Identify the Largest Outlier in an Array - Video Solutions

Problem Overview: You receive an integer array where all elements except one form a special structure: among the remaining numbers, one value equals the sum of several others. The task is to identify the largest number that could be the outlier while the rest of the array still satisfies this condition.

The key observation comes from simple algebra. If o is the outlier and y is the element equal to the sum of the special numbers, then:

totalSum - o - y = y

Rearranging gives y = (totalSum - o) / 2. For any candidate outlier o, the array must contain a valid y. Efficient solutions focus on testing this relationship quickly.

Approach 1: Sorting and Iterative Check (O(n log n) time, O(1) extra space)

Sort the array first, then iterate from the largest value toward the smallest while treating each element as a possible outlier. For each candidate o, compute (totalSum - o) / 2 and check whether that value exists elsewhere in the array. Sorting helps you evaluate larger outliers first and use binary search or pointer checks for existence. The approach is straightforward and works well when the array size is moderate.

This solution relies mainly on Array manipulation and enumeration of candidates. The downside is the O(n log n) cost from sorting.

Approach 2: Hashing for Frequency and Sum (O(n) time, O(n) space)

Use a frequency map to store counts of every value in the array. Compute the total sum once. Then iterate through the array and treat each number as the potential outlier o. Calculate remaining = totalSum - o. If remaining is even, the required element must be y = remaining / 2.

Check the hash map to see if y exists among the remaining elements. Frequency handling matters: if y equals the candidate outlier, ensure its count is greater than one before accepting it. Track the largest valid outlier while scanning the array. Hash lookups keep the validation constant time.

This method uses a Hash Table for frequency counting and efficient lookups. The algorithm performs a single pass after building the map, giving an optimal O(n) time solution with O(n) space.

Recommended for interviews: Start by explaining the equation derived from the sum relationship. A brute reasoning step shows you understand the structure of the problem. Then move to the hash map solution, which interviewers typically expect because it reduces the search to constant-time lookups using Counting and frequency tracking. The O(n) hashing approach is both clean and optimal.