**π‘ Problem Formulation:** You need to determine the position of a specific element within a monotonic sequence, a sequence which is either entirely non-increasing or non-decreasing. For example, given the monotonic sequence [1, 2, 4, 4, 5, 6] and the element 4, the desired output is the position: 2 (using zero-based indexing).

## Method 1: Linear Search

A linear search algorithm iterates through the sequence until it finds the target element, providing its index. This method is straightforward and practical for small sequences but can be slow for large ones due to its O(n) complexity.

Here’s an example:

def linear_search(sequence, target): for index, elem in enumerate(sequence): if elem == target: return index return -1 print(linear_search([1, 2, 4, 4, 5, 6], 4))

Output:

2

This code snippet defines a function named `linear_search`

that takes a sequence and the target element as arguments. It uses a for loop to iterate through the sequence while checking each element against the target. If a match is found, the index is returned; otherwise, -1 is returned to indicate that the element was not found.

## Method 2: Binary Search

Binary search is a much faster algorithm with O(log n) complexity. It requires the sequence to be sorted and repeatedly divides the search interval in half to locate the target element.

Here’s an example:

def binary_search(sequence, target): left, right = 0, len(sequence) - 1 while left <= right: mid = (left + right) // 2 if sequence[mid] target: right = mid - 1 else: return mid return -1 print(binary_search([1, 2, 4, 4, 5, 6], 4))

Output:

2

This code defines a function `binary_search`

that utilizes binary search to find the position of the target element within the sequence. Starting with the full range of indices, the function repeatedly narrows down the search interval by comparing the target with the middle element, until the element is found or the range is exhausted.

## Method 3: Using the bisect Module

The bisect module in Python provides functions for manipulating sorted lists. The `bisect_left()`

function can be used to find the position of an element in a sorted sequence, suited for large data sets.

Here’s an example:

import bisect sequence = [1, 2, 4, 4, 5, 6] target = 4 index = bisect.bisect_left(sequence, target) print(index)

Output:

2

The provided code imports the `bisect`

module and uses the `bisect_left()`

function on the `sequence` list to find the first occurrence of `target`. This function utilizes a binary search internally, hence, providing an efficient way to find the index.

## Method 4: Using numpy’s searchsorted

If you are working with numerical data, NumPy’s `searchsorted()`

function finds indices where elements should be inserted to maintain order. By default, it finds the first suitable position from the left, like `bisect_left()`

.

Here’s an example:

import numpy as np sequence = np.array([1, 2, 4, 4, 5, 6]) target = 4 index = np.searchsorted(sequence, target, side='left') print(index)

Output:

2

This snippet utilizes the Python library NumPy to utilize the `searchsorted()`

function. This is especially useful for sequences that are NumPy arrays. By specifying `side=’left’`, it ensures the first occurrence’s index is returned, much like `bisect_left()`

.

## Bonus One-Liner Method 5: Using List Comprehension

For succinctness, one can utilize list comprehension to find the index of the first occurrence of the target element in the sequence. This method is most effective with small datasets due to O(n) complexity.

Here’s an example:

sequence = [1, 2, 4, 4, 5, 6] target = 4 index = [idx for idx, val in enumerate(sequence) if val == target][0] print(index)

Output:

2

This one-liner creates a list of indices where the element equals the target and then selects the first index. Note that this will raise an `IndexError`

if the target is not in the sequence.

## Summary/Discussion

**Method 1: Linear Search.**Simple and easy to implement. Best for short sequences. Inefficient for large datasets due to linear complexity.**Method 2: Binary Search.**Highly efficient with logarithmic complexity. Requires the sequence to be sorted beforehand. Ideal for large datasets.**Method 3: bisect Module.**Utilizes binary search from standard Python library; it’s optimized and concise. Needs sorted input.**Method 4: NumPy’s searchsorted.**Best suited for numerical data represented as NumPy arrays. Offers performance benefits with sorted arrays.**Bonus Method 5: List Comprehension.**A concise one-liner solution. Prone to errors if the element isn’t found and not suitable for large datasets due to its linear time complexity.