Minimum Cost to Make Array Equal

Minimum Cost to Make Array Equal

1. What is this problem about?

The Minimum Cost to Make Array Equal is a sophisticated optimization problem where you are given two arrays: one containing integers (the elements) and another containing costs. The cost array specifies how much it costs to increase or decrease the corresponding element by 1. The goal is to make all elements in the array equal to some value $X$, such that the total cost is minimized. Since $X$ can be any value present in the array (or even between them), this problem requires finding a "weighted center" that balances the costs associated with moving all other elements to that point.

2. Why is this asked in interviews?

Companies like Google, Microsoft, and Oracle ask the Minimum Cost to Make Array Equal interview question because it tests a candidate's knowledge of mathematical properties and their ability to apply advanced search techniques. It bridges the gap between basic median properties (which solve the problem if all costs are 1) and more complex scenarios where costs vary. This problem evaluates whether you can recognize that the cost function is convex, allowing for optimizations like binary search or ternary search, which are crucial for high-performance computing.

3. Algorithmic pattern used

This problem can be solved using several patterns, most notably Binary Search on the answer or the Weighted Median approach. Because the total cost function is convex (it looks like a U-shape), we can binary search over the possible range of values for $X$. For a chosen value, we calculate the cost; if the cost of $X+1$ is less, we move right; otherwise, we move left. Alternatively, sorting the elements and finding the weighted median—the point where the cumulative cost reaches half the total cost—provides the optimal $X$. This "Array, Sorting, Binary Search, Greedy, Prefix Sum interview pattern" is a powerful toolset for handling cost minimization in linear arrays.

4. Example explanation

Suppose we have: Elements: [1, 3, 5] Costs: [2, 1, 4]

Total cost to move everything to 3:

• Move 1 to 3: $|1-3| * 2 = 4$
• Move 3 to 3: $|3-3| * 1 = 0$
• Move 5 to 3: $|5-3| * 4 = 8$
• Total = 12.

Total cost to move everything to 4:

• Move 1 to 4: $|1-4| * 2 = 6$
• Move 3 to 4: $|3-4| * 1 = 1$
• Move 5 to 4: $|5-4| * 4 = 4$
• Total = 11.

Total cost to move everything to 5:

• Move 1 to 5: $|1-5| * 2 = 8$
• Move 3 to 5: $|3-5| * 1 = 2$
• Move 5 to 5: $|5-5| * 4 = 0$
• Total = 10. In this case, 5 is the optimal target.

5. Common mistakes candidates make

A common pitfall in the Minimum Cost to Make Array Equal coding problem is assuming that the target value must be the simple average of the numbers. The average only minimizes the sum of squared distances, not absolute distances. Another mistake is failing to handle the costs correctly, treating it like a standard median problem without weighting. Candidates also often write an $O(N^2)$ solution by checking every possible value in the array, which will fail for large datasets. Understanding the convexity of the cost function is key to avoiding these inefficient approaches.

6. Interview preparation tip

When you see a problem asking to minimize a sum of absolute differences multiplied by weights, immediately think of the weighted median. If the weights are equal, it's just the median. If you can't recall the weighted median property, remember that binary searching on the target value is a robust alternative for any convex cost function. Practice implementing binary search on continuous or discrete ranges, as it is a versatile "Binary Search interview pattern" that appears in many hard-level algorithmic challenges.