Problem Description

You are provided with two integer arrays, nums1 and nums2, and the goal is to create a data structure that supports two types of queries. Firstly, you should be able to add a positive integer to an element at a given index in nums2. Secondly, you need to count the number of pairs (i, j) where the sum of nums1[i] and nums2[j] is equal to a specified value (tot). The pairs should only be considered if i and j are valid indexes within nums1 and nums2, respectively.

To accomplish the task, you should implement a class FindSumPairs with the following methods:

• FindSumPairs(int[] nums1, int[] nums2): a constructor that initializes the object with the two integer arrays, nums1 and nums2.
• add(int index, int val): a method that adds the integer val to the element at index in nums2.
• count(int tot): a method that returns the number of pairs (i, j) where nums1[i] + nums2[j] equals the integer tot.

Intuition

The intuitive approach to solving this problem involves efficient data retrieval and update methods. If we were to perform a brute force approach for the count operation, we would iterate through all possible (i, j) pairs to find their sum and compare with tot, which would result in a time-consuming process, especially with large arrays.

A better way is to use a hash table to keep track of the frequency of each number in nums2. This allows us to quickly calculate how many times a certain number that can be added to elements of nums1 appears in nums2 to reach the required sum (tot).

Here's the intuition for the methods:

• The constructor initializes the object and also creates a counter (Counter from collections in Python) that maps each number in nums2 to its frequency.
• The add method updates the specific element in nums2 and adjusts the counter correspondingly. If a number's frequency changes (because it's being increased by val), we decrease the count of the old number and increase the count for the new, updated number.
• The count method calculates the required pairs by traversing through nums1 and checking if the complement to reach tot (calculated as tot - nums1[i]) exists in the hash table (counter for nums2). The sum of all the frequencies of these complements gives us the total number of valid pairs that meet the condition.

Solution Approach

The implementation of the FindSumPairs class makes use of hash tables to store elements and their frequencies from nums2. This is essentially a mapping from each unique integer in nums2 to the number of times it appears. The Python Counter class from the collections module is used here for this purpose as it automatically counts the frequency of items in a list.

Here's the breakdown of the implementation:

The __init__ function of the FindSumPairs class initializes two properties, nums1 and nums2, with the corresponding input arrays. Additionally, a Counter object named cnt is created to store the frequency count of elements in nums2.

def __init__(self, nums1: List[int], nums2: List[int]):
    self.nums1 = nums1
    self.nums2 = nums2
    self.cnt = Counter(nums2)

The add function takes in an index and a value val to add to nums2 at the specified index. Before updating nums2[index] with the new value, it decreases the count of the old value in cnt. Then, it increases the count of nums2[index] plus val. After updating the cnt, nums2 is updated with the new value.

def add(self, index: int, val: int) -> None:
    old = self.nums2[index]
    self.cnt[old] -= 1
    if self.cnt[old] == 0:
        del self.cnt[old]  # Remove the entry from the counter if the frequency is zero
    self.cnt[old + val] += 1
    self.nums2[index] += val

The count function takes in a value tot and returns the number of pairs (i, j) where nums1[i] + nums2[j] == tot. To do this, it sums up the counts of tot - nums1[i] in cnt for each element i in nums1. In other words, for every number in nums1, it calculates the complement (the number that needs to be added to it in order to reach tot) and uses this to get the frequency of such complements from the Counter.

def count(self, tot: int) -> int:
    return sum(self.cnt[tot - v] for v in self.nums1)

By utilizing the Counter, the class is able to execute the count function in O(n) time relative to the size of nums1, as it only has to iterate over nums1 and can retrieve the complement counts in constant time from the hash table. This is significantly more efficient than attempting to calculate the pairs through nested loops, which would be O(n * m) where n and m are the sizes of nums1 and nums2.

Example Walkthrough

Let's step through the problem using a small example to illustrate the solution approach.

Suppose nums1 is [1, 2, 3] and nums2 is [1, 4, 5, 2], and we are interested in finding pairs whose sum equals tot = 5.

Upon class initialization:

1. nums1 remains [1, 2, 3].
2. nums2 remains [1, 4, 5, 2].
3. A Counter is created from nums2, resulting in {1: 1, 4: 1, 5: 1, 2: 1}.

Now, let's say we perform an add operation – add(2, 2) – which adds 2 to the element at index 2 in nums2.

1. The old value at nums2[2] is 5, so the counter for 5 is decreased from 1 to 0 and consequently removed since its count is now zero.
2. The value at nums2[2] is updated to 7 (5 + 2), so the counter for 7 is increased from 0 to 1.
3. Now nums2 becomes [1, 4, 7, 2], and the Counter reflects {1: 1, 4: 1, 7: 1, 2: 1}.

Next, we perform a count operation – count(5) to find pairs summing up to 5.

1. We iterate through nums1, looking for values that, when added to values from nums2, yield 5.
2. For nums1[0] = 1, we find tot - nums1[0] = 5 - 1 = 4; cnt[4] equals 1 indicating a pair (1, 4).
3. For nums1[1] = 2, tot - nums1[1] = 5 - 2 = 3; cnt[3] equals 0 indicating no valid pair exists with the second element 2.
4. For nums1[2] = 3, tot - nums1[2] = 5 - 3 = 2; cnt[2] equals 1 indicating a pair (3, 2).
5. Sum up the counts of valid pairs for all elements in nums1, giving us cnt[4] + cnt[3] + cnt[2] = 1 + 0 + 1 = 2.

Therefore, the count(5) operation tells us there are 2 valid pairs (1, 4) and (3, 2) whose sum equals 5.

This example demonstrates the efficiency of the approach using a Counter to avoid iterating over nums2 each time we call count. Instead, we make use of the precomputed frequencies to quickly tally the number of pairs that sum up to tot.

Solution Implementation

Time and Space Complexity

Time Complexity

• __init__(self, nums1: List[int], nums2: List[int]): The constructor initializes two lists nums1 and nums2. It also counts the elements of nums2 using Counter which takes O(n) time where n is the length of nums2.

• add(self, index: int, val: int) -> None: This method updates an element in nums2 and modifies the count of the old and the new value in cnt. The update operation and the changes in the counter have a constant time complexity O(1) since dictionary (counter) operations in Python have an average-case complexity of O(1).

• count(self, tot: int) -> int: The count method iterates over nums1 and for each element v in nums1, it accesses the counter cnt to find the count of (tot - v). If nums1 has m elements, the time complexity is O(m) since each lookup in the counter is O(1) on average.

Space Complexity

• __init__(self, nums1: List[int], nums2: List[int]): Apart from the input lists nums1 and nums2, a counter cnt is created to store the frequency of each element in nums2. The space complexity is O(n) where n is the number of unique elements in nums2.

• add(self, index: int, val: int) -> None: This method uses no extra space and thus has a space complexity of O(1).

• count(self, tot: int) -> int: This method does not use extra space apart from a few variables for its operation, hence the space complexity is O(1).

Overall, the space complexity of the FindSumPairs class is determined by the space required for storing the input lists and the counter, which is O(n) where n is the size of nums2 and the number of unique elements it contains.

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