Problem Description

The problem provides us with four integer arrays nums1, nums2, nums3, and nums4, each of the same length n. Your task is to find the number of quadruplets (i, j, k, l) such that the sum of the elements at these indices from each array equals zero, that is nums1[i] + nums2[j] + nums3[k] + nums4[l] == 0. The indices i, j, k, l vary from 0 to n-1.

Intuition

To find quadruplets that sum up to zero, a naive approach would involve checking all possible combinations of indices (i, j, k, l) from the four arrays, which leads to a solution with a time complexity of O(n^4). However, this isn't efficient when n is large.

Instead, we can first consider pairs of elements from nums1 and nums2. We record the possible sums that these pairs can make and the frequency of each sum. A Counter in Python can be used effectively for this purpose.

Then, we can look for the pairs of elements from nums3 and nums4 that, when added together, create a sum that negates the sum we obtained from nums1 and nums2. This way, the total sum will be zero. We find these complementary sums by iterating through all combinations of nums3 and nums4 and checking if the negated sum is present in our Counter (which is populated by sums of nums1 and nums2).

Using this method significantly reduces the time complexity because we only consider two loops of n^2 operations each, hence O(n^2). Furthermore, since we are storing the sum and frequency in a Counter (which is a hashmap, effectively), our lookups for the complementary sum are done in constant time.

Solution Approach

The implementation of the solution uses a two-step process that takes advantage of the Counter class in Python, which is a type of dictionary designed for counting hashable objects, a subclass of dict. Here's a breakdown of the algorithm:

Step 1: Compute Pair Sums from nums1 and nums2

Firstly, we iterate over all pairs of elements (a, b) where a is from nums1 and b is from nums2. We calculate their sum and store this in a Counter dictionary, which will hold the sum as keys and the frequency of these sums as values. This dictionary will capture all possible pairwise sums along with how many times each sum occurs.

cnt = Counter(a + b for a in nums1 for b in nums2)

Step 2: Find Complementary Sums from nums3 and nums4

Next, for each combination of elements (c, d) from nums3 and nums4, we calculate their sum and look for the complement of this sum in our previously populated Counter (specifically, we're looking for -(c + d)). If found, it means there exists at least one pair from nums1 and nums2 that can be added to this nums3 and nums4 pair to make the total sum zero. The frequency stored in the Counter for -(c + d) represents how many such pairs we have from nums1 and nums2.

return sum(cnt[-(c + d)] for c in nums3 for d in nums4)

By iterating over all pairs from nums3 and nums4 and summing up the counts from the Counter, we get the total number of valid quadruplets that meet our condition.

This approach effectively reduces a complex problem into a simpler one that requires fewer computations by dividing it into two parts. It utilizes the power of hashmaps to speed up lookups for complementary pairs, ensuring an overall time complexity of O(n^2) which is a significant improvement over the brute force approach with O(n^4). Also, because we are storing up to n^2 pairs in the Counter, the space complexity is also O(n^2).

class Solution:
    def fourSumCount(self, nums1: List[int], nums2: List[int], nums3: List[int], nums4: List[int]) -> int:
        cnt = Counter(a + b for a in nums1 for b in nums2)
        return sum(cnt[-(c + d)] for c in nums3 for d in nums4)

Notice how the implementation is cogent and does not require nested loops over all four arrays, significantly enhancing efficiency.

Example Walkthrough

Let's go through a small example using the provided solution approach to understand how it works in practice. We will take four arrays with a smaller length for simplicity.

Suppose nums1 = [1, -1], nums2 = [-1, 1], nums3 = [0, 1], and nums4 = [1, -1], and we need to find the number of quadruplets such that nums1[i] + nums2[j] + nums3[k] + nums4[l] == 0.

Step 1: Compute Pair Sums from nums1 and nums2

Firstly, we calculate all possible sums of pairs (a, b) where a is from nums1 and b is from nums2, and store these sums in the Counter:

• Pair (1, -1): sum is 0
• Pair (1, 1): sum is 2
• Pair (-1, -1): sum is -2
• Pair (-1, 1): sum is 0

Thus, the Counter after step 1 will look like this: {0: 2, 2: 1, -2: 1}, which tells us that the sum 0 appears twice and the sums 2 and -2 appear once.

Step 2: Find Complementary Sums from nums3 and nums4

Now, we need to find pairs (c, d) where c is from nums3 and d is from nums4 such that - (c + d) is in the Counter.

• Pair (0, 1): sum is 1 and its complement -1 is not in the Counter, so this pair does not contribute.
• Pair (0, -1): sum is -1 and its complement 1 is not in the Counter, so no contribution from this pair either.
• Pair (1, 1): sum is 2 and its complement -2 is in the Counter and it appears once, so we have one quadruplet: (1,1,1,1).
• Pair (1, -1): sum is 0 and its complement 0 is in the Counter appearing twice, contributing two quadruplets: (1,-1,1,-1) and (-1,1,1,-1).

By iterating over the last two arrays and summing up the counts for each valid complement in the Counter, we find a total of 3 quadruplets.

Hence, for the given small example, the fourSumCount method would return 3. This illustrates that even for small arrays, the algorithm effectively combines pairs from the first two and last two arrays, utilizing the Counter to manage and efficiently count the complementary pairs, achieving a significant performance gain compared to the brute force approach.

Solution Implementation

Time and Space Complexity

The given Python code is designed to find the number of tuples (i, j, k, l) such that nums1[i] + nums2[j] + nums3[k] + nums4[l] is zero.

Time Complexity:

The time complexity is analyzed based on the number of operations performed by the code:

• The first part of the code creates a Counter object to count the frequency of sums of pairs taken from nums1 and nums2. This involves iterating over each element in nums1 and nums2, which, if the length of the lists is n, results in n * n or n^2 operations.

• The second part computes the sum of frequencies of the complement of each possible sum in nums3 and nums4 that makes the total sum zero. This also requires n * n or n^2 operations.

Combining these two, we get a total of 2 * n^2 operations, but since constants are neglected in Big O notation, the time complexity of the code is O(n^2).

Space Complexity:

The space complexity is considered based on the additional memory used by the program:

• The Counter object holds at most n^2 entries, corresponding to each unique sum of elements from nums1 and nums2.

• No other significant extra space is used for computation since the second sum is computed iteratively and sums are not stored.

Hence, the space complexity of the code is O(n^2).

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