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Subarray/Substring vs Subsequence and Programs to Generate them

Subarray/Substring

A subbarray is a contiguous part of array. An array that is inside another array. For example, consider the array [1, 2, 3, 4], There are 10 non-empty sub-arrays. The subbarays are (1), (2), (3), (4), (1,2), (2,3), (3,4), (1,2,3), (2,3,4) and (1,2,3,4). In general, for an array/string of size n, there are n*(n+1)/2 non-empty subarrays/subsrings.

subseq-vs-subarray



How to generate all subarrays?
We can run two nested loops, the outer loop picks starting element and inner loop considers all elements on right of the picked elements as ending element of subarray.

C++

/*  C++ code to generate all possible subarrays/subArrays
    Complexity- O(n^3) */
#include<bits/stdc++.h>
using namespace std;
  
// Prints all subarrays in arr[0..n-1]
void subArray(int arr[], int n)
{
    // Pick starting point
    for (int i=0; i <n; i++)
    {
        // Pick ending point
        for (int j=i; j<n; j++)
        {
            // Print subarray between current starting
            // and ending points
            for (int k=i; k<=j; k++)
                cout << arr[k] << " ";
  
            cout << endl;
        }
    }
}
  
// Driver program
int main()
{
    int arr[] = {1, 2, 3, 4};
    int n = sizeof(arr)/sizeof(arr[0]);
    cout << "All Non-empty Subarrays ";
    subArray(arr, n);
    return 0;
}

Java

// Java program toto generate all possible subarrays/subArrays
//  Complexity- O(n^3) */
  
class Test
{
    static int arr[] = new int[]{1, 2, 3, 4};
      
    // Prints all subarrays in arr[0..n-1]
    static void subArray( int n)
    {
        // Pick starting point
        for (int i=0; i <n; i++)
        {
            // Pick ending point
            for (int j=i; j<n; j++)
            {
                // Print subarray between current starting
                // and ending points
                for (int k=i; k<=j; k++)
                    System.out.print(arr[k]+" ");
            }
        }
    }
      
    // Driver method to test the above function
    public static void main(String[] args) 
    {
        System.out.println("All Non-empty Subarrays");
        subArray(arr.length);
          
    }
}

Python3

# Python3 code to generate all possible
# subarrays/subArrays
# Complexity- O(n^3) 
  
# Prints all subarrays in arr[0..n-1]
def subArray(arr, n):
  
    # Pick starting point
    for i in range(0,n):
  
        # Pick ending point
        for j in range(i,n):
  
            # Print subarray between
            # current starting
            # and ending points
            for k in range(i,j+1):
                print (arr[k],end=" ")
  
            print (" ",end="")
  
              
# Driver program
arr = [1, 2, 3, 4]
n = len(arr)
print ("All Non-empty Subarrays")
  
subArray(arr, n);
  
# This code is contributed by Shreyanshi.

C#

// C# program toto generate all
// possible subarrays/subArrays
// Complexity- O(n^3)
using System;
  
class GFG
    static int []arr = new int[]{1, 2, 3, 4};
      
    // Prints all subarrays in arr[0..n-1]
    static void subArray( int n)
    {
          
        // Pick starting point
        for (int i = 0; i < n; i++)
        {
              
            // Pick ending point
            for (int j = i; j < n; j++)
            {
                  
                // Print subarray between current 
                // starting and ending points
                for (int k = i; k <= j; k++)
                    Console.Write(arr[k]+" ");
                    Console.WriteLine("");
            }
        }
    }
      
    // Driver Code
    public static void Main() 
    {
        Console.WriteLine("All Non-empty Subarrays");
        subArray(arr.Length);
          
    }
  
}
  
// This code is contributed by Sam007.

PHP

<?php
// PHP code to generate all possible
// subarrays/subArrays Complexity- O(n^3) 
  
// Prints all subarrays 
// in arr[0..n-1]
function subArray($arr, $n)
{
      
    // Pick starting point
    for ($i = 0; $i < $n; $i++)
    {
          
        // Pick ending point
        for ($j = $i; $j < $n; $j++)
        {
              
            // Print subarray between 
            // current starting
            // and ending points
            for ($k = $i; $k <= $j; $k++)
                echo $arr[$k] , " ";
  
            echo " ";
        }
    }
}
  
    // Driver Code
    $arr= array(1, 2, 3, 4);
    $n = sizeof($arr);
    echo "All Non-empty Subarrays ";
    subArray($arr, $n);
      
// This ocde is contributed by m_kit
?>

Output:

All Non-empty Subarrays
1 
1 2 
1 2 3 
1 2 3 4 
2 
2 3 
2 3 4 
3 
3 4 
4 

 
Subsequence
A subsequence is a sequence that can be derived from another sequence by zero or more elements, without changing the order of the remaining elements.
For the same example, there are 15 sub-sequences. They are (1), (2), (3), (4), (1,2), (1,3),(1,4), (2,3), (2,4), (3,4), (1,2,3), (1,2,4), (1,3,4), (2,3,4), (1,2,3,4). More generally, we can say that for a sequence of size n, we can have (2n-1) non-empty sub-sequences in total.

A string example to differentiate:
Consider strings “geeksforgeeks” and “gks”. “gks” is a subsequence of “geeksforgeeks” but not a substring. “geeks” is both a subsequence and subarray. Every subarray is a subsequence. More specifically, Subsequence is a generalization of substring.

How to generate all Subsequences?
We can use algorithm to generate power set for generation of all subsequences.

C++

/*  C++ code to generate all possible subsequences.
    Time Complexity O(n * 2^n) */
#include<bits/stdc++.h>
using namespace std;
  
void printSubsequences(int arr[], int n)
{
    /* Number of subsequences is (2**n -1)*/
    unsigned int opsize = pow(2, n);
  
    /* Run from counter 000..1 to 111..1*/
    for (int counter = 1; counter < opsize; counter++)
    {
        for (int j = 0; j < n; j++)
        {
            /* Check if jth bit in the counter is set
                If set then print jth element from arr[] */
            if (counter & (1<<j))
                cout << arr[j] << " ";
        }
        cout << endl;
    }
}
  
// Driver program
int main()
{
    int arr[] = {1, 2, 3, 4};
    int n = sizeof(arr)/sizeof(arr[0]);
    cout << "All Non-empty Subsequences ";
    printSubsequences(arr, n);
    return 0;
}

Java

/*  Java code to generate all possible subsequences.
    Time Complexity O(n * 2^n) */
  
import java.math.BigInteger;
  
class Test
{
    static int arr[] = new int[]{1, 2, 3, 4};
      
    static void printSubsequences(int n)
    {
        /* Number of subsequences is (2**n -1)*/
        int opsize = (int)Math.pow(2, n);
       
        /* Run from counter 000..1 to 111..1*/
        for (int counter = 1; counter < opsize; counter++)
        {
            for (int j = 0; j < n; j++)
            {
                /* Check if jth bit in the counter is set
                    If set then print jth element from arr[] */
        
                if (BigInteger.valueOf(counter).testBit(j))
                    System.out.print(arr[j]+" ");
            }
            System.out.println();
        }
    }
      
    // Driver method to test the above function
    public static void main(String[] args) 
    {
        System.out.println("All Non-empty Subsequences");
        printSubsequences(arr.length);
    }
}

Python3

# Python 3 code to generate all
# possible subsequences.
# Time Complexity O(n * 2 ^ n) 
import math
  
def printSubsequences(arr, n) :
  
    # Number of subsequences is (2**n -1)
    opsize = math.pow(2, n)
  
    # Run from counter 000..1 to 111..1
    for counter in range( 1, (int)(opsize)) :
        for j in range(0, n) :
              
            # Check if jth bit in the counter
            # is set If set then print jth 
            # element from arr[] 
            if (counter & (1<<j)) :
                print( arr[j], end =" ")
          
        print()
  
# Driver program
arr = [1, 2, 3, 4]
n = len(arr)
print( "All Non-empty Subsequences")
  
printSubsequences(arr, n)
  
# This code is contributed by Nikita Tiwari.

PHP

<?php
// PHP code to generate all 
// possible subsequences.
// Time Complexity O(n * 2^n) 
  
function printSubsequences($arr, $n)
{
    // Number of subsequences 
    // is (2**n -1)
    $opsize = pow(2, $n);
  
    /* Run from counter
    000..1 to 111..1*/
    for ($counter = 1; 
         $counter < $opsize
         $counter++)
    {
        for ( $j = 0; $j < $n; $j++)
        {
             /* Check if jth bit in 
                the counter is set
                If set then print jth
                element from arr[] */
            if ($counter & (1 << $j))
                echo $arr[$j], " ";
        }
        echo " ";
    }
}
  
// Driver Code
$arr = array (1, 2, 3, 4);
$n = sizeof($arr);
  
echo "All Non-empty Subsequences ";
  
printSubsequences($arr, $n);
      
// This code is contributed by ajit
?>

Output:

All Non-empty Subsequences
1 
2 
1 2 
3 
1 3 
2 3 
1 2 3 
4 
1 4 
2 4 
1 2 4 
3 4 
1 3 4 
2 3 4 
1 2 3 4

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This article is attributed to GeeksforGeeks.org

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