all subsequences Algorithm
The all subsequences algorithm is a powerful technique used in computer science and mathematics to generate all possible subsequences of a given sequence. Subsequences are an ordered list of elements derived from the original sequence by deleting some elements without changing the order of the remaining elements. The algorithm plays a crucial role in solving various problems in the fields of computational biology, pattern matching, and data mining, among others.
The all subsequences algorithm can be implemented using either an iterative or a recursive approach. In the iterative method, a loop is used to generate subsequences by considering each element of the given sequence as either included or excluded. This results in a total of 2^n subsequences for a sequence of length n (including the empty subsequence). The recursive approach, on the other hand, involves breaking down the sequence into smaller subsequences until a base case is reached (usually when the sequence is empty or has only one element). Then, the subsequences are combined back together to form the complete set of subsequences. Both methods are efficient in generating all subsequences and can be tailored to specific applications depending on the problem requirements.
"""
In this problem, we want to determine all possible subsequences
of the given sequence. We use backtracking to solve this problem.
Time complexity: O(2^n),
where n denotes the length of the given sequence.
"""
def generate_all_subsequences(sequence):
create_state_space_tree(sequence, [], 0)
def create_state_space_tree(sequence, current_subsequence, index):
"""
Creates a state space tree to iterate through each branch using DFS.
We know that each state has exactly two children.
It terminates when it reaches the end of the given sequence.
"""
if index == len(sequence):
print(current_subsequence)
return
create_state_space_tree(sequence, current_subsequence, index + 1)
current_subsequence.append(sequence[index])
create_state_space_tree(sequence, current_subsequence, index + 1)
current_subsequence.pop()
"""
remove the comment to take an input from the user
print("Enter the elements")
sequence = list(map(int, input().split()))
"""
sequence = [3, 1, 2, 4]
generate_all_subsequences(sequence)
sequence = ["A", "B", "C"]
generate_all_subsequences(sequence)
