Section 1.1: Method 1 - Nested Loop Approach

The first technique involves employing a nested loop to compare each element against every other element in the list. When a duplicate is detected, it gets appended to a list that is ultimately returned by the function.

def find_duplicates(nums):

duplicates = []

n = len(nums)

for i in range(n):

for j in range(i+1, n):

if nums[i] == nums[j] and nums[i] not in duplicates:

duplicates.append(nums[i])

return duplicates

• Time Complexity: O(n^2) (Worst case scenario)
• Space Complexity: O(n) for storing duplicates.

Section 1.2: Method 2 - Sorting and Checking

This method involves sorting the list first and then checking adjacent elements for duplicates. If duplicates are identified, they are stored in a set for return.

def find_duplicates(nums):

duplicates = set()

nums.sort()

for i in range(1, len(nums)):

if nums[i] == nums[i - 1]:

duplicates.add(nums[i])

return list(duplicates)

• Time Complexity: O(n log n)
• Space Complexity: O(n) for the duplicates set.

Section 1.3: Method 3 - Utilizing a Hash Map

This approach leverages a hash map to keep track of the occurrences of each item in the list. Duplicates are added to a set for later retrieval.

def find_duplicates(nums):

duplicates = set()

num_dict = {}

for num in nums:

if num in num_dict:

duplicates.add(num)

else:

num_dict[num] = 1

return list(duplicates)

• Time Complexity: O(n) (Optimal scenario)
• Space Complexity: O(n) for the duplicates set and the hash map.

Section 1.4: Method 4 - Using Collections' Counter

In this method, Python’s collections module is utilized to track the count of each item in the list via the Counter class. Duplicates are then filtered and returned.

from collections import Counter

def find_duplicates(nums):

duplicates = [item for item, count in Counter(nums).items() if count > 1]

return duplicates

• Time Complexity: O(n) (Optimal scenario)
• Space Complexity: O(n) due to the Counter object and duplicates list.

Section 1.5: Method 5 - Employing Sets

The final method uses two sets to monitor duplicates and the items in the list. Duplicates are identified and stored in a set for return.

def find_duplicates(nums):

duplicates = set()

seen = set()

for num in nums:

if num in seen:

duplicates.add(num)

else:

seen.add(num)

return list(duplicates)

• Time Complexity: O(n) (Optimal scenario)
• Space Complexity: O(n) for the seen and duplicates sets.