Find the Integer Added to Array II

Find the Integer Added to Array II

1. What is this problem about?

The Find the Integer Added to Array II coding problem is a more complex variation of the "array shift" challenge. Here, you have two arrays, nums1 and nums2. nums2 was formed by removing exactly two elements from nums1 and then adding a constant integer $x$ to every remaining element. You need to find the minimum possible value of $x$ that satisfies this condition.

2. Why is this asked in interviews?

Meta and other high-growth companies ask this Sorting and Two Pointers interview question to see if a candidate can handle "noise" in data. The removal of two elements adds a layer of complexity—you can no longer rely on simple sums or minimums. It tests your ability to enumerate possibilities and validate them efficiently, which is a key skill in debugging and data reconciliation.

3. Algorithmic pattern used

The primary topics interview pattern is Sorting and Enumeration with Two Pointers. By sorting both arrays, you can limit the number of candidates for $x$. Since only two elements were removed from nums1, the minimum element of nums2 must have come from one of the first three smallest elements of nums1. You can try these three possibilities for $x$ and, for each, use a two-pointer approach to check if nums2 can be formed by adding $x$ to a subset of nums1.

4. Example explanation

nums1 = [10, 5, 2, 8] nums2 = [5, 11] Sorted nums1: [2, 5, 8, 10] Sorted nums2: [5, 11]

Possible candidates for $x$ (using nums2[0] - nums1[0, 1, or 2]):

1. $x = 5 - 2 = 3$. Check if [2+3, 5+3, 8+3, 10+3] contains [5, 11]. It contains 5, 8, 11, 13. [5, 11] is a subset. $x=3$ is valid.
2. $x = 5 - 5 = 0$. Check if [2, 5, 8, 10] contains [5, 11]. No.
3. $x = 5 - 8 = -3$. Check if [-1, 2, 5, 7] contains [5, 11]. No. The minimum valid $x$ is 3.

5. Common mistakes candidates make

• Trying all combinations: There are $O(N^2)$ ways to remove two elements. Trying all of them is too slow. Enumerating the first three possible $x$ values is much faster.
• Two-Pointer Logic Errors: Failing to correctly skip elements in nums1 when they don't match the current target in nums2.
• Not Sorting: Without sorting, the two-pointer approach and the limited enumeration of $x$ become much more difficult to implement.

6. Interview preparation tip

When a problem says "all elements but $K$ follow a rule," think about how those $K$ removals affect the boundaries (min/max). In this case, $K=2$, so the original minimum must be one of the three smallest in the new array. This kind of "constrained search" is a powerful technique for optimizing problems with a small number of outliers.