Morgan Stanley·Software Engineer·Technical Phone Screen·Intermediate

Intermediate

Jun 2026

Summary

Morgan Stanley software engineer interview with a pretty meaty algorithms problem that had more layers to it than I expected. The question looked like a basic array problem but pulling apart the optimal strategy took a while to articulate clearly.

Questions Asked (1)

Given an array of positive integers, you can repeatedly pick any two numbers, pay a cost equal to their sum, and replace them with that sum until one number remains. What algorithm minimizes the total cost, why is it optimal, what are the time and space complexities, and can you implement it in C++? Also handle edge cases like a single element, duplicates, and very large integers.

Algorithms & Data StructuresTechnical Trade-offs

Author's notes

I knew a min-heap was involved pretty quickly but stumbled when asked to actually prove why greedy works here.

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AI HintsAI Generated

Suggested Approach

Recognize this as the optimal merge pattern problem and propose a greedy algorithm using a min-heap: repeatedly extract the two smallest numbers, sum them, add the sum to the total cost, and insert the sum back. Explain that this is optimal because merging the smallest elements first minimizes the contribution of larger elements to subsequent sums, analogous to Huffman coding. Then analyze time and space complexity and provide a C++ implementation with edge case handling.

Pro tip: Mention that this is equivalent to building a Huffman tree and that the total cost equals the weighted external path length. Also, proactively discuss integer overflow and suggest using 64-bit integers or arbitrary-precision libraries for very large inputs.

1. Identify the problem and optimal strategy

State that the problem is to minimize the total cost of merging numbers, and the optimal strategy is a greedy approach using a min-heap to always merge the two smallest elements.

2. Explain why greedy is optimal

Argue that merging the smallest elements first ensures that larger elements are added fewer times in subsequent merges, minimizing the overall sum. This is a classic exchange argument or can be related to Huffman coding optimality.

3. Analyze time and space complexity

With a min-heap, each extraction and insertion takes O(log n), and we perform n-1 merges, leading to O(n log n) time. Space is O(n) for the heap.

4. Implement in C++ with edge cases

Write a function using std::priority_queue with greater<int> as min-heap. Handle edge cases: if array size <= 1, return 0; use long long to avoid overflow; discuss duplicates (handled naturally) and very large integers (suggest big integers if needed).

5. Test and verify with examples

Walk through a small example (e.g., [1,2,3,4]) to demonstrate the algorithm and verify the total cost. Mention that the algorithm works for any positive integers.

Key Points to Mention

Greedy algorithm using a min-heap (priority queue) to always merge the two smallest elements.
Optimality proof: exchange argument or equivalence to Huffman coding; merging smallest first minimizes total cost.
Time complexity: O(n log n) due to heap operations; space complexity: O(n).
C++ implementation using std::priority_queue with std::greater<long long> for min-heap.
Edge cases: single element (cost 0), duplicates (handled naturally), very large integers (use long long or big integers to avoid overflow).
Connection to Huffman coding and weighted external path length.

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.