Rotate Function - Problem
Imagine you have an array of numbers that can be rotated clockwise - where elements move to the right and the last element wraps around to the front. For each possible rotation, we calculate a special weighted sum where each element is multiplied by its position index.
Given an integer array nums of length n, we define:
- Rotation k: Array obtained by rotating
numsclockwise bykpositions - Function F(k):
0 × arr[0] + 1 × arr[1] + 2 × arr[2] + ... + (n-1) × arr[n-1]
Your goal is to find the maximum value among all possible rotation functions F(0), F(1), F(2), ..., F(n-1).
Example: For [4,3,2,6]:
• F(0) = 0×4 + 1×3 + 2×2 + 3×6 = 25
• F(1) = 0×6 + 1×4 + 2×3 + 3×2 = 16
• F(2) = 0×2 + 1×6 + 2×4 + 3×3 = 23
• F(3) = 0×3 + 1×2 + 2×6 + 3×4 = 26
Maximum = 26
Input & Output
example_1.py — Basic rotation
$
Input:
[4,3,2,6]
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Output:
26
💡 Note:
F(0)=25, F(1)=16, F(2)=23, F(3)=26. Maximum is 26.
example_2.py — Single element
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Input:
[100]
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Output:
0
💡 Note:
Only one element, so F(0) = 0 × 100 = 0.
example_3.py — Negative numbers
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Input:
[-2,1,-3,4]
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Output:
3
💡 Note:
F(0)=-7, F(1)=3, F(2)=-3, F(3)=-13. Maximum is 3.
Constraints
- 1 ≤ nums.length ≤ 105
- -100 ≤ nums[i] ≤ 100
- The answer is guaranteed to fit in a 32-bit integer
Visualization
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Understanding the Visualization
1
Initial Setup
Wheel has positions with multipliers 0×, 1×, 2×, 3× and numbers [4,3,2,6]
2
First Spin
Number 6 moves from 3× position to 0× position, affecting the total score
3
Pattern Recognition
Each spin adds the sum of all numbers but subtracts n times the number that moved to 0×
4
Optimal Solution
Apply the formula F(k+1) = F(k) + sum - n × nums[n-k-1] for each rotation
Key Takeaway
🎯 Key Insight: Instead of recalculating each rotation from scratch, we can use the mathematical relationship between consecutive rotations to solve this in O(n) time with O(1) space.
💡
Explanation
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