Minimum Operations to Make a Uni-Value Grid

Minimum Operations to Make a Uni-Value Grid

1. What is this problem about?

The Minimum Operations to Make a Uni-Value Grid problem involves transforming all elements of a 2D matrix into a single, uniform value. In one operation, you can add or subtract a specific integer x from any element in the grid. The goal is to determine the minimum total number of operations required to make all elements equal, or to identify if such a transformation is impossible. It combines elements of matrix traversal, modular arithmetic, and statistical properties.

2. Why is this asked in interviews?

Companies like Meta and Google ask this question to evaluate a candidate's ability to apply mathematical concepts to algorithmic problems. It tests your understanding of the "Median" property: in a set of numbers, the value that minimizes the sum of absolute differences to all other points is the median. It also requires a solid grasp of the properties of remainders, as all elements must share the same remainder when divided by x for a solution to exist.

3. Algorithmic pattern used

The algorithmic pattern involves Sorting and Math. First, you must check if a solution is possible by ensuring that the difference between every element and the first element is divisible by x. If not, return -1. Next, flatten the 2D grid into a 1D array and sort it. The optimal value to which all elements should be converted is the median of this sorted array. Finally, iterate through the array and sum up the absolute differences between each element and the median, dividing each difference by x.

4. Example explanation

Consider a 2x2 grid [[2, 4], [6, 8]] and x = 2.

• Check feasibility: All elements (2, 4, 6, 8) have the same remainder when divided by 2 (all are even). So it's possible.
• Flatten and sort: [2, 4, 6, 8].
• Find median: The median can be 4 or 6. Let's pick 4.
• Calculate operations:
  • |2 - 4| / 2 = 1
  • |4 - 4| / 2 = 0
  • |6 - 4| / 2 = 1
  • |8 - 4| / 2 = 2
• Total operations: 1 + 0 + 1 + 2 = 4.

5. Common mistakes candidates make

Candidates often try to use the average (mean) instead of the median. While the mean minimizes the sum of squared differences, the median minimizes the sum of absolute differences, which is what "minimum operations" usually implies in linear distance problems. Another common error is forgetting to check the divisibility condition first, leading to incorrect calculations when elements cannot actually be made equal using the given step size x.

6. Interview preparation tip

Understand the difference between mean and median in the context of optimization. Whenever you see a problem asking to minimize the sum of distances to a point, your first thought should be the median. Also, practice handling 2D matrices by flattening them into 1D arrays when the spatial relationship between elements (neighbors) doesn't matter for the final calculation.