Problem Description

The problem provides us with two integer arrays, nums1 and nums2, which are both 0-indexed. The goal is to create a list called answer that consists of two sublists:

• answer[0] should include all unique integers that are found in nums1 but not in nums2.
• answer[1] should include all unique integers that are found in nums2 but not in nums1.

These lists of unique integers should exclude any duplicates and can be returned in any order.

Intuition

To solve this problem, we start by understanding the requirement for unique elements that are present in one array but not the other. This naturally leads us to think of set operations because sets automatically handle uniqueness and provide efficient operations for difference and intersection.

The intuitive approach is to:

• Convert both nums1 and nums2 into sets to eliminate any repeated numbers within each array.
• Use set subtraction (-) to find elements that are in one set but not the other.

Set subtraction s1 - s2 yields a set of elements that are in s1 but not in s2. Here are the steps:

1. Convert nums1 to a set s1 to obtain unique elements from nums1.
2. Convert nums2 to a set s2 to get unique elements from nums2.
3. Perform s1 - s2 to find all elements that are in nums1 but not in nums2.
4. Do the reverse and compute s2 - s1 to find all elements in nums2 but not in nums1.
5. Convert these results back into lists and return them as the answer list with two sublists as described above.

This solution is concise, and the key is leveraging set operations to efficiently answer the question without having to implement complex logic for handling uniqueness or element comparison.

Solution Approach

The solution consists of a class Solution with a method findDifference which takes two lists, nums1 and nums2, and returns a list containing two lists representing the differences between the two original lists. Here is a step-by-step explanation of the implementation:

• First, we need to create two sets: s1 is created from nums1 and s2 is created from nums2. Using sets is a critical choice here because:

  • Sets automatically remove duplicate values.
  • They provide efficient operations for set difference (-), which is directly needed for this problem.

• s1 is our set representing all unique elements from nums1, and s2 for nums2. Now, we use set subtraction to get the elements that are unique to each set:

  • s1 - s2 will give us a new set of elements that are present in s1 (unique to nums1) but not in s2.
  • s2 - s1 will give us a new set of elements that are present in s2 (unique to nums2) but not in s1.

• Finally, these sets are converted back to lists. This is necessary because the problem asks us to return a list of lists; sets wouldn’t be an acceptable return type according to the problem constraints.

• The two lists are then packed into one list and returned. The order in which the lists are returned is as per the problem requirements: the first sublist represents numbers in nums1 not present in nums2, and the second sublist for the opposite.

The algorithm's complexity is beneficial because set operations like the difference are generally O(n), where n is the number of elements in the set. This efficiency comes from the fact that sets are usually implemented as hash tables, allowing for constant-time average performance for add, remove, and check-for-existence operations, which is much faster than trying to perform these operations on an unsorted list.

The code is thus:

class Solution:
    def findDifference(self, nums1: List[int], nums2: List[int]) -> List[List[int]]:
        s1, s2 = set(nums1), set(nums2)
        return [list(s1 - s2), list(s2 - s1)]

Using sets not only makes the code cleaner and more readable but also ensures the operations are done as efficiently as possible for the problem's requirements.

Example Walkthrough

Let's consider the arrays nums1 = [1, 2, 3, 3, 4] and nums2 = [3, 4, 4, 5, 6]. We want to use the solution approach to find the differences between these arrays.

Here is the step-by-step walkthrough of the solution:

1. Convert the nums1 array to a set s1 to remove duplicates and gain unique elements. The conversion results in s1 = {1, 2, 3, 4}.

2. Similarly, convert nums2 array to a set s2 yielding s2 = {3, 4, 5, 6}.

3. Subtract the set s2 from s1 to find elements that are unique to nums1. Performing s1 - s2 gives us {1, 2} since numbers 3 and 4 are present in both sets and are hence not part of the set difference.

4. Do the reverse subtraction, s2 - s1, to find elements unique to nums2. We get {5, 6} because again, 3 and 4 are present in both sets.

5. These set differences are converted back to lists: list(s1 - s2) becomes [1, 2] and list(s2 - s1) becomes [5, 6]. Note that the order of elements in these lists doesn't matter.

6. Finally, return these lists as sublists in a single list: [[1, 2], [5, 6]].

Using this approach ensures the returned lists have no duplicates and only contain elements that are unique to each of the original lists, nums1 and nums2. The final answer for our example input is therefore [[1, 2], [5, 6]].

Solution Implementation

Time and Space Complexity

The given Python code defines a function findDifference that takes two lists of integers, nums1 and nums2, and returns two lists:

• The first list contains all elements that are in nums1 but not in nums2.
• The second list contains all elements that are in nums2 but not in nums1.

To achieve this, the function converts both lists to sets, which are then used to find the difference between them.

Time Complexity

The time complexity of the function is determined by multiple operations:

• Converting nums1 to a set: O(n) where n is the length of nums1.
• Converting nums2 to a set: O(m) where m is the length of nums2.
• Finding the difference s1 - s2: This is O(len(s1)) because it essentially involves checking each element in s1 to see if it is not in s2.
• Finding the difference s2 - s1: Analogously, this is O(len(s2)).

Assuming n is the length of nums1 and m is the length of nums2, the overall time complexity is O(n + m) as the set differences are proportional to the size of the sets.

Space Complexity

The space complexity is also determined by multiple factors:

• The space used by s1: O(n) where n is the number of unique elements in nums1.
• The space used by s2: O(m) where m is the number of unique elements in nums2.
• The space used by the output lists: This largely depends on how many elements are unique to each list after the set difference operation. However, in the worst case (where all elements are unique), this would again be O(n + m).

The overall space complexity is thus O(n + m) where n is the number of unique elements in nums1 and m is the number of unique elements in nums2.

Learn more about how to find time and space complexity quickly using problem constraints.