Identify the Largest Outlier in an Array

Identify the Largest Outlier in an Array

What is this problem about?

In the Identify the Largest Outlier in an Array interview question, you are given an array of $n$ integers. You are told that $n-2$ elements are "data" elements, one element is the sum of all those data elements, and the remaining element is an "outlier." Your goal is to find the largest possible value for the outlier.

Why is this asked in interviews?

Companies like Goldman Sachs and Google ask this to test your ability to derive mathematical relationships from word problems. It evaluation your mastery of Hash Table interview patterns and linear time search. The problem requires you to transform the description into a single equation that can be checked efficiently for every element in the array.

Algorithmic pattern used

This problem relies on Mathematical Derivation and Frequency Counting.

1. Let $S$ be the total sum of all elements in the array.
2. Let $O$ be the outlier and $D$ be the sum of the $n-2$ data elements.
3. We know $S = D + O + ( ext{the element that is the sum of data elements})$.
4. Since the "element that is the sum of data elements" is itself $D$, the equation is: $S = D + O + D o S = 2D + O$.
5. This means $O = S - 2D$, or more usefully, $2D = S - O$.
6. For each element in the array, assume it is the outlier ($O$). Calculate the potential sum $D = (S - O) / 2$.
7. Check if $D$ exists in the array (being careful if $D$ and $O$ are the same element).

Example explanation

Array: [2, 3, 5, 100]. Total Sum $S = 110$.

1. Assume 100 is the outlier. $D = (110 - 100) / 2 = 5$.
2. Does 5 exist in the array? Yes.
3. So 100 is a valid outlier.
4. Assume 5 is the outlier. $D = (110 - 5) / 2 = 52.5$. (Not an integer, ignore). The largest valid outlier is 100.

Common mistakes candidates make

• Brute Force: Trying to pick every pair of elements as (Outlier, Sum) which leads to $O(N^2)$.
• Parity: Forgetting that $(S - O)$ must be even for $D$ to be an integer.
• Frequency Logic: Not correctly checking if $D$ is in the array when $D$ and $O$ have the same value (you need at least two instances of that value).

Interview preparation tip

Always calculate the global sum of an array first in problems involving "sub-sums" or "excluded elements." It usually reduces the complexity from $O(N^2)$ to $O(N)$ by allowing you to calculate the "remainder" in constant time.