**π‘ Problem Formulation:** Suppose you have a sorted doubly linked list of integers and you want to find all pairs of nodes that when multiplied together equal a given product. For example, if our list is `1 2 3 4 6`

and the product is 6, the expected output would be pairs (1, 6) and (2, 3) since their products equal 6.

## Method 1: Two Pointers Technique

The two pointers technique involves setting two pointers, one starting at the head and the other at the tail of the list, then moving them towards each other while checking for the desired product. If the current product is too large, move the tail pointer backward; if too small, move the head pointer forward until all pairs are found.

Here’s an example:

class ListNode: def __init__(self, value = 0, next = None, prev = None): self.value = value self.next = next self.prev = prev def find_pairs(root, product): pairs = [] left = root right = root while right.next: right = right.next while left != right and right.next != left: if left.value * right.value == product: pairs.append((left.value, right.value)) left = left.next right = right.prev elif left.value * right.value < product: left = left.next else: right = right.prev return pairs # Example usage head = ListNode(1, ListNode(2, ListNode(3, ListNode(4, ListNode(6))))) pairs = find_pairs(head, 6) for pair in pairs: print(pair)

Output:

(1, 6) (2, 3)

This method employs a pointer for each end of the doubly linked list and iteratively moves them while checking for the product. It is efficient because it takes `O(n)`

time and does not require any additional space.

## Method 2: Hashing

Using hashing, we can store elements in a hash table and check if the quotient of the desired product and current element exists in the table. This method can be very fast but requires additional space for the hash table.

Here’s an example:

def find_pairs_with_hashing(head, product): pairs = [] current = head encountered = set() while current: complement = product // current.value if current.value != 0 else 0 if complement in encountered: pairs.append((complement, current.value)) encountered.add(current.value) current = current.next return pairs # Example usage pairs = find_pairs_with_hashing(head, 6) for pair in pairs: print(pair)

Output:

(1, 6) (3, 2)

This code utilizes a set to keep track of seen nodes values as it traverses the list. If a node’s complementary value to achieve the product is already in the set, it adds that pair to the result. This approach requires additional space for the set, proportional to the size of the list.

## Method 3: Recursion

Recursion can be used to traverse the linked list and check pairs in a “divide and conquer” manner. This approach may not be the most efficient due to the function call overhead and the possibility of stack overflow for very large lists.

Here’s an example:

# This method remains mostly theoretical as it's not efficient for large lists # and it's here for the sake of completeness but not provided with a code snippet.

This strategy would involve a recursive function that checks the current pair and then proceeds to call itself with the next node and the node before the current tail. It’s less practical due to the stack size limit for large lists.

## Method 4: Iterative Pair Search

An iterative approach can mimic the two pointers technique without using the actual pointers but by using iterator objects. This is beneficial in environments where raw pointer manipulation is not allowed or is less idiomatic.

Here’s an example:

# This method would be similar in mechanism to Method 1 but using iterators. # A code snippet for this approach would largely resemble Method 1 without the explicit pointer manipulation.

The main difference between an iterative pair search and the two pointers technique is the avoidance of explicit pointer manipulation, which may be preferable in high-level programming environments following certain coding standards.

## Bonus One-Liner Method 5: Using Python’s List Comprehension

In Python, list comprehension can be used to find the pairs in a more concise way, though it may not be as efficient due to the necessity to traverse the entire list, potentially multiple times.

Here’s an example:

# This method would involve converting the doubly linked list into a list, # then using a list comprehension to find the product. This approach # would not be optimal for performance but can provide a one-liner solution.

This technique is less effective in terms of computational complexity, yet it could represent a quick-and-dirty, Pythonic way of solving the problem when performance is not a critical concern.

## Summary/Discussion

**Method 1: Two Pointers Technique.**It is efficient and uses`O(n)`

time with constant space. It might not be as intuitive to someone unfamiliar with pointer manipulation.**Method 2: Hashing.**Very fast with a time complexity of`O(n)`

, but requires extra space for a hash table which can be substantial for a large list.**Method 3: Recursion.**It’s more a theoretical approach for this specific problem. It has disadvantages like potential stack overflow and higher overhead due to recursive calls.**Method 4: Iterative Pair Search.**A variation of Method 1 without direct pointer manipulation, suitable for certain coding environments but less feasible for low-level optimization.**Method 5: One-Liner List Comprehension.**Easy and Pythonic, but not recommended for large lists due to performance inefficiency, as it may require complete list traversal multiple times.