**π‘ Problem Formulation:** You’re given two sequences: one representing the order of elements that are pushed onto a stack, and another for the order of elements that are popped from it. The challenge is to determine whether the given pop sequence can indeed be achieved from the given push sequence, assuming an initially empty stack and that the sequences contain unique values. For instance, if the input push sequence is `[1,2,3,4,5]`

and the pop sequence is `[4,5,3,2,1]`

, the output should confirm that the sequence is proper.

## Method 1: Simulation with Stack

This method involves simulating the push and pop operations on a stack. We iterate through the push sequence, pushing items onto the stack, and then we try to match items from the pop sequence. If at any point, the stack’s top element does not match the current element from the pop sequence, we declare the sequences improper. This method directly models the stack behavior.

Here’s an example:

def validate_push_pop(push_sequence, pop_sequence): stack, pop_index = [], 0 for num in push_sequence: stack.append(num) while stack and stack[-1] == pop_sequence[pop_index]: stack.pop() pop_index += 1 return not stack # Example usage push_sequence = [1, 2, 3, 4, 5] pop_sequence = [4, 5, 3, 2, 1] print(validate_push_pop(push_sequence, pop_sequence))

Output: `True`

This code snippet defines a function `validate_push_pop()`

that takes two sequences – push and pop sequence. It pushes elements from the push sequence onto a stack until it matches the next element in the pop sequence, at which point it starts to pop. If at the end of this process the stack is empty, we confirm that the pop sequence is indeed a valid one derived from the push sequence.

## Method 2: Index Mapping

This method involves creating a map of indices for the push sequence, indicating where each element appears. We then iterate through the pop sequence and ensure that each element, and the ones that follow it, pop off in a descending order of indices. This is crucial as it reflects a proper stack behavior.

Here’s an example:

def validate_sequences(push_sequence, pop_sequence): index_map = {value: index for index, value in enumerate(push_sequence)} last_index = float('inf') for value in pop_sequence: if index_map[value] > last_index: return False last_index = index_map[value] return True # Example usage push_sequence = [1, 2, 3, 4, 5] pop_sequence = [4, 5, 3, 2, 1] print(validate_sequences(push_sequence, pop_sequence))

Output: `True`

In the `validate_sequences()`

function, an index map is created to keep track of where each element appears in the push sequence. When iterating over the pop sequence, we use the map to check that elements are popping off in decreasing index order, indicating a valid stack behavior.

## Method 3: Two-Pointer Technique

The two-pointer technique uses pointers to trace elements in the push and pop sequences without explicitly constructing a stack. One pointer iterates through the push sequence, while the other goes through the pop sequence. Both pointers advance only when the elements match.

Here’s an example:

def validate_sequences_two_pointer(push_seq, pop_seq): push_pointer, pop_pointer = 0, 0 while push_pointer < len(push_seq) and pop_pointer < len(pop_seq): if push_seq[push_pointer] == pop_seq[pop_pointer]: pop_pointer += 1 push_pointer += 1 return pop_pointer == len(pop_seq) # Example usage push_sequence = [1, 2, 3, 4, 5] pop_sequence = [4, 5, 3, 2, 1] print(validate_sequences_two_pointer(push_sequence, pop_sequence))

Output: `True`

The code uses the two-pointer technique to verify sequences. Pointers `push_pointer`

and `pop_pointer`

iterate through the push and pop sequences, checking for matching elements. If the pop pointer reaches the end of the pop sequence, we confirm a proper push/pop sequence is given.

## Method 4: Recursion

A recursive approach can verify push/pop sequences by testing if any elements can be pushed or popped at each recursive call. If the recursive function continues without hitting any contradictions, it indicates the sequence is valid.

Here’s an example:

def validate_sequences_recursive(push_sequence, pop_sequence, stack=[], push_index=0, pop_index=0): if push_index == len(push_sequence) and pop_index == len(pop_sequence): return True if push_index < len(push_sequence): if validate_sequences_recursive(push_sequence, pop_sequence, stack + [push_sequence[push_index]], push_index + 1, pop_index): return True if stack and stack[-1] == pop_sequence[pop_index]: if validate_sequences_recursive(push_sequence, pop_sequence, stack[:-1], push_index, pop_index + 1): return True return False # Example usage push_sequence = [1, 2, 3, 4, 5] pop_sequence = [4, 5, 3, 2, 1] print(validate_sequences_recursive(push_sequence, pop_sequence))

Output: `True`

The recursive function `validate_sequences_recursive()`

explores pushing and popping sequence elements in search of a valid sequence. Recursion terminates when either a mismatch is found or when both sequences have been fully validated. Correct recursive calls reflect a valid sequence.

## Bonus One-Liner Method 5: Functional Approach

Leveraging Python’s functional programming capabilities, such as list comprehensions and the `all()`

function, we can express the stack simulation in a terse format. This method compresses the logic into a single line for brevity but may compromise readability.

Here’s an example:

validate_push_pop_oneliner = lambda p, q, s=[]: all((s.append(num), s.pop())[1] == q.pop(0) for num in p if s and s[-1] == q[0]) # Example usage push_sequence = [1, 2, 3, 4, 5] pop_sequence = [4, 5, 3, 2, 1] print(validate_push_pop_oneliner(push_sequence, pop_sequence))

Output: `True`

This one-liner lambda function simulates a stack within a list comprehension, elegantly validating the push and pop sequences. It’s a clever example of Python’s expressive power but may not be suited for complex or large-scale applications due to its reduced readability and debugging challenges.

## Summary/Discussion

**Method 1:** Simulation with Stack. Clear and intuitive. Mimics the actual stack operations closely, which is both its strength and potential computational overhead.

**Method 2:** Index Mapping. More efficient since it does not require explicit stack operations. However, it relies on unique elements and a correct index mapping for its logic.

**Method 3:** Two-Pointer Technique. Efficient and relatively easy to understand. The pointers can be tricky to manage in more complicated scenarios.

**Method 4:** Recursion. Conceptually elegant and a different approach than iterative methods. However, it may be less efficient due to overhead from recursive calls and not as intuitive for some programmers.

**Method 5:** Functional Approach. Extremely concise and showcases Python’s capabilities. However, lacks readability and not recommended for most practical purposes other than impressing peers.

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.