

Sparse Matrix Array Representation

C++

// 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;
}

Java

// Java program for Sparse Matrix Representation
// using Array
class GFG 
{
 
    public static void main(String[] args)
    {
        int sparseMatrix[][]
                = {
                    {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[][] = new int[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++) 
            {
                System.out.printf("%d ", compactMatrix[i][j]);
            }
            System.out.printf("\n");
        }
    }
} 
 
/* This code contributed by PrinciRaj1992 */

Python3

# Python program for Sparse Matrix Representation
# using arrays
 
# assume a sparse matrix of order 4*5 
# let assume another matrix compactMatrix 
# now store the value,row,column of arr1 in sparse matrix compactMatrix
 
sparseMatrix = [[0,0,3,0,4],[0,0,5,7,0],[0,0,0,0,0],[0,2,6,0,0]]
 
# initialize size as 0
size = 0
 
for i in range(4):
    for j in range(5):
        if (sparseMatrix[i][j] != 0):
            size += 1
 
# number of columns in compactMatrix(size) should
# be equal to number of non-zero elements in sparseMatrix
rows, cols = (3, size)
compactMatrix = [[0 for i in range(cols)] for j in range(rows)]
 
k = 0
for i in range(4):
    for j in range(5):
        if (sparseMatrix[i][j] != 0):
            compactMatrix[0][k] = i
            compactMatrix[1][k] = j
            compactMatrix[2][k] = sparseMatrix[i][j]
            k += 1
 
for i in compactMatrix:
    print(i)
 
# This code is contributed by MRINALWALIA

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

Sparse-Matrix-Linked-List 2

C

// 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;
}

Python3

# Python Program for Representation of
# Sparse Matrix into Linked List
 
# Node Class to represent Linked List Node
class Node:
 
    # Making the slots for storing row,
    # column, value, and address
    __slots__ = "row", "col", "data", "next"
 
    # Constructor to initialize the values
    def __init__(self, row=0, col=0, data=0, next=None):
 
        self.row = row
        self.col = col
        self.data = data
        self.next = next
 
 
# Class to convert Sparse Matrix
# into Linked List
class Sparse:
 
    # Initialize Class Variables
    def __init__(self):
        self.head = None
        self.temp = None
        self.size = 0
 
    # Function which returns the size
    # of the Linked List
    def __len__(self):
        return self.size
 
    # Check the Linked List is
    # Empty or not
    def isempty(self):
        return self.size == 0
 
    # Responsible function to create
    # Linked List from Sparse Matrix
    def create_new_node(self, row, col, data):
 
        # Creating New Node
        newNode = Node(row, col, data, None)
 
        # Check whether the List is
        # empty or not
        if self.isempty():
            self.head = newNode
        else:
            self.temp.next = newNode
        self.temp = newNode
 
        # Incrementing the size
        self.size += 1
 
    # Function display the contents of
    # Linked List
    def PrintList(self):
        temp = r = s = self.head
        print("row_position:", end=" ")
        while temp != None:
            print(temp.row, end=" ")
            temp = temp.next
        print()
        print("column_postion:", end=" ")
        while r != None:
            print(r.col, end=" ")
            r = r.next
        print()
        print("Value:", end=" ")
        while s != None:
            print(s.data, end=" ")
            s = s.next
        print()
 
# Driver Code
if __name__ == "__main__":
 
    # Creating Object
    s = Sparse()
 
    # Assuming 4x5 Sparse Matrix
    sparseMatric = [[0, 0, 3, 0, 4],
                    [0, 0, 5, 7, 0],
                    [0, 0, 0, 0, 0],
                    [0, 2, 6, 0, 0]]
    for i in range(4):
        for j in range(5):
 
            # Creating Linked List by only those
            # elements which are non-zero
            if sparseMatric[i][j] != 0:
                s.create_new_node(i, j, sparseMatric[i][j])
 
    # Printing the Linked List Representation
    # of the sparse matrix
    s.PrintList()
 
    # This code is contributed by Naveen Rathore

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)
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