Rotate Function

Rotate Function

What is this problem about?

The Rotate Function interview question defines F(k) as the sum of i * nums[(i + k) % n] for all indices i, where nums is a given array. You must find the maximum value of F(k) over all valid values of k (from 0 to n-1). The key challenge is computing all n values of F(k) efficiently rather than recomputing each from scratch.

Why is this asked in interviews?

Amazon and Google ask this problem because it tests the ability to derive a mathematical recurrence between consecutive F(k) values, eliminating the need to recompute each F(k) from scratch. It combines mathematical reasoning with dynamic programming and array manipulation — skills relevant to signal processing, rotation optimization, and scheduling problems.

Algorithmic pattern used

The pattern is mathematical recurrence with prefix sums. Observe that F(k) - F(k-1) = sum(nums) - n * nums[n-k]. This is derived by noting that rotating by one step adds the array sum once and subtracts n times the element that "wraps around." Compute F(0) directly, then compute each subsequent F(k) using the recurrence: F(k) = F(k-1) + total_sum - n * nums[n-k]. Track the maximum as you go.

Example explanation

nums = [4, 3, 2, 6], n = 4, total = 15.

F(0) = 0×4 + 1×3 + 2×2 + 3×6 = 0 + 3 + 4 + 18 = 25.

Recurrence: F(k) = F(k-1) + 15 - 4 × nums[n-k].

• F(1) = 25 + 15 - 4×6 = 25 + 15 - 24 = 16.
• F(2) = 16 + 15 - 4×2 = 16 + 15 - 8 = 23.
• F(3) = 23 + 15 - 4×3 = 23 + 15 - 12 = 26.

Maximum: 26.

Common mistakes candidates make

• Recomputing F(k) from scratch for each k — O(n^2) time, too slow.
• Deriving the recurrence incorrectly — carefully verify F(k) - F(k-1) by expanding a small example.
• Off-by-one in the index nums[n-k] — test with a concrete 3- or 4-element array.
• Forgetting to include F(0) as a candidate for the maximum.

Interview preparation tip

For the Rotate Function coding problem, the array, math, and dynamic programming interview pattern is the efficient approach. Spend time deriving the recurrence on paper before coding — it is the core insight. Google interviewers appreciate seeing the algebraic derivation written out explicitly. Once the recurrence is correct, the code is just a simple loop. Practice the derivation: expand F(k) and F(k-1) in terms of array elements, then simplify.