Finding Pairs With a Certain Sum

Finding Pairs With a Certain Sum

1. What is this problem about?

The Finding Pairs With a Certain Sum interview question asks you to design a data structure that manages two arrays, nums1 and nums2. You need to support two operations: add(index, val), which adds a value to an element in the second array, and count(tot), which returns the number of pairs $(i, j)$ such that nums1[i] + nums2[j] == tot.

2. Why is this asked in interviews?

Companies like Meta and Amazon use the Finding Pairs coding problem to test a candidate's ability to optimize for the more frequent operation. In most cases, nums1 is much smaller than nums2. The goal is to realize that maintaining a frequency map for the larger array makes the search for pairs extremely efficient. It evaluation your Hash Table interview pattern skills.

3. Algorithmic pattern used

This problem follows the Frequency Map Optimization pattern.

1. Storage: Store nums1 as-is. Store nums2 in a Hash Map freq2 where the key is the number and the value is its frequency.
2. add Operation:
   • Update the value in the array nums2.
   • Update the freq2 map by decrementing the count of the old value and incrementing the count of the new value.
3. count Operation:
   • Iterate through each number $x$ in nums1.
   • The required value from nums2 is target = tot - x.
   • Add freq2[target] to your result.

4. Example explanation

nums1 = [1, 1], nums2 = [2, 3]. freq2 = {2:1, 3:1}.

• count(4):
  • For first 1: need 3. freq2[3] = 1.
  • For second 1: need 3. freq2[3] = 1.
  • Total: 2 pairs.
• add(1, 1): nums2 becomes [2, 4]. freq2 = {2:1, 4:1}.
• count(4):
  • For first 1: need 3. freq2[3] = 0.
  • ... Total: 0.

5. Common mistakes candidates make

• Nested Loops: Using $O(N \cdot M)$ for every count operation, which is too slow.
• Inefficient Add: Re-building the entire frequency map for nums2 every time a single element changes.
• Ignoring Array Sizes: Failing to iterate over the smaller array during the count operation.

6. Interview preparation tip

Always identify which dataset is "static" or "smaller." Pre-processing the larger or more dynamic dataset into a Hash Map is a standard way to solve "Two-Sum" style problems in a Design interview pattern.