From c65e3ef0d8eaec50350cf1a4ad9e47741877bf50 Mon Sep 17 00:00:00 2001 From: CyC2018 Date: Fri, 30 Oct 2026 20:18:31 +0800 Subject: [PATCH] =?UTF-8?q?Update=20Leetcode=20=E9=A2=98=E8=A7=A3=20-=20?= =?UTF-8?q?=E4=BA=8C=E5=88=86=E6=9F=A5=E6=89=BE.md?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit 补充旋转数组最小值、字符查找等题解的分析说明,更新两个有序数组中位数的排除逻辑并新增配图,同时补充参考资料并进行文字校对。 --- notes/Leetcode 题解 - 二分查找.md | 43 +++++++++++++++++++++--------- notes/pics/lc-bs-median-kth.png | Bin 0 -> 209457 bytes 2 files changed, 31 insertions(+), 12 deletions(-) create mode 100644 notes/pics/lc-bs-median-kth.png diff --git a/notes/Leetcode 题解 - 二分查找.md b/notes/Leetcode 题解 - 二分查找.md index 29ec8365..4b9be940 100644 --- a/notes/Leetcode 题解 - 二分查找.md +++ b/notes/Leetcode 题解 - 二分查找.md @@ -8,6 +8,7 @@ * [5. 旋转数组的最小数字](#5-旋转数组的最小数字) * [6. 查找区间](#6-查找区间) * [7. 两个有序数组的中位数](#7-两个有序数组的中位数) + * [参考资料](#参考资料) @@ -47,14 +48,14 @@ public int binarySearch(int[] nums, int key) { - m = (l + h) / 2 - m = l + (h - l) / 2 -l + h 可能出现加法溢出,也就是说加法的结果大于整型能够表示的范围。但是 l 和 h 都为正数,因此 h - l 不会出现加法溢出问题。所以,最好使用第二种计算方法。 +l + h 可能出现加法溢出,也就是说加法的结果大于整型能够表示的范围。但是 l 和 h 都不小于 0,因此 h - l 不会出现溢出问题。所以,最好使用第二种计算方法。 #### 未成功查找的返回值 循环退出时如果仍然没有查找到 key,那么表示查找失败。可以有两种返回值: -- -1:以一个错误码表示没有查找到 key -- l:将 key 插入到 nums 中的正确位置 +- -1:以一个错误码表示没有查找到 key; +- l:将 key 插入到 nums 中的正确位置。 #### 变种 @@ -77,9 +78,9 @@ public int binarySearch(int[] nums, int key) { 该实现和正常实现有以下不同: -- h 的赋值表达式为 h = m -- 循环条件为 l \< h -- 最后返回 l 而不是 -1 +- h 的赋值表达式为 h = m; +- 循环条件为 l \< h; +- 最后返回 l 而不是 -1。 在 nums[m] \>= key 的情况下,可以推导出最左 key 位于 [l, m] 区间中,这是一个闭区间。h 的赋值表达式为 h = m,因为 m 位置也可能是解。 @@ -95,7 +96,7 @@ l m h ... ``` -当循环体退出时,不表示没有查找到 key,因此最后返回的结果不应该为 -1。为了验证有没有查找到,需要在调用端判断一下返回位置上的值和 key 是否相等。 +当循环体退出时,不表示没有查找到 key,因此最后返回的结果不应该为 -1。为了验证有没有查找到,需要在调用端先判断返回值小于 nums.length,再判断该位置上的值和 key 是否相等。 ## 1. 求开方 @@ -112,7 +113,7 @@ Output: 2 Explanation: The square root of 8 is 2.82842..., and since we want to return an integer, the decimal part will be truncated. ``` -一个数 x 的开方 sqrt 一定在 0 \~ x 之间,并且满足 sqrt == x / sqrt。可以利用二分查找在 0 \~ x 之间查找 sqrt。 +要找的是满足 r * r \<= x 的最大整数 r,它一定在 1 \~ x 之间(x \>= 2 时),可以用二分查找。比较 mid 和 x / mid 而不是计算 mid * mid,是为了避免乘法溢出。 对于 x = 8,它的开方是 2.82842...,最后应该返回 2 而不是 3。在循环条件为 l \<= h 并且循环退出时,h 总是比 l 小 1,也就是说 h = 2,l = 3,因此最后的返回值应该为 h 而不是 l。 @@ -157,6 +158,8 @@ Output: "c" 题目描述:给定一个有序的字符数组 letters 和一个字符 target,要求找出 letters 中大于 target 的最小字符,如果找不到就返回第 1 个字符。 +letters[m] \<= target 时答案一定在 m 的右边,令 l = m + 1;否则 m 可能就是答案,令 h = m - 1 继续在左边找更小的。循环结束时 l = h + 1,指向第一个大于 target 的字符,l 等于 n 表示所有字符都不大于 target。 + ```java public char nextGreatestLetter(char[] letters, char target) { int n = letters.length; @@ -190,6 +193,8 @@ Output: 2 令 index 为 Single Element 在数组中的位置。在 index 之后,数组中原来存在的成对状态被改变。如果 m 为偶数,并且 m + 1 \< index,那么 nums[m] == nums[m + 1];m + 1 \>= index,那么 nums[m] != nums[m + 1]。 +例如 [1, 1, 2, 3, 3, 4, 4, 8, 8] 中 index = 2:m = 0 时 nums[0] == nums[1],还没到 index;m = 4 时 nums[4] = 3、nums[5] = 4,成对状态已经被打乱,index 在 m 或它的左边。 + 从上面的规律可以知道,如果 nums[m] == nums[m + 1],那么 index 所在的数组位置为 [m + 2, h],此时令 l = m + 2;如果 nums[m] != nums[m + 1],那么 index 所在的数组位置为 [l, m],此时令 h = m。 因为 h 的赋值表达式为 h = m,那么循环条件也就只能使用 l \< h 这种形式。 @@ -250,6 +255,12 @@ Input: [3,4,5,1,2], Output: 1 ``` +题目描述:一个升序数组在某个位置被旋转,数组中的元素互不相同,求最小值。 + +旋转后的数组分为两段升序,后一段的元素都小于前一段,最小值是后一段的第一个元素。用 nums[m] 和区间末尾的 nums[h] 比较:nums[m] \<= nums[h] 说明 m 已经在后一段,最小值在 [l, m] 中,令 h = m;否则 m 还在前一段,最小值在 [m + 1, h] 中,令 l = m + 1。 + +不能改为和 nums[l] 比较:nums[m] \> nums[l] 时,可能是 m 在前一段,也可能是 [l, h] 这一段本身没有旋转、最小值就是 nums[l],两种情况无法区分。 + ```java public int findMin(int[] nums) { int l = 0, h = nums.length - 1; @@ -267,7 +278,7 @@ public int findMin(int[] nums) { ## 6. 查找区间 -34\. Find First and Last Position of Element in Sorted Array +34\. Find First and Last Position of Element in Sorted Array (Medium) [Leetcode](https://leetcode.com/problems/find-first-and-last-position-of-element-in-sorted-array/) / [力扣](https://leetcode-cn.com/problems/find-first-and-last-position-of-element-in-sorted-array/) @@ -308,13 +319,13 @@ private int findFirst(int[] nums, int target) { } ``` -在寻找第一个位置的二分查找代码中,需要注意 h 的取值为 nums.length,而不是 nums.length - 1。先看以下示例: +在寻找第一个位置的二分查找代码中,需要注意 h 的初始值为 nums.length,而不是 nums.length - 1。先看以下示例: ```text nums = [2,2], target = 2 ``` -如果 h 的取值为 nums.length - 1,那么 last = findFirst(nums, target + 1) - 1 = 1 - 1 = 0。这是因为 findFirst 只会返回 [0, nums.length - 1] 范围的值,对于 findFirst([2,2], 3),希望返回 3 应该插入 nums 中的位置,也就是数组最后一个位置再往后一个位置,即 nums.length。所以需要将 h 取值为 nums.length,从而使得 findFirst 返回的区间更大,能够覆盖 target 大于 nums 最后一个元素的情况。 +findFirst(nums, 3) 要返回 3 应该插入的位置,也就是 nums.length = 2,于是 last = 2 - 1 = 1。如果 h 的初始值为 nums.length - 1,findFirst 的返回值只能在 [0, nums.length - 1] 中,这里会返回 1,得到错误的 last = 0。 ## 7. 两个有序数组的中位数 @@ -324,7 +335,11 @@ nums = [2,2], target = 2 题目描述:求两个有序数组合并后的中位数,要求时间复杂度为 O(log(m + n))。 -把问题转化为求两个数组合并后第 k 小的数。比较两个数组各自的第 k / 2 个元素,较小的那个元素以及它前面的元素都不可能是第 k 小,可以一次排除,然后在剩下的部分里寻找第 k - 排除个数 小的数。k 每轮减半,时间复杂度为 O(log(m + n))。 +把问题转化为求两个数组合并后第 k 小的数。比较两个数组各自的第 k / 2 个元素,假设 nums1 的不大于 nums2 的。合并时让相等的元素中 nums1 的排在前面,那么排在它前面的最多只有 nums1 中的 k / 2 - 1 个和 nums2 中的 k / 2 - 1 个,它最多排在第 k - 1 位,因此 nums1 的前 k / 2 个元素都排在第 k 小之前,可以一次排除。 + +排除了 x 个数之后,在剩下的部分里寻找第 k - x 小的数;某个数组剩余的长度不足 k / 2 时,取它的最后一个元素比较。k 每轮减半,时间复杂度为 O(log(m + n))。 + +

```java public double findMedianSortedArrays(int[] nums1, int[] nums2) { @@ -354,3 +369,7 @@ private int kth(int[] nums1, int i, int[] nums2, int j, int k) { return kth(nums1, i, nums2, b + 1, k - (b - j + 1)); } ``` + +## 参考资料 + +- [LeetCode 题库](https://leetcode.com/problemset/) diff --git a/notes/pics/lc-bs-median-kth.png b/notes/pics/lc-bs-median-kth.png new file mode 100644 index 0000000000000000000000000000000000000000..e665a0f20d8271fc9dd9eb5e2173e8eed65d3147 GIT binary patch literal 209457 zcmXV21z43^(>)*(f`D{`N(m_4B}xbaf=VOZ-HoK8q=M2V-6`E5CEZ;j4FV!v|D5~% zK0fy%9NzQ3d(W&{v)1hMQAtte_AQEA2n6D`>@&%i2*k}j1mZ>xCMtaAj>;2XG88C6y|lpE61I8 zkNJC}x3fwq``7mOr>tCJwzucvJQvQBkL`lf5EA%kzRbaq^xb8>_Ed;Hk>ZgJx8$l* zJ=sXaV|+9rn;$(-l!>Cc1JyVs@Nv)s8iKmZ*m}yCg9Y-Ys(BOoBI)-julAAekzBLwj{BLrL9cV zpi@k;BBlI8<90pspBuhxJ)6Sr$rD?Lj8RzZ_fCYUI%Ok&=mZwwqDg#8eTpC0>HF%Q z11tU?@d#3>JG>M;G2c)+R|loqG1mT!YvMh^GW)5BymSmAq`h?$XHOGhZV}#Y(84WV zM@Sn$irI%ns}P^YoW>w?S-gJe5ym2kkA_0Ojt`G3xoUbKFw?=pQjibpxB6R4nMg`m 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