|
| 1 | +--- |
| 2 | +title: 396.旋转函数:求diff(增量法) |
| 3 | +date: 2026-05-01 21:35:54 |
| 4 | +tags: [题解, LeetCode, 中等, 数组, 数学, 动态规划] |
| 5 | +categories: [题解, LeetCode] |
| 6 | +--- |
| 7 | + |
| 8 | +# 【LetMeFly】396.旋转函数:求diff(增量法) |
| 9 | + |
| 10 | +力扣题目链接:[https://leetcode.cn/problems/rotate-function/](https://leetcode.cn/problems/rotate-function/) |
| 11 | + |
| 12 | +<p>给定一个长度为 <code>n</code> 的整数数组 <code>nums</code> 。</p> |
| 13 | + |
| 14 | +<p>假设 <code>arr<sub>k</sub></code> 是数组 <code>nums</code> 顺时针旋转 <code>k</code> 个位置后的数组,我们定义 <code>nums</code> 的 <strong>旋转函数</strong> <code>F</code> 为:</p> |
| 15 | + |
| 16 | +<ul> |
| 17 | + <li><code>F(k) = 0 * arr<sub>k</sub>[0] + 1 * arr<sub>k</sub>[1] + ... + (n - 1) * arr<sub>k</sub>[n - 1]</code></li> |
| 18 | +</ul> |
| 19 | + |
| 20 | +<p>返回 <em><code>F(0), F(1), ..., F(n-1)</code>中的最大值 </em>。</p> |
| 21 | + |
| 22 | +<p>生成的测试用例让答案符合 <strong>32 位</strong> 整数。</p> |
| 23 | + |
| 24 | +<p> </p> |
| 25 | + |
| 26 | +<p><strong>示例 1:</strong></p> |
| 27 | + |
| 28 | +<pre> |
| 29 | +<strong>输入:</strong> nums = [4,3,2,6] |
| 30 | +<strong>输出:</strong> 26 |
| 31 | +<strong>解释:</strong> |
| 32 | +F(0) = (0 * 4) + (1 * 3) + (2 * 2) + (3 * 6) = 0 + 3 + 4 + 18 = 25 |
| 33 | +F(1) = (0 * 6) + (1 * 4) + (2 * 3) + (3 * 2) = 0 + 4 + 6 + 6 = 16 |
| 34 | +F(2) = (0 * 2) + (1 * 6) + (2 * 4) + (3 * 3) = 0 + 6 + 8 + 9 = 23 |
| 35 | +F(3) = (0 * 3) + (1 * 2) + (2 * 6) + (3 * 4) = 0 + 2 + 12 + 12 = 26 |
| 36 | +所以 F(0), F(1), F(2), F(3) 中的最大值是 F(3) = 26 。 |
| 37 | +</pre> |
| 38 | + |
| 39 | +<p><strong>示例 2:</strong></p> |
| 40 | + |
| 41 | +<pre> |
| 42 | +<strong>输入:</strong> nums = [100] |
| 43 | +<strong>输出:</strong> 0 |
| 44 | +</pre> |
| 45 | + |
| 46 | +<p> </p> |
| 47 | + |
| 48 | +<p><strong>提示:</strong></p> |
| 49 | + |
| 50 | +<ul> |
| 51 | + <li><code>n == nums.length</code></li> |
| 52 | + <li><code>1 <= n <= 10<sup>5</sup></code></li> |
| 53 | + <li><code>-100 <= nums[i] <= 100</code></li> |
| 54 | +</ul> |
| 55 | + |
| 56 | + |
| 57 | + |
| 58 | +## 解题方法:增量法 |
| 59 | + |
| 60 | +以样例1为例: |
| 61 | + |
| 62 | +> $nums = [4, 3, 2, 6]$ |
| 63 | +> |
| 64 | +> $F(0) = (0 * 4) + (1 * 3) + (2 * 2) + (3 * 6) = 0 + 3 + 4 + 18 = 25$ |
| 65 | +> |
| 66 | +> $F(1) = (0 * 6) + (1 * 4) + (2 * 3) + (3 * 2) = 0 + 4 + 6 + 6 = 16$ |
| 67 | +
|
| 68 | +如何由$F(0)$得到$F(1)$呢?很简单: |
| 69 | + |
| 70 | +> $F(1) = F(0) + (4 + 3 + 2 + 6) - 4\times 6$ |
| 71 | +> |
| 72 | +> 即:$F(1)=F(0)+\sum nums[i] - len(nums)\times nums[-1]$ |
| 73 | +
|
| 74 | +我们只需要遍历一遍$nums$数组,得到$F(0)$、$\sum nums[i]$,就能在$O(1)$的时间内推出$F(1)$了,递推推到$F(n-1)$总耗时$O(n)$。 |
| 75 | + |
| 76 | ++ 时间复杂度$O(n)$ |
| 77 | ++ 空间复杂度$O(1)$ |
| 78 | + |
| 79 | +### AC代码 |
| 80 | + |
| 81 | +#### C++ |
| 82 | + |
| 83 | +```cpp |
| 84 | +/* |
| 85 | + * @LastEditTime: 2026-05-01 21:34:50 |
| 86 | + */ |
| 87 | +typedef long long ll; |
| 88 | +class Solution { |
| 89 | +public: |
| 90 | + int maxRotateFunction(vector<int>& nums) { |
| 91 | + ll now = 0, sum = 0; |
| 92 | + int n = nums.size(); |
| 93 | + for (int i = 0; i < n; i++) { |
| 94 | + now += i * nums[i]; |
| 95 | + sum += nums[i]; |
| 96 | + } |
| 97 | + |
| 98 | + ll ans = now; |
| 99 | + for (int i = 1; i < n; i++) { |
| 100 | + now = now + sum - n * nums[n - i]; |
| 101 | + ans = max(ans, now); |
| 102 | + } |
| 103 | + return ans; |
| 104 | + } |
| 105 | +}; |
| 106 | +``` |
| 107 | +
|
| 108 | +> 同步发文于[CSDN](https://letmefly.blog.csdn.net/article/details/160694129)和我的[个人博客](https://blog.letmefly.xyz/),原创不易,转载经作者同意后请附上[原文链接](https://blog.letmefly.xyz/2026/05/01/LeetCode%200396.%E6%97%8B%E8%BD%AC%E5%87%BD%E6%95%B0/)哦~ |
| 109 | +> |
| 110 | +> 千篇源码题解[已开源](https://github.com/LetMeFly666/LeetCode) |
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