From 232cfbd5a6d8370b000ee44b8c97f1457631dc1a Mon Sep 17 00:00:00 2001 From: CyC2018 Date: Fri, 30 Oct 2026 20:18:37 +0800 Subject: [PATCH] =?UTF-8?q?Update=20Leetcode=20=E9=A2=98=E8=A7=A3=20-=20?= =?UTF-8?q?=E6=95=B0=E5=AD=A6.md?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit 补充多数投票算法的原理说明并新增配图。补充除自身以外数组乘积、进制转换等题目的推导过程与分析,增补参考资料并完成文字校对。 --- notes/Leetcode 题解 - 数学.md | 45 ++++++++++++++++++++------- notes/pics/lc-math-majority-vote.png | Bin 0 -> 80765 bytes 2 files changed, 34 insertions(+), 11 deletions(-) create mode 100644 notes/pics/lc-math-majority-vote.png diff --git a/notes/Leetcode 题解 - 数学.md b/notes/Leetcode 题解 - 数学.md index 5269fb11..a2ec3928 100644 --- a/notes/Leetcode 题解 - 数学.md +++ b/notes/Leetcode 题解 - 数学.md @@ -26,6 +26,7 @@ * [3. 乘积数组](#3-乘积数组) * [4. 找出数组中的乘积最大的三个数](#4-找出数组中的乘积最大的三个数) * [5. 用 Rand7() 实现 Rand10()](#5-用-rand7-实现-rand10) + * [参考资料](#参考资料) @@ -91,14 +92,14 @@ int lcm(int a, int b) { ### 3. 使用位操作和减法求解最大公约数 -编程之美:2.7 +《编程之美》2.7 对于 a 和 b 的最大公约数 f(a, b),有: -- 如果 a 和 b 均为偶数,f(a, b) = 2\*f(a/2, b/2); -- 如果 a 是偶数 b 是奇数,f(a, b) = f(a/2, b); -- 如果 b 是偶数 a 是奇数,f(a, b) = f(a, b/2); -- 如果 a 和 b 均为奇数,f(a, b) = f(b, a-b); +- 如果 a 和 b 均为偶数,f(a, b) = 2\*f(a/2, b/2); +- 如果 a 是偶数 b 是奇数,f(a, b) = f(a/2, b); +- 如果 b 是偶数 a 是奇数,f(a, b) = f(a, b/2); +- 如果 a 和 b 均为奇数,f(a, b) = f(b, a-b)。 乘 2 和除 2 都可以转换为移位操作。 @@ -213,7 +214,7 @@ public String toHex(int num) { 28 -> AB ``` -因为是从 1 开始计算的,而不是从 0 开始,因此需要对 n 执行 -1 操作。 +这种表示没有 0,A \~ Z 对应 1 \~ 26,每一位的取值是 1 \~ 26 而不是 0 \~ 25。先把 n 减 1,最低位就变成 0 \~ 25,对应 A \~ Z,再对 (n - 1) / 26 递归处理更高位。例如 52:51 % 26 = 25 对应 Z,51 / 26 = 1 对应 A,结果为 AZ。 ```java public String convertToTitle(int n) { @@ -243,7 +244,7 @@ public int trailingZeroes(int n) { } ``` -如果统计的是 N! 的二进制表示中最低位 1 的位置,只要统计有多少个 2 即可,该题目出自 编程之美:2.2 。和求解有多少个 5 一样,2 的个数为 N/2 + N/22 + N/23 + ... +如果统计的是 N! 的二进制表示中最低位 1 的位置,只要统计有多少个 2 即可,该题目出自《编程之美》2.2。和求解有多少个 5 一样,2 的个数为 N/2 + N/22 + N/23 + ... ## 字符串加法减法 @@ -348,7 +349,7 @@ public int minMoves2(int[] nums) { #### 解法 2 -使用快速选择找到中位数,平均时间复杂度 O(N) +使用快速选择找到中位数,平均时间复杂度 O(N)。下面的实现固定以第一个元素为基准,输入已经有序时每次只能排除一个元素,退化为 O(N2),可以先随机打乱数组或随机选取基准。 ```java public int minMoves2(int[] nums) { @@ -414,7 +415,11 @@ public int majorityElement(int[] nums) { } ``` -可以利用 Boyer-Moore Majority Vote Algorithm 来解决这个问题,使得时间复杂度为 O(N)。可以这么理解该算法:使用 cnt 来统计一个元素出现的次数,当遍历到的元素和统计元素不相等时,令 cnt--。如果前面查找了 i 个元素,且 cnt == 0,说明前 i 个元素没有 majority,或者有 majority,但是出现的次数少于 i / 2,因为如果多于 i / 2 的话 cnt 就一定不会为 0。此时剩下的 n - i 个元素中,majority 的数目依然多于 (n - i) / 2,因此继续查找就能找出 majority。 +Boyer-Moore 多数投票算法(Boyer-Moore Majority Vote Algorithm)可以在 O(N) 时间、O(1) 空间内解决这个问题。它维护一个候选元素 majority 和计数 cnt:遇到和候选相同的元素 cnt 加一,不同的元素 cnt 减一,相当于让两个不同的元素互相抵消;cnt 为 0 时把当前元素作为新的候选。 + +每次抵消的是一对不同的元素,其中最多有一个是多数元素,所以抵消掉若干对之后,多数元素在剩下的元素中出现的次数仍然多于一半。cnt 回到 0 时,从上次 cnt 为 0 到现在扫过的一段恰好两两抵消完,可以整段丢掉,从下一个元素重新开始;扫描结束时没有被抵消的候选就是多数元素。本题保证多数元素存在,否则还要再遍历一次,确认候选的出现次数确实多于 n / 2。 + +

```java public int majorityElement(int[] nums) { @@ -444,7 +449,7 @@ Returns: True 间隔:3,5,7,... -间隔为等差数列,使用这个特性可以得到从 1 开始的平方序列。 +间隔为等差数列,使用这个特性可以得到从 1 开始的平方序列:依次减去 1、3、5……,num 是完全平方数当且仅当恰好减到 0。这种做法的时间复杂度为 O(sqrt(N))。 ```java public boolean isPerfectSquare(int num) { @@ -463,6 +468,8 @@ public boolean isPerfectSquare(int num) { [Leetcode](https://leetcode.com/problems/power-of-three/description/) / [力扣](https://leetcode-cn.com/problems/power-of-three/description/) +1162261467 是 int 范围内最大的 3 的幂(319)。3 是素数,319 的正约数只有 30 \~ 319,因此能整除它的正整数一定是 3 的幂。 + ```java public boolean isPowerOfThree(int n) { return n > 0 && (1162261467 % n == 0); @@ -483,6 +490,15 @@ For example, given [1,2,3,4], return [24,12,8,6]. 要求时间复杂度为 O(N),并且不能使用除法。 +除了 nums[i] 之外所有元素的乘积,等于它左边所有元素的乘积乘以右边所有元素的乘积。第一遍从左往右把左边的乘积 left 写进 products,第二遍从右往左再乘上右边的乘积 right,输出数组之外只用了两个变量。例如 [2, 3, 4, 5]: + +```text +下标 0 1 2 3 +左边的乘积 1 2 6 24 +右边的乘积 60 20 5 1 +结果 60 40 30 24 +``` + ```java public int[] productExceptSelf(int[] nums) { int n = nums.length; @@ -513,6 +529,8 @@ Input: [1,2,3,4] Output: 24 ``` +数组中可能有负数,乘积最大的三个数只有两种可能:最大的三个数,或者最大的数乘以最小的两个数(两个负数相乘为正)。遍历一次记录最大的三个数和最小的两个数,取两种乘积中较大的一个。 + ```java public int maximumProduct(int[] nums) { int max1 = Integer.MIN_VALUE, max2 = Integer.MIN_VALUE, max3 = Integer.MIN_VALUE, min1 = Integer.MAX_VALUE, min2 = Integer.MAX_VALUE; @@ -547,7 +565,7 @@ public int maximumProduct(int[] nums) { 题目描述:已知 rand7() 等概率返回 1 \~ 7,只用它实现等概率返回 1 \~ 10 的 rand10()。 -把两次 rand7() 的结果看成一个两位的 7 进制数,(rand7() - 1) * 7 + rand7() 能等概率地生成 1 \~ 49。只保留 1 \~ 40,它们对 10 取余之后 0 \~ 9 各出现 4 次,仍然是等概率的;落在 41 \~ 49 时重新生成,这种方法称为拒绝采样。 +把两次 rand7() 的结果看成一个两位的 7 进制数,(rand7() - 1) * 7 + rand7() 能等概率地生成 1 \~ 49。只保留 1 \~ 40,它们对 10 取余之后 0 \~ 9 各出现 4 次,仍然是等概率的;落在 41 \~ 49 时重新生成,这种方法称为拒绝采样。每轮成功的概率为 40 / 49,平均需要 49 / 40 ≈ 1.23 轮,也就是约 2.45 次 rand7()。 ```java public int rand10() { @@ -559,3 +577,8 @@ public int rand10() { } } ``` + +## 参考资料 + +- [LeetCode 题库](https://leetcode.com/problemset/) +- 《编程之美》小组. 编程之美 [M]. 电子工业出版社, 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