Print all subsequences of a string in Python

Given a string s of size n (1 ≤ n ≤ 20), the task is to print all subsequences of string. A subsequence is a sequence that can be derived from another sequence by deleting some or no elements without changing the order of the remaining elements.

Examples: 

Input: s = "abc"
Output: ["", "a", "b", "c", "ab", "ac", "bc", "abc"]
Explanation: All possible combinations of the characters in the string are returned, including the empty string.

Input: S = "ab"
Output: ["", "a", "b", "ab"]
Explanation: All subsequences of the string ab are returned in any order.

Using Recursion

The recursive approach works by considering two options for each character:

• Include the character in the current subsequence.
• Exclude the character from the current subsequence.

By recursively generating subsequences for all characters, we can obtain the complete set of subsequences.

# Recursive function to generate subsequences.
def subseq_rec(idx, curr, s, res):
  
    # Base case: If we've processed all characters,
    # add current subsequence
    if idx == len(s):
        res.append(curr)
        return
      
    # Include current character in subsequence
    subseq_rec(idx + 1, curr + s[idx], s, res)
    
    # Exclude current character from subsequence
    subseq_rec(idx + 1, curr, s, res)


# Wrapper function to generate all subsequences recursively.
def all_subseq_rec(s):
   
    # List to store results
    res = []  
    
    # Start recursion from index 0
    subseq_rec(0, "", s, res)  
    return res

s = "abc"
print(all_subseq_rec(s))

['abc', 'ab', 'ac', 'a', 'bc', 'b', 'c', '']

Time Complexity: O(n * 2^n), where n is length of the string. At each step we make two recursive calls resulting in 2^n subsequences.
Auxiliary Space: O(2^n) for storing all subsequences and O(n) recursion stack.

Using Iterative Bit Masking

The iterative approach uses bit masking to generate all subsequences. Each bit in a number represents whether a character is included in the subsequence.

How it Works:

• For a string of length n there are 2^n subsets.
• Each number from 0 to 2^n - 1 represents a subset, where binary representation of number determines which characters are included in the subsequence.
• If the j-th bit is 1 then include the j-th character of string in subsequence.

# Generate subsequences using bit masking.
def all_subseq(s):

    n = len(s)  
    
    # List to store results
    res = []  
    
    # Loop over all possible masks from 0 to 2^n - 1
    for mask in range(1 << n):
      	
        # Current subsequence being formed
        curr = ""  
        
        # Check each bit in mask
        for i in range(n):
          
            # If i-th bit is set, include s[i]
            if mask & (1 << i):  
                curr += s[i]
                
        # Add current subsequence to result list
        res.append(curr)
    return res

s = "abc"
print(all_subseq(s))

['', 'a', 'b', 'ab', 'c', 'ac', 'bc', 'abc']

Time Complexity: O(n * 2^n), where 2^n is the number of subsequences and n is the maximum length of each subsequence.
Auxiliary Space: O(2^n), as all subsequences are stored in a list.

Using Python's Built-in Libraries (Combinations)

Using itertools.combinations library we can generate all subsequences of lengths 0 to n.

from itertools import combinations

# Generate subsequences using combinations from itertools.
def all_subseq(s):
	
    # Start with the empty subsequence
    res = [""]  
    
    # Generate combinations of lengths 1 to n
    for r in range(1, len(s) + 1):
      
        # Add all combinations of length r to result
        res.extend([''.join(comb) for comb in combinations(s, r)])
    return res

s = "abc"
print(all_subseq(s))

['', 'a', 'b', 'c', 'ab', 'ac', 'bc', 'abc']

Time Complexity: O(n * 2^n), as it generates all subsets.
Auxiliary Space: O(2^n), as all subsequences are stored.

kartik

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Article Tags :

• Misc
• Python
• subsequence
• Python list-programs

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