**💡 Problem Formulation:** You have an array where initially, the sequence of elements decreases and then strictly increases, or vice versa—formulating a “turning point” or “peak” in the sequence. The task is to identify the index of this turning element. For instance, with an input array [5, 4, 3, 2, 3, 4, 5], the output should be index 3, where the value 2 represents the turning point.

## Method 1: Linear Search

The simplest approach is a linear search that iterates over the array from the beginning, checking for the turning point where the sequence switches from decreasing to increasing or vice versa. This method is straightforward but may not be optimal for very large arrays.

Here’s an example:

def find_turning_point(arr): for i in range(1, len(arr)-1): if arr[i-1] > arr[i] < arr[i+1] or arr[i-1] arr[i+1]: return i return None print(find_turning_point([5, 4, 3, 2, 3, 4, 5]))

Output: `3`

This code defines a function `find_turning_point()`

, which loops through the elements of an array to find the index where the sequence either starts increasing or decreasing, indicating the turning point. Though effective for small arrays, its linear time complexity isn’t ideal for large datasets.

## Method 2: Binary Search

A more efficient approach uses binary search principles, tailored for peak finding. Assuming the array has one peak element, we can recursively find the point where the change in sequence direction occurs. This is more efficient for larger arrays with a time complexity of O(log n).

Here’s an example:

def find_peak(arr, left, right): mid = (left + right) // 2 if arr[mid-1] arr[mid+1]: return mid elif arr[mid-1] > arr[mid]: return find_peak(arr, left, mid-1) else: return find_peak(arr, mid+1, right) arr = [5, 4, 3, 2, 3, 4, 5] print(find_peak(arr, 0, len(arr)-1))

Output: `3`

In this snippet, the function `find_peak()`

performs a binary search to locate the peak’s index efficiently. The function is called recursively, halving the search range in each iteration until the peak is found, offering a significant performance benefit for large arrays.

## Method 3: Divide and Conquer

The divide and conquer algorithm is a refinement of the binary search for this specific problem. The array is divided into two halves to locate the point where the change in direction occurs. Its refinement comes in the way it reduces the problem space.

Here’s an example:

def find_turning_point_divide_conquer(arr): def search(low, high): if low == high: return low mid = (low + high) // 2 if arr[mid] > arr[mid+1]: return search(low, mid) return search(mid+1, high) return search(0, len(arr) - 1) arr = [5, 4, 3, 2, 3, 4, 5] print(find_turning_point_divide_conquer(arr))

Output: `3`

This code uses a divide and conquer strategy, effectively a more tailored approach of Method 2, where the binary search is used recursively to find the exact index of the turning point. This approach is efficient for large arrays and has a similar complexity to binary search.

## Method 4: Python Libraries (NumPy)

Using Python’s NumPy library provides a set of optimized functions which can be utilized to solve this problem efficiently. While potentially overkill for small datasets, this method would be advantageous for arrays with complex numerical operations.

Here’s an example:

import numpy as np def find_turning_point_numpy(arr): arr_diff = np.diff(arr) return np.argmax((arr_diff[:-1] 0)) + 1 arr = np.array([5, 4, 3, 2, 3, 4, 5]) print(find_turning_point_numpy(arr))

Output: `3`

This example uses NumPy’s `diff`

function to compute the discrete difference between elements and then identifies the turning point index. This takes advantage of NumPy’s optimized C backend for performance boosts on large datasets.

## Bonus One-Liner Method 5: List Comprehension

If succinctness is the goal, Python’s list comprehensions can be used to find the turning point in a one-liner, at the expense of readability and possibly performance for large arrays.

Here’s an example:

arr = [5, 4, 3, 2, 3, 4, 5] print([i for i in range(1, len(arr)-1) if arr[i-1] > arr[i] < arr[i+1] or arr[i-1] arr[i+1]][0])

Output: `3`

This single line defines and executes a list comprehension that iterates through array indexes, captures the index where a turning point occurs, and prints out the first such index it finds. This method trades off efficiency and readability for brevity.

## Summary/Discussion

**Method 1: Linear Search.**Simple and straightforward. Best for small arrays. Time complexity is O(n).**Method 2: Binary Search.**Efficient for large arrays. Time complexity is O(log n). Requires the understanding of recursive functions.**Method 3: Divide and Conquer.**Similar efficiency to binary search, with potentially slight performance boost due to tailored approach. Suitable for large arrays with time complexity of O(log n).**Method 4: Python Libraries (NumPy).**Best for numerical operations and optimized for performance. Simplicity can come at the cost of over-reliance on third-party libraries.**Bonus Method 5: List Comprehension.**Quick and compact. Less readable and potentially less performant for larger arrays.