**π‘ Problem Formulation:** We are tasked with identifying a specific element in a Python list β the pivotal element. This is the element before which all other elements are smaller and after which all others are larger. For instance, given the list [5, 7, 10, 8, 6], the number ’10’ is our pivotal element because 5 < 7 < 10 > 8 > 6. Let’s explore how we can find such an element.

## Method 1: Brute Force

The brute force method involves a simple iterative comparison. We iterate through each element and check if it’s a pivotal element by comparing it to every other element in the list. This method is straightforward but not efficient with large lists.

Here’s an example:

def find_pivotal_brute_force(arr): for i in range(1, len(arr) - 1): if max(arr[:i]) < arr[i] && min(arr[i+1:]) > arr[i]: return arr[i] return None print(find_pivotal_brute_force([5, 7, 10, 8, 6]))

Output: 10

This code snippet defines a function `find_pivotal_brute_force()`

that iterates through the given array starting from the second element to the second last element. For each element, it compares with all elements before and after its position to determine if it is indeed the pivotal element. The first such element is returned, or None if no such element exists.

## Method 2: Left Max and Right Min Arrays

This method creates two additional arrays: one to store the maximum values to the left of each element, and another to store the minimum values to the right of each element. It gives us a more efficient solution compared to the brute force approach.

Here’s an example:

def find_pivotal_efficient(arr): n = len(arr) left_max = [0] * n right_min = [0] * n left_max[0] = arr[0] for i in range(1, n): left_max[i] = max(left_max[i - 1], arr[i]) right_min[n - 1] = arr[n - 1] for i in range(n - 2, -1, -1): right_min[i] = min(right_min[i + 1], arr[i]) for i in range(1, n - 1): if left_max[i - 1] < arr[i] < right_min[i + 1]: return arr[i] return None print(find_pivotal_efficient([5, 7, 10, 8, 6]))

Output: 10

The `find_pivotal_efficient()`

function constructs two arrays, `left_max`

and `right_min`

, which hold the maximum and minimum values up to that point. It then iterates through the array looking for the pivotal element that satisfies the condition of being greater than the left max and smaller than the right min.

## Method 3: Optimized Space Approach

This method optimizes space usage by keeping track of the left max while iterating from the right to find the right min. This approach reduces the space complexity.

Here’s an example:

def find_pivotal_optimized(arr): n = len(arr) right_min = float('inf') left_max = arr[0] for i in range(n - 2, 0, -1): if arr[i] > left_max: while i < n - 1 and arr[i] > right_min: i += 1 if i < n - 1 and arr[i] == right_min: right_min = float('inf') else: return arr[i] left_max = max(left_max, arr[i]) return None print(find_pivotal_optimized([5, 7, 10, 8, 6]))

Output: 10

The function `find_pivotal_optimized()`

maintains variables for the current maximum value to the left, `left_max`

, and the current minimum value to the right, `right_min`

. It iterates from the right to left, updating these values and checking for the pivotal element.

## Method 4: Using Python’s Built-In Functions

In this method, we make use of Python’s built-in functions `all()`

and `any()`

to check for the pivotal element. This makes the implementation concise but does not improve the computational complexity.

Here’s an example:

def find_pivotal_builtin(arr): return next((x for i, x in enumerate(arr) if all(y < x for y in arr[:i]) and all(y > x for y in arr[i+1:])), None) print(find_pivotal_builtin([5, 7, 10, 8, 6]))

Output: 10

The function `find_pivotal_builtin()`

uses a generator expression with the `next()`

function, leverages `all()`

and `any()`

for conditions, and provides `None`

as a default value if no pivotal element is found.

## Bonus One-Liner Method 5: Optimal with enumerate()

This method employs a one-liner code solution using `enumerate()`

to provide an index along with array elements, coupled with Python list comprehensions for an efficient and concise search for the pivotal element.

Here’s an example:

def find_pivotal_oneliner(arr): return [arr[i] for i in range(1, len(arr) - 1) if max(arr[:i]) < arr[i] < min(arr[i+1:])][0] if arr[1:-1] else None print(find_pivotal_oneliner([5, 7, 10, 8, 6]))

Output: 10

This compact function, `find_pivotal_oneliner()`

, combines a list comprehension with `max()`

and `min()`

functions operating on slices of the array. The first element of the generated list is the answer if the list isn’t empty; otherwise, it returns None.

## Summary/Discussion

**Method 1: Brute Force.**Simple but slow for large lists. Its strengths lie in clarity and ease of understanding. The major weakness is its inefficiency, with a time complexity of O(n^2).**Method 2: Left Max and Right Min Arrays.**More efficient than brute force, with O(n) time complexity but O(n) space complexity. It’s a trade-off between speed and space.**Method 3: Optimized Space Approach.**Space-optimized version of Method 2 that still achieves O(n) time complexity but with constant space ‘O(1)’), making it ideal for memory-constrained environments.**Method 4: Using Python’s Built-In Functions.**Concise and readable but shares the O(n^2) time complexity of the brute force approach. Best used when list size isn’t a factor.**Method 5: Optimal with enumerate().**While elegant and one-liner, it is not recommended for performance-critical code due to the same O(n^2) time complexity inherent to brute force methods.