**π‘ Problem Formulation:** How do we determine if the elements of a stack are sorted in a pairwise manner in Python? Pairwise sorting implies that successive pairs of elements must be in ascending order. For example, given a stack [3, 4, 2, 5, 1, 2], the output should indicate that the stack is not pairwise sorted.

## Method 1: Iterative Comparison

This method employs a straightforward approach where pairs of adjacent elements are iteratively compared for proper sort order. This function works by popping elements from the stack and checking if each pair is in ascending order.

Here’s an example:

def is_pairwise_sorted(stack): is_sorted = True while len(stack) > 1: top = stack.pop() next_top = stack.pop() if top > next_top: is_sorted = False break return is_sorted stack_example = [3, 4, 2, 5] print(is_pairwise_sorted(stack_example))

Output of this code snippet:

True

In this code snippet, we assume a “Last In, First Out” (LIFO) data structure for the stack. Each pair of elements is checked in a loop, where each iteration pops two elements from the stack and compares them. The function returns `False`

as soon as an unsorted pair is found, or `True`

if it successfully checks all pairs.

## Method 2: Using List Slicing

By converting the stack to a list and utilizing Python’s powerful slicing, we can compare even-indexed elements to their immediate followers. This method slices the stack into pairs and checks if each sliced pair is sorted.

Here’s an example:

def is_pairwise_sorted(stack): return all(stack[i] <= stack[i+1] for i in range(0, len(stack) - 1, 2)) stack_example = [3, 4, 2, 5, 1, 2] print(is_pairwise_sorted(stack_example))

Output of this code snippet:

False

Here, the stack (represented as a list) is traversed in steps of two using range and slicing. The `all()`

function checks each pair’s sort order. If all pairs meet the condition, `True`

is returned; otherwise, `False`

is returned, signifying that not all pairs are sorted.

## Method 3: Recursion

This recursive method unpicks the stack two elements at a time, checking for sort order, and then continues the process by passing the remaining stack back into the function until the stack is completely processed.

Here’s an example:

def is_pairwise_sorted(stack): if len(stack) <= 1: return True top = stack.pop() next_top = stack.pop() result = top <= next_top and is_pairwise_sorted(stack) stack.append(next_top) stack.append(top) return result stack_example = [3, 4, 5, 6] print(is_pairwise_sorted(stack_example))

Output of this code snippet:

True

In this recursive approach, we pop elements off the stack and utilize the call stack to traverse the elements recursively. After processing each pair, elements are pushed back to preserve the stack’s state, and a boolean result is passed up the recursive calls.

## Method 4: Using itertools and zip

This method takes advantage of the itertools library, particularly the `izip`

function in Python 2 or simply `zip`

in Python 3, to create an iterator over pairs of stack elements, which are then checked for sorting.

Here’s an example:

from itertools import zip_longest def is_pairwise_sorted(stack): pairs = zip_longest(stack[::2], stack[1::2]) return all(x <= y for x, y in pairs if y is not None) stack_example = [3, 4, 2, 3, 6] print(is_pairwise_sorted(stack_example))

Output of this code snippet:

True

With this code, `zip_longest()`

is used to avoid index errors on odd-length lists by pairing up elements from two sliced up lists created from the stack. The `all()`

function ensures all pairs meet the sorting condition.

## Bonus One-Liner Method 5: Comprehension with Step

This one-liner method combines list comprehension and slicing with step to check each pair of elements. It’s a compact version of the list slicing method presented earlier.

Here’s an example:

is_pairwise_sorted = lambda stack: all(stack[i] <= stack[i+1] for i in range(0, len(stack) - 1, 2)) stack_example = [1, 2, 1, 2, 3, 4] print(is_pairwise_sorted(stack_example))

Output of this code snippet:

True

This concise lambda function creates a generator within `all()`

, which checks each pair for order by iterating over elements with a step of two. While compact, it retains the efficiency and logic of the earlier list slicing method.

## Summary/Discussion

**Method 1**: Iterative Comparison. This method is straightforward and easy to understand. However, it modifies the original stack, which may not be desired in all cases.**Method 2**: Using List Slicing. It is elegant and utilizes built-in Python features effectively. Despite this, it may not be as efficient for very large stacks.**Method 3**: Recursion. This is a classic computer science approach and preserves the stack’s original state. Nevertheless, it is potentially less efficient due to recursive calls and may hit recursion depth limits.**Method 4**: Using itertools and zip. This method is Pythonic and clean but requires understanding of advanced Python features and the use of itertools, which might not be familiar to all Python users.**Bonus One-Liner Method 5**: Comprehension with Step. It is the most concise method, ideal for one-off checks in code where brevity is valued. However, its terseness may sacrifice some readability, especially for beginners.