How to solve a Google coding interview question

Mock Technical Interview at Google: Summary

Key Concepts:

• Matrix representation of land (1s for good land, 0s for bad land)
• Finding the maximum area square of good land
• Brute force approach vs. optimized solutions
• Recursion with memoization
• Dynamic programming (bottom-up approach)
• Space-time complexity analysis

1. Problem Definition and Clarification:

• The problem is to find the largest square area of "good land" (represented by 1s) within a matrix of 1s and 0s.
• The candidate, Sami, clarifies that the desired area must be a square, not a rectangle.
• Juliana, the interviewer, confirms this requirement.

2. Naive (Brute Force) Solution:

• Sami initially proposes a brute-force approach:
  • Iterate through every position in the matrix.
  • For each position, attempt to expand a square outwards, checking if all cells within the square are 1s.
  • Keep track of the maximum square size found.
• Sami acknowledges that this approach has a time complexity of O(n^4) (where n is the dimension of the square), making it inefficient.
• Juliana asks Sami to visualize the brute force approach using the example provided.
• Sami explains how, starting from a given index [i][j], the algorithm would check for 1x1, 2x2, 3x3 squares, and so on, by expanding the boundary and verifying that all new cells are 1s.

3. Recursive Solution:

• Sami suggests a recursive approach:
  • If a cell is 0, return 0 (no square possible).
  • If a cell is 1, recursively check the size of the largest square that can be formed to the right, below, and diagonally.
  • The size of the square at the current cell is determined by the minimum of the sizes of the squares to the right, below, and diagonally, plus 1.
• Juliana guides Sami to consider the implications of a zero value in the neighboring cells.
• Sami explains that the recursion would involve asking the neighbors how many contiguous ones they have, forming a square.
• The base cases for the recursion are:
  • If a cell is 0, return 0.
  • If a cell is 1, recursively call the function for the right and bottom neighbors and return their minimum.
• Sami notes that the recursive solution can be optimized using memoization to avoid redundant calculations.
• Memoization would reduce the time complexity to at most O(n^2).

4. Dynamic Programming (Bottom-Up) Solution:

• Sami proposes a dynamic programming solution:
  • Create a DP matrix of the same size as the input matrix, initialized with zeros.
  • Iterate through the input matrix.
  • If a cell in the input matrix is 0, set the corresponding cell in the DP matrix to 0.
  • If a cell in the input matrix is 1, set the corresponding cell in the DP matrix to the minimum of the values in the DP matrix to the top, left, and diagonal-left, plus 1.
  • The final result is the maximum value in the DP matrix.
• Juliana points out that the DP array will contain the height of the square instead of the area.
• Sami confirms that the DP array will be populated incrementally, looking at the surrounding positions in both the input and DP arrays.

5. Coding the Dynamic Programming Solution:

• Sami starts coding the dynamic programming solution in Python.
• He defines a function largest_square that takes a 2D array of integers (bin_array) as input and returns an integer representing the maximum area.
• He initializes a DP array with all zeros, considering the dimensions of the input array (n x m).
• He iterates through the input array using nested for loops.
• If an entry in the input array is zero, the corresponding entry in the DP array remains zero (using continue).
• If an entry in the input array is one, the algorithm calculates the values of the left, top, and diagonal neighbors in the DP array, handling boundary conditions using boolean expressions.
• The DP array is updated with the minimum of the left, top, and diagonal values plus one.
• Sami initially uses separate if statements to handle boundary conditions but then simplifies the code using boolean multiplication.
• He tracks the maximum value encountered in the DP array using a variable max_num and updates it within the for loop.
• Finally, the function returns the max_num.

6. Code Refinement and Optimization:

• Juliana suggests handling boundary conditions before entering the main loop.
• She also points out that checking if all neighboring values are greater than zero is unnecessary.
• Sami removes the redundant if statement, simplifying the code.
• Juliana suggests tracking the maximum within the for loop instead of calculating it separately at the end.
• Sami implements this optimization, further improving the code's efficiency.

7. Final Solution and Conclusion:

• Sami presents the final dynamic programming solution, which efficiently finds the maximum square area of good land.
• The solution has a time complexity of O(n*m), where n and m are the dimensions of the input matrix.
• The interviewer, Juliana, confirms that the solution is correct and well-explained.
• Sami acknowledges Juliana's guidance and helpful corrections throughout the interview.

Main Takeaways:

• Clearly communicate your thought process during technical interviews.
• Don't hesitate to ask clarifying questions to ensure you understand the problem correctly.
• Start with a naive solution and then work towards a more optimized approach.
• Consider different algorithmic approaches (e.g., recursion, dynamic programming).
• Pay attention to boundary conditions and potential optimizations.
• Be receptive to feedback and guidance from the interviewer.