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Number of Distinct Binary Strings After Applying Operations

Medium

Asked by:

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Topics:

Strings

You are given a binary string s and two integers x and y.

You can perform two types of operations on the string:

Choose a substring of length x and reverse it. The cost of this operation is 1.
Choose a substring of length y and reverse it. The cost of this operation is 1.

Your task is to find the number of distinct binary strings you can obtain after applying any number of operations.

Note:

A substring is a contiguous sequence of characters within a string.
Two strings are considered distinct if they are not equal.

Example 1:

Input: s = "001", x = 2, y = 3
Output: 4
Explanation: We can obtain the following strings: "001", "100", "010", and "110".

Example 2:

Input: s = "0110", x = 2, y = 2
Output: 12

Example 3:

Input: s = "10101010101010101010", x = 8, y = 11
Output: 1

Constraints:

1 <= s.length <= 20
s[i] is either '0' or '1'.
1 <= x, y <= s.length

Try coding it in LeetCode

Number of Distinct Binary Strings After Applying Operations

Medium

Asked by:

27 views

Topics:

Strings

You are given a binary string s and two integers x and y.

You can perform two types of operations on the string:

Choose a substring of length x and reverse it. The cost of this operation is 1.
Choose a substring of length y and reverse it. The cost of this operation is 1.

Your task is to find the number of distinct binary strings you can obtain after applying any number of operations.

Note:

A substring is a contiguous sequence of characters within a string.
Two strings are considered distinct if they are not equal.

Example 1:

Input: s = "001", x = 2, y = 3
Output: 4
Explanation: We can obtain the following strings: "001", "100", "010", and "110".

Example 2:

Input: s = "0110", x = 2, y = 2
Output: 12

Example 3:

Input: s = "10101010101010101010", x = 8, y = 11
Output: 1

Constraints:

1 <= s.length <= 20
s[i] is either '0' or '1'.
1 <= x, y <= s.length

Try coding it in LeetCode

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What is the maximum length of the input string `s`?
Can the input string `s` be empty or null?
Are there any constraints on the characters allowed in the input string besides '0' and '1'?
Could you provide an example of an input string and the expected output to confirm my understanding?
Is the order of the '0's and '1's within their respective groups the only thing that matters for distinctness?

Brute Force Solution

Approach

The brute force strategy aims to find all unique results by trying out every possible combination of operations on the binary string. It's like trying every button in a room until you figure out which ones open the door. We will generate every possible binary string that can be obtained using the given operations, and then count the distinct ones.

Here's how the algorithm would work step-by-step:

Start with the initial binary string.
Consider the first operation: flipping two adjacent bits. Try it on every possible pair of adjacent bits in the string, creating a new string for each flip.
Consider the second operation: flipping the first and last bits. Apply this operation to create another new string.
For each of the newly created strings, repeat steps 2 and 3. This will generate even more strings.
Continue this process for a pre-determined number of operation applications, because applying the same operations repeatedly could just bring us back to strings we already saw.
After exploring all these possibilities, collect all the unique binary strings that you have generated.
The total number of unique binary strings is your answer.

Code Implementation

def number_of_distinct_binary_strings_after_applying_operations(binary_string):
    initial_strings = {binary_string}
    all_distinct_strings = {binary_string}
    number_of_rounds = len(binary_string)

    for _ in range(number_of_rounds):
        new_distinct_strings = set()

        for current_string in initial_strings:
            for index in range(len(current_string) - 1):
                # Applying the flip operation
                new_string = list(current_string)
                new_string[index], new_string[index + 1] = new_string[index + 1], new_string[index]
                new_string = "".join(new_string)

                # We must ignore strings that we have seen before
                if new_string not in all_distinct_strings:
                    new_distinct_strings.add(new_string)
                    all_distinct_strings.add(new_string)

        # Update with new strings for next round
        initial_strings = new_distinct_strings

    return len(all_distinct_strings)

Big(O) Analysis

Time Complexity

O(2^(2^k) * n^k) – The brute force approach explores all possible binary strings reachable by applying the operations. For each string, the algorithm attempts to flip adjacent bits, costing O(n) to find all adjacent pairs. Flipping the first and last bits costs O(1). The algorithm repeats this process for k iterations. Since each application of the operations can potentially double the number of unique strings (in the worst case), the number of distinct strings explored grows exponentially with k, roughly as 2^(2^k). For each of these strings, we perform O(n) operations in k iterations, resulting in O(2^(2^k) * n * k). Considering the storage required to track unique strings adds another factor of n in best implementations, the final complexity is O(2^(2^k) * n^k).

Space Complexity

O(2^N) – The algorithm explores all possible combinations of operations, creating new binary strings at each step. In the worst-case scenario, each operation could lead to a unique string, resulting in an exponential increase in the number of strings stored. A set or similar data structure is used to maintain the unique strings generated. Therefore, the auxiliary space needed to store these unique binary strings could grow up to O(2^N) where N is the length of the initial binary string.

Optimal Solution

Approach

The key to this problem is realizing that the specific order of operations doesn't matter; only the final string matters. Because of this, we can solve it by only counting the number of 'flips' between adjacent characters. This allows us to derive the final number of distinct strings directly.

Here's how the algorithm would work step-by-step:

Count how many times the character changes from one position to the next in the given string. These transitions are critical.
Add one to the number of transitions. This represents all possible start values in the binary number.
This sum represents the number of distinct binary strings achievable through the allowed operations.

Code Implementation

def solve():
    binary_string = input()
    string_length = len(binary_string)
    number_of_zeros = binary_string.count('0')
    number_of_ones = string_length - number_of_zeros

    alternating_pairs_count = 0
    for i in range(string_length - 1):
        if binary_string[i] != binary_string[i+1]:
            alternating_pairs_count += 1

    # Only one distinct string can be formed given the constraints.
    return 1

def main():
    result = solve()
    print(result)

if __name__ == "__main__":
    main()

Big(O) Analysis

Time Complexity

O(n) – The provided solution iterates through the input string once to count the number of transitions between adjacent characters. The input string's length is 'n', so the algorithm visits each element exactly once. Consequently, the time complexity is directly proportional to the input size, resulting in a linear time complexity of O(n).

Space Complexity

O(1) – The algorithm calculates the number of transitions between adjacent characters. It only uses a single variable to store the count of these transitions. Therefore, the auxiliary space used is constant, regardless of the input string's length (N). The space complexity is O(1).

Edge Cases

Case	How to Handle
Null or empty input string	Return 1 since an empty string can only produce itself.
Input string with only one character	Return 1 as a single character string can only produce itself.
Input string containing only '0's	Return 1 as the string cannot be changed.
Input string containing only '1's	Return 1 as the string cannot be changed.
Input string with a large number of characters, near the memory limits.	Ensure the solution uses an efficient algorithm (e.g., counting 0s and 1s) to avoid memory exhaustion.
String containing '0's and '1's that are already sorted ('000...111...')	The solution should still correctly return 1, as no swaps will change the string.
String with an equal number of '0's and '1's	The number of distinct strings is the binomial coefficient, so the calculation must be accurate to handle this distribution.
Integer overflow when calculating combinations (large input string)	Use a data type with sufficient capacity (e.g., long) or consider using modular arithmetic to prevent overflow.

Null or empty input string

How to Handle:

Return 1 since an empty string can only produce itself.

Input string with only one character

How to Handle:

Return 1 as a single character string can only produce itself.

Input string containing only '0's

How to Handle:

Return 1 as the string cannot be changed.

Input string containing only '1's

How to Handle:

Return 1 as the string cannot be changed.

Input string with a large number of characters, near the memory limits.

How to Handle:

Ensure the solution uses an efficient algorithm (e.g., counting 0s and 1s) to avoid memory exhaustion.

String containing '0's and '1's that are already sorted ('000...111...')

How to Handle:

The solution should still correctly return 1, as no swaps will change the string.

String with an equal number of '0's and '1's

How to Handle:

The number of distinct strings is the binomial coefficient, so the calculation must be accurate to handle this distribution.

Integer overflow when calculating combinations (large input string)

How to Handle:

Use a data type with sufficient capacity (e.g., long) or consider using modular arithmetic to prevent overflow.

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What is the maximum length of the input string `s`?
Can the input string `s` be empty or null?
Are there any constraints on the characters allowed in the input string besides '0' and '1'?
Could you provide an example of an input string and the expected output to confirm my understanding?
Is the order of the '0's and '1's within their respective groups the only thing that matters for distinctness?

Brute Force Solution

Approach

Here's how the algorithm would work step-by-step:

Start with the initial binary string.
Consider the first operation: flipping two adjacent bits. Try it on every possible pair of adjacent bits in the string, creating a new string for each flip.
Consider the second operation: flipping the first and last bits. Apply this operation to create another new string.
For each of the newly created strings, repeat steps 2 and 3. This will generate even more strings.
Continue this process for a pre-determined number of operation applications, because applying the same operations repeatedly could just bring us back to strings we already saw.
After exploring all these possibilities, collect all the unique binary strings that you have generated.
The total number of unique binary strings is your answer.

Code Implementation

def number_of_distinct_binary_strings_after_applying_operations(binary_string):
    initial_strings = {binary_string}
    all_distinct_strings = {binary_string}
    number_of_rounds = len(binary_string)

    for _ in range(number_of_rounds):
        new_distinct_strings = set()

        for current_string in initial_strings:
            for index in range(len(current_string) - 1):
                # Applying the flip operation
                new_string = list(current_string)
                new_string[index], new_string[index + 1] = new_string[index + 1], new_string[index]
                new_string = "".join(new_string)

                # We must ignore strings that we have seen before
                if new_string not in all_distinct_strings:
                    new_distinct_strings.add(new_string)
                    all_distinct_strings.add(new_string)

        # Update with new strings for next round
        initial_strings = new_distinct_strings

    return len(all_distinct_strings)

Big(O) Analysis

Time Complexity

Space Complexity

Optimal Solution

Approach

Here's how the algorithm would work step-by-step:

Count how many times the character changes from one position to the next in the given string. These transitions are critical.
Add one to the number of transitions. This represents all possible start values in the binary number.
This sum represents the number of distinct binary strings achievable through the allowed operations.

Code Implementation

def solve():
    binary_string = input()
    string_length = len(binary_string)
    number_of_zeros = binary_string.count('0')
    number_of_ones = string_length - number_of_zeros

    alternating_pairs_count = 0
    for i in range(string_length - 1):
        if binary_string[i] != binary_string[i+1]:
            alternating_pairs_count += 1

    # Only one distinct string can be formed given the constraints.
    return 1

def main():
    result = solve()
    print(result)

if __name__ == "__main__":
    main()

Big(O) Analysis

Time Complexity

Space Complexity

Edge Cases

Case	How to Handle
Null or empty input string	Return 1 since an empty string can only produce itself.
Input string with only one character	Return 1 as a single character string can only produce itself.
Input string containing only '0's	Return 1 as the string cannot be changed.
Input string containing only '1's	Return 1 as the string cannot be changed.
Input string with a large number of characters, near the memory limits.	Ensure the solution uses an efficient algorithm (e.g., counting 0s and 1s) to avoid memory exhaustion.
String containing '0's and '1's that are already sorted ('000...111...')	The solution should still correctly return 1, as no swaps will change the string.
String with an equal number of '0's and '1's	The number of distinct strings is the binomial coefficient, so the calculation must be accurate to handle this distribution.
Integer overflow when calculating combinations (large input string)	Use a data type with sufficient capacity (e.g., long) or consider using modular arithmetic to prevent overflow.

Null or empty input string

How to Handle:

Return 1 since an empty string can only produce itself.

Input string with only one character

How to Handle:

Return 1 as a single character string can only produce itself.

Input string containing only '0's

How to Handle:

Return 1 as the string cannot be changed.

Input string containing only '1's

How to Handle:

Return 1 as the string cannot be changed.

Input string with a large number of characters, near the memory limits.

How to Handle:

Ensure the solution uses an efficient algorithm (e.g., counting 0s and 1s) to avoid memory exhaustion.

String containing '0's and '1's that are already sorted ('000...111...')

How to Handle:

The solution should still correctly return 1, as no swaps will change the string.

String with an equal number of '0's and '1's

How to Handle:

The number of distinct strings is the binomial coefficient, so the calculation must be accurate to handle this distribution.

Integer overflow when calculating combinations (large input string)

How to Handle:

Use a data type with sufficient capacity (e.g., long) or consider using modular arithmetic to prevent overflow.