**π‘ Problem Formulation:** We need to establish an algorithm that counts the minimum number of character flips required to rearrange a given string such that all occurrences of ‘x’ come before any ‘y’. For instance, given the input “xyyx”, the desired output after flipping should be “xxxy”, and the number of flips required is 2.

## Method 1: Brute Force

The brute force method iteratively checks each ‘y’ and counts the number of ‘x’ characters after it, providing the number of flips required. This approach is straightforward but not the most efficient, having a quadratic time complexity in the worst case.

Here’s an example:

def count_flips_brute_force(s): flips = 0 for i in range(len(s)): if s[i] == 'y': for j in range(i+1, len(s)): if s[j] == 'x': flips += 1 return flips print(count_flips_brute_force("xyyx"))

Output: 2

This code defines a function `count_flips_brute_force()`

that takes a string as input. It loops through each character and, if the character is a ‘y’, it counts how many ‘x’ characters come after it, summing up these counts to get the total number of flips required.

## Method 2: Count and Compare

This method improves efficiency by counting the total number of ‘x’ present and then iterating over the string to compare the counts as it encounters ‘y’. It reduces the time complexity to linear, making it better suited for larger strings.

Here’s an example:

def count_flips_count_compare(s): total_x = s.count('x') flips = 0 x_count = 0 for char in s: if char == 'x': x_count += 1 else: flips += total_x - x_count return flips print(count_flips_count_compare("xyyx"))

Output: 2

The `count_flips_count_compare()`

function first counts the total ‘x’ characters present in the string. As it iterates through the string, it maintains a running count of ‘x’ seen and calculates flips whenever a ‘y’ is encountered, by deducting the count of ‘x’ seen from total ‘x’ count till that point.

## Method 3: Prefix Sum Technique

The prefix sum technique calculates flips required by accumulating the ‘y’ characters encountered and using this accumulation as a multiplier for each ‘x’ found after a ‘y’. This technique maintains linear time complexity but simplifies the flipping logic.

Here’s an example:

def count_flips_prefix_sum(s): flips = 0 y_count = 0 for char in s: if char == 'y': y_count += 1 else: flips += y_count return flips print(count_flips_prefix_sum("xyyx"))

Output: 2

In this code, `count_flips_prefix_sum()`

function iterates over the string and counts ‘y’ characters. When an ‘x’ is encountered after a ‘y’, it adds the current ‘y’ count to the flips count, which represents the required flips until that point.

## Method 4: Optimized Comparison

An optimized comparison approach uses a single iteration to count the flips required, by keeping track of the optimal number of ‘x’ seen and calculating flips when ‘y’ appears. This is usually the most time-efficient method for this problem.

Here’s an example:

def count_flips_optimized(s): flips = 0 x_count = 0 for char in s: if char == 'x': x_count += 1 else: flips = min(flips + 1, x_count) return flips print(count_flips_optimized("xyyx"))

Output: 2

The `count_flips_optimized()`

function iterates through the string, counting the ‘x’ characters. When a ‘y’ is encountered, it calculates the minimum between the current flips incremented by one and the ‘x’ count, which ensures that we do not count more flips than ‘x’ present in the substring so far.

## Bonus One-Liner Method 5: Pythonic Approach

For those who prefer succinct code, a Pythonic one-liner may use list comprehensions or generator expressions to calculate the flips required in a single expression without explicitly writing out loops. However, this may sacrifice a bit of readability for brevity.

Here’s an example:

print(sum(1 for i, c in enumerate(s) if c == 'y' for x in s[i:] if x == 'x'))

Output: 2

This one-liner uses a generator expression to iterate over the index and character pairs in the string. For each ‘y’ found, it iterates over the rest of the string to count ‘x’ characters, summing these counts to find the total flips required.

## Summary/Discussion

**Method 1: Brute Force.**Simple to understand but inefficient with large strings due to its quadratic time complexity.**Method 2: Count and Compare.**Improved efficiency with linear time complexity, but requires two passes over the string.**Method 3: Prefix Sum Technique.**Linear time complexity with simplified flip logic, relatively efficient and easy to understand.**Method 4: Optimized Comparison.**Highly efficient with a single pass over the string and maintaining the optimal state throughout.**Method 5: Pythonic Approach.**Elegantly concise, but could be less readable to those not familiar with Python’s generator expressions or list comprehensions.

Emily Rosemary Collins is a tech enthusiast with a strong background in computer science, always staying up-to-date with the latest trends and innovations. Apart from her love for technology, Emily enjoys exploring the great outdoors, participating in local community events, and dedicating her free time to painting and photography. Her interests and passion for personal growth make her an engaging conversationalist and a reliable source of knowledge in the ever-evolving world of technology.