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Sparse Matrix and its representations | Set 1 (Using Arrays and Linked Lists)
  • Difficulty Level : Hard
  • Last Updated : 25 Mar, 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: 

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

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: 



  1. Array representation
  2. 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 Link list
#include<iostream>
using namespace std;
 
// Node class to represent link list
class Node
{
    public:
    int row;
    int col;
    int data;
    Node *next;
};
 
// Function to create new node
void create_new_node(Node **p, int row_index,
                     int col_index, int x)
{
    Node *temp = *p;
    Node *r;
     
    // If link list is empty then
    // create first node and assign value.
    if (temp == NULL)
    {
        temp = new Node();
        temp->row = row_index;
        temp->col = col_index;
        temp->data = x;
        temp->next = NULL;
        *p = temp;
    }
     
    // If link list is already created
    // then append newly created node
    else
    {
        while (temp->next != NULL)  
            temp = temp->next;
             
        r = new Node();
        r->row = row_index;
        r->col = col_index;
        r->data = x;
        r->next = NULL;
        temp->next = r;
    }
}
 
// Function prints contents of linked list
// starting from start
void printList(Node *start)
{
    Node *ptr = start;
    cout << "row_position:";
    while (ptr != NULL)
    {
        cout << ptr->row << " ";
        ptr = ptr->next;
    }
    cout << endl;
    cout << "column_position:";
 
    ptr = start;
    while (ptr != NULL)
    {
        cout << ptr->col << " ";
        ptr = ptr->next;
    }
    cout << endl;
    cout << "Value:";
    ptr = start;
     
    while (ptr != NULL)
    {
        cout << ptr->data << " ";
        ptr = ptr->next;
    }
}
 
// Driver Code
int main()
{
     
    // 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 } };
     
    // Creating head/first node of list as NULL
    Node *first = NULL;
    for(int i = 0; i < 4; i++)
    {
        for(int j = 0; j < 5; j++)
        {
             
            // Pass only those values which
            // are non - zero
            if (sparseMatrix[i][j] != 0)
                create_new_node(&first, i, j,
                                sparseMatrix[i][j]);
        }
    }
    printList(first);
 
    return 0;
}
 
// This code is contributed by ronaksuba

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)

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.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
 

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