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

Last Updated : 23 Jul, 2025

Prerequisite : Sparse Matrix and its representations Set 1 (Using Arrays and Linked Lists)
In this post other two methods of sparse matrix representation are discussed.

List of Lists
Dictionary

List of Lists (LIL)

One of the possible representation of sparse matrix is List of Lists (LIL). Where one list is used to represent the rows and each row contains the list of triples: Column index, Value(non - zero element) and address field, for non - zero elements. For the best performance both lists should be stored in order of ascending keys.

Sparse-Matrix-List-of-Lists

Implementation:

C++

// C++ program for Sparse Matrix Representation
// using List Of Lists
#include<bits/stdc++.h>
using namespace std;
#define R 4
#define C 5

// Node to represent row - list
struct row_list
{
    int row_number;
    struct row_list *link_down;
    struct value_list *link_right;
};

// Node to represent triples
struct value_list
{
    int column_index;
    int value;
    struct value_list *next;
};

// Function to create node for non - zero elements
void create_value_node(int data, int j, struct row_list **z)
{
    struct value_list *temp, *d;

    // Create new node dynamically
    temp = new value_list();
    temp->column_index = j+1;
    temp->value = data;
    temp->next = NULL;

    // Connect with row list
    if ((*z)->link_right==NULL)
        (*z)->link_right = temp;
    else
    {
        // d points to data list node
        d = (*z)->link_right;
        while(d->next != NULL)
            d = d->next;
        d->next = temp;
    }
}

// Function to create row list
void create_row_list(struct row_list **start, int row,
                    int column, int Sparse_Matrix[R][C])
{
    // For every row, node is created
    for (int i = 0; i < row; i++)
    {
        struct row_list *z, *r;

        // Create new node dynamically
        z = new row_list();
        z->row_number = i+1;
        z->link_down = NULL;
        z->link_right = NULL;
        if (i==0)
            *start = z;
        else
        {
            r = *start;
            while (r->link_down != NULL)
                r = r->link_down;
            r->link_down = z;
        }

        // Firstly node for row is created,
        // and then traversing is done in that row
        for (int j = 0; j < 5; j++)
        {
            if (Sparse_Matrix[i][j] != 0)
            {
                create_value_node(Sparse_Matrix[i][j], j, &z);
            }
        }
    }
}

//Function display data of LIL
void print_LIL(struct row_list *start)
{
    struct row_list *r;
    struct value_list *z;
    r = start;

    // Traversing row list
    while (r != NULL)
    {
        if (r->link_right != NULL)
        {
            cout<<"row="<<r->row_number<<endl;
            z = r->link_right;

            // Traversing data list
            while (z != NULL)
            {
                cout<<"column="<<z->column_index<<" value="<<z->value<<endl;
                z = z->next;
            }
        }
        r = r->link_down;
    }
}

//Driver of the program
int main()
{
    // Assume 4x5 sparse matrix
    int Sparse_Matrix[R][C] =
    {
        {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 of lists
    struct row_list* start = NULL;

    //Function creating List of Lists
    create_row_list(&start, R, C, Sparse_Matrix);

    // Display data of List of lists
    print_LIL(start);
    return 0;
}

// This code is contributed by rutvik_56.

// C program for Sparse Matrix Representation
// using List Of Lists
#include<stdio.h>
#include<stdlib.h>
#define R 4
#define C 5

// Node to represent row - list
struct row_list
{
    int row_number;
    struct row_list *link_down;
    struct value_list *link_right;
};

// Node to represent triples
struct value_list
{
    int column_index;
    int value;
    struct value_list *next;
};

// Function to create node for non - zero elements
void create_value_node(int data, int j, struct row_list **z)
{
    struct value_list *temp, *d;

    // Create new node dynamically
    temp = (struct value_list*)malloc(sizeof(struct value_list));
    temp->column_index = j+1;
    temp->value = data;
    temp->next = NULL;

    // Connect with row list
    if ((*z)->link_right==NULL)
        (*z)->link_right = temp;
    else
    {
        // d points to data list node
        d = (*z)->link_right;
        while(d->next != NULL)
            d = d->next;
        d->next = temp;
    }
}

// Function to create row list
void create_row_list(struct row_list **start, int row,
                    int column, int Sparse_Matrix[R][C])
{
    // For every row, node is created
    for (int i = 0; i < row; i++)
    {
        struct row_list *z, *r;

        // Create new node dynamically
        z = (struct row_list*)malloc(sizeof(struct row_list));
        z->row_number = i+1;
        z->link_down = NULL;
        z->link_right = NULL;
        if (i==0)
            *start = z;
        else
        {
            r = *start;
            while (r->link_down != NULL)
                r = r->link_down;
            r->link_down = z;
        }

        // Firstly node for row is created,
        // and then traversing is done in that row
        for (int j = 0; j < 5; j++)
        {
            if (Sparse_Matrix[i][j] != 0)
            {
                create_value_node(Sparse_Matrix[i][j], j, &z);
            }
        }
    }
}

//Function display data of LIL
void print_LIL(struct row_list *start)
{
    struct row_list *r;
    struct value_list *z;
    r = start;

    // Traversing row list
    while (r != NULL)
    {
        if (r->link_right != NULL)
        {
            printf("row=%d \n", r->row_number);
            z = r->link_right;

            // Traversing data list
            while (z != NULL)
            {
                printf("column=%d value=%d \n",
                     z->column_index, z->value);
                z = z->next;
            }
        }
        r = r->link_down;
    }
}

//Driver of the program
int main()
{
    // Assume 4x5 sparse matrix
    int Sparse_Matrix[R][C] =
    {
        {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 of lists
    struct row_list* start = NULL;

    //Function creating List of Lists
    create_row_list(&start, R, C, Sparse_Matrix);

    // Display data of List of lists
    print_LIL(start);
    return 0;
}

Java

// Java program for Sparse Matrix Representation
// using List Of Lists

class GFG
{
    static int R = 4;
    static int C = 5;
    
    // Node to represent row - list
    static class row_list
    {
        int row_number;
        row_list link_down;
        value_list link_right;
    };
    
    // Node to represent triples
    static class value_list
    {
        int column_index;
        int value;
        value_list next;
    };
    
    // Function to create node for non - zero elements
    static row_list create_value_node(int data, int j,  row_list z)
    {
        value_list temp, d;
    
        // Create new node dynamically
        temp = new value_list();
        temp.column_index = j+1;
        temp.value = data;
        temp.next = null;
    
        // Connect with row list
        if (z.link_right==null)
            z.link_right = temp;
        else
        {
            // d points to data list node
            d = z.link_right;
            while(d.next != null)
                d = d.next;
            d.next = temp;
        }
        return z;
    }
    
    // Function to create row list
    static row_list create_row_list(row_list start, int row,
                        int column, int Sparse_Matrix[][])
    {
        // For every row, node is created
        for (int i = 0; i < row; i++)
        {
            row_list z, r;
    
            // Create new node dynamically
            z = new row_list();
            z.row_number = i+1;
            z.link_down = null;
            z.link_right = null;
            if (i==0)
                start = z;
            else
            {
                r = start;
                while (r.link_down != null)
                    r = r.link_down;
                r.link_down = z;
            }
    
            // Firstly node for row is created,
            // and then traversing is done in that row
            for (int j = 0; j < 5; j++)
            {
                if (Sparse_Matrix[i][j] != 0)
                {
                    z = create_value_node(Sparse_Matrix[i][j], j, z);
                }
            }
        }
        return start;
    }
    
    //Function display data of LIL
    static void print_LIL(row_list start)
    {
        row_list r;
        value_list z;
        r = start;
    
        // Traversing row list
        while (r != null)
        {
            if (r.link_right != null)
            {
                System.out.println("row=" + r.row_number);
                z = r.link_right;
    
                // Traversing data list
                while (z != null)
                {
                    System.out.println("column="+z.column_index+" value="+z.value);
                    z = z.next;
                }
            }
            r = r.link_down;
        }
    }
    
    //Driver of the program
    public static void main(String[] args)
    {
        // Assume 4x5 sparse matrix
        int Sparse_Matrix[][] =
        {
            {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 of lists
        row_list start = null;
    
        //Function creating List of Lists
        start = create_row_list(start, R, C, Sparse_Matrix);
    
        // Display data of List of lists
        print_LIL(start);
    }
}


// This code is contributed by phasing17.

Python3

# Python3 program for Sparse Matrix Representation
# using List Of Lists
R = 4;
C = 5;

# Node to represent row - list
class row_list :
    
    def __init__(self):
    
        self.row_number = None;
        self.link_down = None;
        self.link_right = None;
    
# Node to represent triples
class value_list :
    def __init__(self):
    
        self.column_index = None;
        self.value = None;
        self.next = None;
    

# Function to create node for non - zero elements
def create_value_node(data, j, z):

    # Create node dynamically
    temp = value_list();
    temp.column_index = j + 1;
    temp.value = data;
    temp.next = None;

    # Connect with row list
    if (z.link_right == None):
        z.link_right = temp;
    else :
        # d points to data list node
        d = z.link_right;
        while (d.next != None):
            d = d.next;
        d.next = temp;
    
    return z;


# Function to create row list
def create_row_list(start, row, column, Sparse_Matrix):

    # For every row, node is created
    for i in range(row):


        # Create node dynamically
        z = row_list();
        z.row_number = i + 1;
        z.link_down = None;
        z.link_right = None;
        if (i == 0):
            start = z;
        else :
            r = start;
            while (r.link_down != None):
                r = r.link_down;
            r.link_down = z;
        

        # Firstly node for row is created,
        # and then traversing is done in that row
        for j in range(5):
            if (Sparse_Matrix[i][j] != 0) :
                z = create_value_node(Sparse_Matrix[i][j],j, z);
    
    return start;


# Function display data of LIL
def print_LIL(start):

    r = start;

    # Traversing row list
    while (r != None) :
        if (r.link_right != None) :
            print("row=", r.row_number);
            z = r.link_right;

            # Traversing data list
            while (z != None) :
                print("column=", z.column_index, " value=", z.value);
                z = z.next;
            
        
        r = r.link_down;
    

# Driver of the program

# Assume 4x5 sparse matrix
Sparse_Matrix = [[ 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 of lists
start = None;

# Function creating List of Lists
start = create_row_list(start, R, C, Sparse_Matrix);

# Display data of List of lists
print_LIL(start);

# This code is contributed by phasing17.

// C# program for Sparse Matrix Representation
// using List Of Lists
using System;

// Node to represent row - list
class row_list {
    public int row_number;
    public row_list link_down;
    public value_list link_right;
};

// Node to represent triples
class value_list {
    public int column_index;
    public int value;
    public value_list next;
};

class GFG {
    static int R = 4;
    static int C = 5;

    // Function to create node for non - zero elements
    static row_list create_value_node(int data, int j,
                                      row_list z)
    {
        value_list temp, d;

        // Create new node dynamically
        temp = new value_list();
        temp.column_index = j + 1;
        temp.value = data;
        temp.next = null;

        // Connect with row list
        if (z.link_right == null)
            z.link_right = temp;
        else {
            // d points to data list node
            d = z.link_right;
            while (d.next != null)
                d = d.next;
            d.next = temp;
        }
        return z;
    }

    // Function to create row list
    static row_list create_row_list(row_list start, int row,
                                    int column,
                                    int[, ] Sparse_Matrix)
    {
        // For every row, node is created
        for (int i = 0; i < row; i++) {
            row_list z, r;

            // Create new node dynamically
            z = new row_list();
            z.row_number = i + 1;
            z.link_down = null;
            z.link_right = null;
            if (i == 0)
                start = z;
            else {
                r = start;
                while (r.link_down != null)
                    r = r.link_down;
                r.link_down = z;
            }

            // Firstly node for row is created,
            // and then traversing is done in that row
            for (int j = 0; j < 5; j++) {
                if (Sparse_Matrix[i, j] != 0) {
                    z = create_value_node(
                        Sparse_Matrix[i, j], j, z);
                }
            }
        }
        return start;
    }

    // Function display data of LIL
    static void print_LIL(row_list start)
    {
        row_list r;
        value_list z;
        r = start;

        // Traversing row list
        while (r != null) {
            if (r.link_right != null) {
                Console.WriteLine("row=" + r.row_number);
                z = r.link_right;

                // Traversing data list
                while (z != null) {
                    Console.WriteLine(
                        "column=" + z.column_index
                        + " value=" + z.value);
                    z = z.next;
                }
            }
            r = r.link_down;
        }
    }

    // Driver of the program
    public static void Main(string[] args)
    {
        // Assume 4x5 sparse matrix
        int[, ] Sparse_Matrix = { { 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 of lists
        row_list start = null;

        // Function creating List of Lists
        start = create_row_list(start, R, C, Sparse_Matrix);

        // Display data of List of lists
        print_LIL(start);
    }
}

// This code is contributed by phasing17.

JavaScript

// JavaScript program for Sparse Matrix Representation
// using List Of Lists

let R = 4;
let C = 5;

// Node to represent row - list
class row_list {
    constructor()
    {
        this.row_number;
        this.link_down;
        this.link_right;
    }
};

// Node to represent triples
class value_list {
    constructor()
    {
        this.column_index;
        this.value;
        this.next;
    }
};

// Function to create node for non - zero elements
function create_value_node(data, j, z)
{
    let temp, d;

    // Create new node dynamically
    temp = new value_list();
    temp.column_index = j + 1;
    temp.value = data;
    temp.next = null;

    // Connect with row list
    if (z.link_right == null)
        z.link_right = temp;
    else {
        // d points to data list node
        d = z.link_right;
        while (d.next != null)
            d = d.next;
        d.next = temp;
    }
    return z;
}

// Function to create row list
function create_row_list(start, row, column, Sparse_Matrix)
{
    // For every row, node is created
    for (var i = 0; i < row; i++) {
        let z, r;

        // Create new node dynamically
        z = new row_list();
        z.row_number = i + 1;
        z.link_down = null;
        z.link_right = null;
        if (i == 0)
            start = z;
        else {
            r = start;
            while (r.link_down != null)
                r = r.link_down;
            r.link_down = z;
        }

        // Firstly node for row is created,
        // and then traversing is done in that row
        for (var j = 0; j < 5; j++) {
            if (Sparse_Matrix[i][j] != 0) {
                z = create_value_node(Sparse_Matrix[i][j],
                                      j, z);
            }
        }
    }
    return start;
}

// Function display data of LIL
function print_LIL(start)
{
    let r;
    let z;
    r = start;

    // Traversing row list
    while (r != null) {
        if (r.link_right != null) {
            console.log("row=" + r.row_number);
            z = r.link_right;

            // Traversing data list
            while (z != null) {
                console.log("column=" + z.column_index
                            + " value=" + z.value);
                z = z.next;
            }
        }
        r = r.link_down;
    }
}

// Driver of the program

// Assume 4x5 sparse matrix
let Sparse_Matrix = [
    [ 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 of lists
let start = null;

// Function creating List of Lists
start = create_row_list(start, R, C, Sparse_Matrix);

// Display data of List of lists
print_LIL(start);

// This code is contributed by phasing17.

Output

row=1
column=3 value=3
column=5 value=4
row=2
column=3 value=5
column=4 value=7
row=4
column=2 value=2
column=3 value=6

Time Complexity: O(RC × max(R, C))

Auxiliary Space: O(RC + max(R, C))

Dictionary of Keys

An alternative representation of sparse matrix is Dictionary. For the key field of the dictionary, pair of row and column index is used that maps with the non - zero element of the matrix. This method saves space but sequential access of items is costly.
In C++, dictionary is defined as map class of STL(Standard Template Library). To know more about map click the link below:
Basics of map

Implementation:

CPP

// C++ program for Sparse Matrix Representation
// using Dictionary
#include<bits/stdc++.h>
using namespace std;
#define R 4
#define C 5

// Driver of the program
int main()
{
    // Assume 4x5 sparse matrix
    int Sparse_Matrix[R][C] =
    {
        {0 , 0 , 3 , 0 , 4 },
        {0 , 0 , 5 , 7 , 0 },
        {0 , 0 , 0 , 0 , 0 },
        {0 , 2 , 6 , 0 , 0 }
    };

    /* Declaration of map where first field(pair of
       row and column) represent key and second
       field represent value */
    map< pair<int,int>, int > new_matrix;

    for (int i = 0; i < R; i++)
        for (int j = 0; j < C; j++)
            if (Sparse_Matrix[i][j] != 0)
                new_matrix[make_pair(i+1,j+1)] =
                                Sparse_Matrix[i][j] ;

    int c = 0;

    // Iteration over map
    for (auto i = new_matrix.begin(); i != new_matrix.end(); i++ )
    {
        if (c != i->first.first)
        {
            cout << "row = " << i->first.first << endl ;
            c = i->first.first;
        }
        cout << "column = " << i->first.second <<" ";
        cout << "value = " << i->second << endl;
    }

    return 0;
}

Java

// Java program for Sparse Matrix Representation
// using Dictionary
import java.util.*;
import java.util.concurrent.*;

class GFG
{
  static int R = 4;
  static int C = 5;

  // Driver of the program
  public static void main(String[] args)
  {
    // Assume 4x5 sparse matrix
    int[][] Sparse_Matrix =
    {
      {0 , 0 , 3 , 0 , 4 },
      {0 , 0 , 5 , 7 , 0 },
      {0 , 0 , 0 , 0 , 0 },
      {0 , 2 , 6 , 0 , 0 }
    };

    /* Declaration of map where first field(pair of
           row and column) represent key and second
           field represent value */
    TreeMap< List<Integer>, Integer>  new_matrix = new TreeMap< List<Integer>, Integer>(
      new Comparator<List<Integer>>() {
        public int compare(List<Integer> lst1, List<Integer> lst2) {
          if (lst1.get(0) < lst2.get(0))
            return -1;
          if (lst1.get(0) > lst2.get(0))
            return 1;
          if (lst1.get(1) < lst2.get(1))
            return -1;
          if (lst1.get(1) > lst2.get(1))
            return 1;
          return 0;
        }
      });;

    for (int i = 0; i < R; i++)
      for (int j = 0; j < C; j++)
        if (Sparse_Matrix[i][j] != 0)
          new_matrix.put(Collections.unmodifiableList(Arrays.asList(i + 1, j + 1)), Sparse_Matrix[i][j]);

    int c = 0;

    // Iteration over map

    for (var i : new_matrix.entrySet())
    {
      if (c != i.getKey().get(0))
      {
        System.out.println("row = " +  i.getKey().get(0) )  ;
        c =  i.getKey().get(0);
      }
      System.out.print( "column = " +  i.getKey().get(1) +" ");
      System.out.println("value = " + i.getValue());
    }

  }
}

// This code is contributed by phasing17.

Python3

# Python program for Sparse Matrix Representation
# using Dictionary
R = 4
C = 5

# Driver of the program

# Assume 4x5 sparse matrix
Sparse_Matrix=[[0 , 0 , 3 , 0 , 4] ,
    [0 , 0 , 5 , 7 , 0] ,
    [0 , 0 , 0 , 0 , 0] ,
    [0 , 2 , 6 , 0 , 0]] 


''' Declaration of map where first field(pair of
   row and column) represent key and second
   field represent value '''
new_matrix = {}

for i in range(R):
    for j in range(C):
        if (Sparse_Matrix[i][j] != 0):
            new_matrix[(i + 1, j + 1)] = Sparse_Matrix[i][j] 
c = 0

# Iteration over map
for i in new_matrix:
    if (c != i[0]):
        print("row =", i[0])
        c = i[0]
        
    print("column =", i[1], end = " ")
    print("value =", new_matrix[i])

# This code is contributed by Shubham Singh

// C# program for Sparse Matrix Representation
// using Dictionary
using System;
using System.Collections.Generic;

class GFG
{
  static int R = 4;
  static int C = 5;

  // Driver of the program
  public static void Main(string[] args)
  {
    // Assume 4x5 sparse matrix
    int[, ] Sparse_Matrix =
    {
      {0 , 0 , 3 , 0 , 4 },
      {0 , 0 , 5 , 7 , 0 },
      {0 , 0 , 0 , 0 , 0 },
      {0 , 2 , 6 , 0 , 0 }
    };

    /* Declaration of map where first field(pair of
           row and column) represent key and second
           field represent value */
    Dictionary< Tuple<int,int>, int > new_matrix = new Dictionary< Tuple<int,int>, int >();

    for (int i = 0; i < R; i++)
      for (int j = 0; j < C; j++)
        if (Sparse_Matrix[i, j] != 0)
          new_matrix[Tuple.Create(i+1,j+1)] =
          Sparse_Matrix[i,j] ;

    int c = 0;

    // Iteration over map
    foreach (var i in new_matrix)
    {
      if (c != i.Key.Item1)
      {
        Console.WriteLine("row = " + i.Key.Item1)  ;
        c = i.Key.Item1;
      }
      Console.Write( "column = " + i.Key.Item2 +" ");
      Console.WriteLine("value = " + i.Value);
    }

  }
}

// This code is contributed by phasing17.

JavaScript

// JS program for Sparse Matrix Representation
// using Dictionary
let R = 4
let C = 5

// Driver of the program

// Assume 4x5 sparse matrix
let Sparse_Matrix=[[0 , 0 , 3 , 0 , 4] ,
    [0 , 0 , 5 , 7 , 0] ,
    [0 , 0 , 0 , 0 , 0] ,
    [0 , 2 , 6 , 0 , 0]] 


// Declaration of map where first field(pair of
// row and column) represent key and second
// field represent value '''
let new_matrix = {}

for (var i = 0; i < R; i++)
    for (var j = 0; j < C; j++)
        if (Sparse_Matrix[i][j] != 0)
            new_matrix[ i + 1 + "#" + j + 1] = Sparse_Matrix[i][j] 
let c = 0

// Iteration over map
for ( let [key, val] of Object.entries(new_matrix))
{
    let i = key.split("#")
    if (c != i[0])
    {
        console.log("row =", i[0])
        c = i[0]
    }
        
    console.log("column = " + i[1] + " value = " + val)
}

// This code is contributed by phasing17

Output

row = 1
column = 3 value = 3
column = 5 value = 4
row = 2
column = 3 value = 5
column = 4 value = 7
row = 4
column = 2 value = 2
column = 3 value = 6

Time Complexity: O(R*C)
Auxiliary Space: O(R*C)

kartik

Improve

Article Tags :

Practice Tags :

Matrix

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

Similar Reads

Basics & Prerequisites

Data Structures

Algorithms

Advanced

Interview Preparation

Practice Problem

Thank You!

What kind of Experience do you want to share?