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:
- 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++
// 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
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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