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Google·Software Engineer·Onsite - Coding / Algorithms·Senior

SeniorPrefer not to say
May 2026

Summary

Coding round for a Gemini team SWE role at Google. One meaty algorithmic problem that looked like a histogram variant but had enough wrinkles to trip you up if you hadn't seen the recursive painter pattern before.

Questions Asked (1)

Q1

Given N adjacent fence planks each with a height, find the minimum number of paint-roller operations to cover the entire fence. Each operation costs 1 and is either a full vertical stroke on one plank, or a horizontal stroke across a contiguous range of planks at some height h (only valid if every plank in that range is at least h tall).

Algorithms & Data StructuresTechnical Trade-offs
Author's notes

I recognized the histogram shape pretty fast but got stuck translating that into the recursion.

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AI HintsAI Generated

Suggested Approach

Recognize this as a classic divide-and-conquer problem (similar to the 'Skyline' or 'Fence Painting' problem). The key insight is that the optimal strategy either uses all vertical strokes (N operations) or uses horizontal strokes at the minimum height across the entire range, then recursively solves the subproblems above that height. Implement a recursive function that computes the minimum operations for any subarray by considering the minimum height and splitting at positions where the height equals that minimum.

Pro tip: During the interview, explicitly discuss the trade-off between the recursive divide-and-conquer approach (O(N) time with careful implementation) and a simpler O(N^2) approach, and mention that you can optimize by using a stack to avoid recursion overhead. Also, clarify that horizontal strokes can be at any height, not just integer heights, but the optimal heights are always the plank heights.

1. Understand the problem and identify the optimal substructure

Restate the problem to ensure clarity: we need to cover all planks with minimum operations, where each operation is either a vertical stroke on one plank or a horizontal stroke across a contiguous range at a height not exceeding the minimum height in that range. Recognize that the problem exhibits optimal substructure: the minimum operations for a range can be computed from the minimum operations of its subranges.

2. Define the recursive strategy

For a given range [l, r], find the minimum height m. One option is to use (r-l+1) vertical strokes. Another option is to use m horizontal strokes (one for each level from 1 to m) across the entire range, plus recursively solve the subranges that are above height m. The answer is the minimum of these options.

3. Implement the recursion with base cases

Write a recursive function solve(l, r, base_height) that computes the minimum operations for the subarray from l to r, assuming that all planks in this range have already been painted up to base_height. The function finds the minimum height in the range, computes the cost of horizontal strokes (min_height - base_height) plus the sum of recursive calls on segments separated by planks of minimum height, and compares it with the cost of vertical strokes (r-l+1).

4. Analyze time and space complexity

The naive recursive approach can be O(N^2) in the worst case (e.g., strictly increasing heights). However, by using a stack-based approach or by precomputing the next smaller element, we can achieve O(N) time. Discuss the trade-offs and choose an approach that balances simplicity and efficiency.

5. Test with edge cases and validate

Test with small cases: N=1, all planks same height, strictly increasing/decreasing heights, and random heights. Verify that the algorithm returns the correct minimum operations. Also, consider if the fence can be painted with horizontal strokes at non-integer heights (it can, but optimal heights are always integers from the set of plank heights).

Key Points to Mention

  • The problem is equivalent to finding the minimum number of strokes to paint a histogram, which is a known problem (e.g., LeetCode 1105 or similar).
  • The optimal strategy often involves using horizontal strokes at the minimum height of a segment, then recursively solving the segments above that height.
  • The base case for recursion is when the range is empty (cost 0) or when the minimum height equals the base height (cost 0 for horizontal, but vertical strokes may still be needed for planks above base height).
  • Time complexity can be improved from O(N^2) to O(N) using a stack to keep track of increasing heights, similar to the largest rectangle in histogram problem.
  • The choice between vertical and horizontal strokes depends on the heights: if the minimum height is large, horizontal strokes may be beneficial; if heights vary widely, vertical strokes may be better.
  • Edge cases: all planks same height (answer is min(N, height)), strictly increasing heights (answer is N), strictly decreasing heights (answer is N).

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