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Microsoft·Software Engineer·Technical Phone Screen·Intermediate

Intermediate
Apr 2026

Summary

Microsoft SWE interview with a pretty gnarly combinatorics/DP problem. Not a lot of fluff, they went straight into the technical question and wanted both a working algorithm and a complexity analysis.

Questions Asked (1)

Q1

Given an array of required digit sums of length n, count how many non-decreasing integer arrays of length n exist such that each element has the specified digit sum, each element is at most 5000, and the array is non-decreasing. Return the count modulo 1,000,000,007. Describe an efficient algorithm and analyze its time and space complexity.

Algorithms & Data StructuresTechnical Trade-offs
Author's notes

Took me a while to even parse what was being asked.

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AI HintsAI Generated

Suggested Approach

First, clarify the problem constraints and edge cases, then propose a dynamic programming solution that counts valid non-decreasing sequences by processing elements in order and tracking the last value. Optimize using prefix sums and precomputed counts of numbers with each digit sum up to 5000, and analyze the time and space complexity.

Pro tip: Demonstrate awareness of potential pitfalls: the non-decreasing constraint can be handled by iterating values in increasing order and using prefix sums, but be careful with modulo arithmetic and memory limits. Also, mention that precomputing digit sums for all numbers up to 5000 is trivial and can be done once.

1. Clarify the problem and constraints

Restate the problem to ensure understanding: count non-decreasing arrays of length n where each element has a given digit sum and is ≤5000. Discuss edge cases like n=0, impossible digit sums, and modulo requirements.

2. Precompute valid numbers by digit sum

For each possible digit sum s (0 to 36), precompute a sorted list of all numbers ≤5000 with that digit sum. This allows quick access to possible values for each position.

3. Define DP state and transition

Let dp[i][v] be the number of valid non-decreasing sequences of length i ending with value v. Transition: dp[i][v] = sum_{u ≤ v} dp[i-1][u] for v having the required digit sum. Use prefix sums over v to compute efficiently.

4. Optimize with prefix sums and rolling array

Since n can be large, use a rolling array for the previous DP state and compute prefix sums to achieve O(n * 5000) time. Space can be reduced to O(5000) by keeping only the current and previous DP arrays.

5. Analyze complexity and discuss trade-offs

Time complexity: O(n * 5000) due to iterating over positions and values, with prefix sum optimization. Space: O(5000) for DP arrays. Mention that precomputation takes O(5000) time and space. Discuss potential improvements if n is very large.

Key Points to Mention

  • Dynamic programming with state (position, last value) and transition using prefix sums.
  • Precomputation of numbers by digit sum up to 5000 to quickly filter valid values.
  • Modulo arithmetic to handle large counts (mod 1,000,000,007).
  • Time complexity O(n * 5000) and space O(5000) with rolling arrays.
  • Edge cases: n=0, digit sums that cannot be formed by numbers ≤5000, and non-decreasing constraint.
  • Potential optimization: if n is very large, consider matrix exponentiation or other techniques, but DP is sufficient for typical constraints.

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