Divide an Array Into Subarrays With Minimum Cost I

Divide an Array Into Subarrays With Minimum Cost I

What is this problem about?

The Divide an Array Into Subarrays With Minimum Cost I coding problem asks you to divide an array into 3 contiguous subarrays. The "cost" of a subarray is its first element. You want to minimize the total cost, which is the sum of the first elements of these 3 subarrays. Crucially, the first subarray must start at index 0.

Why is this asked in interviews?

American Express and other financial firms use this question to test basic Greedy interview patterns and sorting logic. It evaluation your ability to realize that since the first element is fixed (index 0), the problem simplifies to finding the two smallest elements in the rest of the array to serve as the "heads" of the remaining two subarrays.

Algorithmic pattern used

This problem uses a Greedy approach with Sorting (or finding the top two minimums).

1. The first subarray's cost is always nums[0].
2. You need to pick two more indices $i$ and $j$ ($0 < i < j < n$) to start the second and third subarrays.
3. To minimize the sum nums[0] + nums[i] + nums[j], you simply need to find the two smallest values in the array nums[1...n-1].

Example explanation

nums = [10, 3, 1, 15, 2]

1. Fixed cost: nums[0] = 10.
2. Remaining elements: [3, 1, 15, 2].
3. The two smallest elements are 1 and 2.
4. Total cost: $10 + 1 + 2 = 13$.

Common mistakes candidates make

• Brute Force: Trying all combinations of $i$ and $j$ ($O(n^2)$), which is unnecessary.
• Ignoring the fixed index: Thinking they can pick any three elements, forgetting that one must be the very first element of the array.
• Subarray constraints: Forgetting that each subarray must be non-empty, which is naturally satisfied if you pick two distinct indices from the remaining elements.

Interview preparation tip

Read carefully! In "minimum cost" problems, check if any part of the cost is "fixed." If it is, focus entirely on optimizing the remaining variable parts. Finding the two smallest numbers in an array is an $O(n)$ operation using two variables, or $O(n \log n)$ using sorting.