From 4e1fb5c07a5c75611658a143c361e5811149d6a1 Mon Sep 17 00:00:00 2001 From: labuladong Date: Sun, 1 Mar 2020 14:30:28 +0800 Subject: [PATCH] =?UTF-8?q?=E4=B8=BA=E4=BA=86=E5=AD=A6=E4=BC=9A=E4=BA=8C?= =?UTF-8?q?=E5=88=86=E6=9F=A5=E6=89=BE=EF=BC=8C=E6=88=91tm=E5=86=99?= =?UTF-8?q?=E4=BA=86=E9=A6=96=E8=AF=97?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../1.jpg" | Bin 0 -> 58764 bytes .../2.jpg" | Bin 0 -> 62216 bytes .../3.jpg" | Bin 0 -> 69027 bytes .../4.jpg" | Bin 0 -> 61417 bytes .../poem.png" | Bin 0 -> 131635 bytes ...45\346\211\276\350\257\246\350\247\243.md" | 280 ++++++++++++++---- 6 files changed, 223 insertions(+), 57 deletions(-) create mode 100644 "pictures/\344\272\214\345\210\206\346\237\245\346\211\276/1.jpg" create mode 100644 "pictures/\344\272\214\345\210\206\346\237\245\346\211\276/2.jpg" create mode 100644 "pictures/\344\272\214\345\210\206\346\237\245\346\211\276/3.jpg" create mode 100644 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-答:因为初始化 right 的赋值是 nums.length - 1,即最后一个元素的索引,而不是 nums.length。 +答:因为初始化 `right` 的赋值是 `nums.length - 1`,即最后一个元素的索引,而不是 `nums.length`。 -这二者可能出现在不同功能的二分查找中,区别是:前者相当于两端都闭区间 [left, right],后者相当于左闭右开区间 [left, right),因为索引大小为 nums.length 是越界的。 +这二者可能出现在不同功能的二分查找中,区别是:前者相当于两端都闭区间 `[left, right]`,后者相当于左闭右开区间 `[left, right)`,因为索引大小为 `nums.length` 是越界的。 -我们这个算法中使用的是前者 [left, right] 两端都闭的区间。**这个区间其实就是每次进行搜索的区间**。 +我们这个算法中使用的是前者 `[left, right]` 两端都闭的区间。**这个区间其实就是每次进行搜索的区间**。 什么时候应该停止搜索呢?当然,找到了目标值的时候可以终止: @@ -80,11 +80,11 @@ int binarySearch(int[] nums, int target) { 但如果没找到,就需要 while 循环终止,然后返回 -1。那 while 循环什么时候应该终止?**搜索区间为空的时候应该终止**,意味着你没得找了,就等于没找到嘛。 -`while(left <= right)` 的终止条件是 left == right + 1,写成区间的形式就是 [right + 1, right],或者带个具体的数字进去 [3, 2],可见**这时候区间为空**,因为没有数字既大于等于 3 又小于等于 2 的吧。所以这时候 while 循环终止是正确的,直接返回 -1 即可。 +`while(left <= right)` 的终止条件是 `left == right + 1`,写成区间的形式就是 `[right + 1, right]`,或者带个具体的数字进去 `[3, 2]`,可见**这时候区间为空**,因为没有数字既大于等于 3 又小于等于 2 的吧。所以这时候 while 循环终止是正确的,直接返回 -1 即可。 -`while(left < right)` 的终止条件是 left == right,写成区间的形式就是 [left, right],或者带个具体的数字进去 [2, 2],**这时候区间非空**,还有一个数 2,但此时 while 循环终止了。也就是说这区间 [2, 2] 被漏掉了,索引 2 没有被搜索,如果这时候直接返回 -1 就是错误的。 +`while(left < right)` 的终止条件是 `left == right`,写成区间的形式就是 `[left, right]`,或者带个具体的数字进去 `[2, 2]`,**这时候区间非空**,还有一个数 2,但此时 while 循环终止了。也就是说这区间 `[2, 2]` 被漏掉了,索引 2 没有被搜索,如果这时候直接返回 -1 就是错误的。 -当然,如果你非要用 while(left < right) 也可以,我们已经知道了出错的原因,就打个补丁好了: +当然,如果你非要用 `while(left < right)` 也可以,我们已经知道了出错的原因,就打个补丁好了: ```java //... @@ -95,28 +95,28 @@ int binarySearch(int[] nums, int target) { ``` -2、为什么 left = mid + 1,right = mid - 1?我看有的代码是 right = mid 或者 left = mid,没有这些加加减减,到底怎么回事,怎么判断? +**2、为什么 `left = mid + 1`,`right = mid - 1`?我看有的代码是 `right = mid` 或者 `left = mid`,没有这些加加减减,到底怎么回事,怎么判断**? 答:这也是二分查找的一个难点,不过只要你能理解前面的内容,就能够很容易判断。 -刚才明确了「搜索区间」这个概念,而且本算法的搜索区间是两端都闭的,即 [left, right]。那么当我们发现索引 mid 不是要找的 target 时,如何确定下一步的搜索区间呢? +刚才明确了「搜索区间」这个概念,而且本算法的搜索区间是两端都闭的,即 `[left, right]`。那么当我们发现索引 `mid` 不是要找的 `target` 时,下一步应该去搜索哪里呢? -当然是 [left, mid - 1] 或者 [mid + 1, right] 对不对?因为 mid 已经搜索过,应该从搜索区间中去除。 +当然是去搜索 `[left, mid-1]` 或者 `[mid+1, right]` 对不对?**因为 `mid` 已经搜索过,应该从搜索区间中去除**。 -3、此算法有什么缺陷? +**3、此算法有什么缺陷**? 答:至此,你应该已经掌握了该算法的所有细节,以及这样处理的原因。但是,这个算法存在局限性。 -比如说给你有序数组 nums = [1,2,2,2,3],target = 2,此算法返回的索引是 2,没错。但是如果我想得到 target 的左侧边界,即索引 1,或者我想得到 target 的右侧边界,即索引 3,这样的话此算法是无法处理的。 +比如说给你有序数组 `nums = [1,2,2,2,3]`,`target` 为 2,此算法返回的索引是 2,没错。但是如果我想得到 `target` 的左侧边界,即索引 1,或者我想得到 `target` 的右侧边界,即索引 3,这样的话此算法是无法处理的。 -这样的需求很常见。你也许会说,找到一个 target,然后向左或向右线性搜索不行吗?可以,但是不好,因为这样难以保证二分查找对数级的复杂度了。 +这样的需求很常见,**你也许会说,找到一个 target,然后向左或向右线性搜索不行吗?可以,但是不好,因为这样难以保证二分查找对数级的复杂度了**。 我们后续的算法就来讨论这两种二分查找的算法。 ### 二、寻找左侧边界的二分搜索 -直接看代码,其中的标记是需要注意的细节: +以下是最常见的代码形式,其中的标记是需要注意的细节: ```java int left_bound(int[] nums, int target) { @@ -138,25 +138,29 @@ int left_bound(int[] nums, int target) { } ``` -1、为什么 while(left < right) 而不是 <= ? +**1、为什么 while 中是 `<` 而不是 `<=`**? + +答:用相同的方法分析,因为 `right = nums.length` 而不是 `nums.length - 1`。因此每次循环的「搜索区间」是 `[left, right)` 左闭右开。 -答:用相同的方法分析,因为 right = nums.length 而不是 nums.length - 1 。因此每次循环的「搜索区间」是 [left, right) 左闭右开。 +`while(left < right)` 终止的条件是 `left == right`,此时搜索区间 `[left, left)` 为空,所以可以正确终止。 -while(left < right) 终止的条件是 left == right,此时搜索区间 [left, left) 为空,所以可以正确终止。 +PS:这里先要说一个搜索左右边界和上面这个算法的一个区别,也是很多读者问的:**刚才的 `right` 不是 `nums.length - 1` 吗,为啥这里非要写成 `nums.length` 使得「搜索区间」变成左闭右开呢**? -2、为什么没有返回 -1 的操作?如果 nums 中不存在 target 这个值,怎么办? +因为对于搜索左右侧边界的二分查找,这种写法比较普遍,我就拿这种写法举例了,保证你以后遇到这类代码可以理解。你非要用两端都闭的写法反而更简单,我会在后面写相关的代码,把三种二分搜索都用一种两端都闭的写法统一起来,你耐心往后看就行了。 + +**2、为什么没有返回 -1 的操作?如果 `nums` 中不存在 `target` 这个值,怎么办**? 答:因为要一步一步来,先理解一下这个「左侧边界」有什么特殊含义: -![](../pictures/%E4%BA%8C%E5%88%86%E6%9F%A5%E6%89%BE/binarySearch1.png) +![](../pictures/%E4%BA%8C%E5%88%86%E6%9F%A5%E6%89%BE/1.jpg) -对于这个数组,算法会返回 1。这个 1 的含义可以这样解读:nums 中小于 2 的元素有 1 个。 +对于这个数组,算法会返回 1。这个 1 的含义可以这样解读:`nums` 中小于 2 的元素有 1 个。 -比如对于有序数组 nums = [2,3,5,7], target = 1,算法会返回 0,含义是:nums 中小于 1 的元素有 0 个。 +比如对于有序数组 `nums = [2,3,5,7]`, `target = 1`,算法会返回 0,含义是:`nums` 中小于 1 的元素有 0 个。 -再比如说 nums 不变,target = 8,算法会返回 4,含义是:nums 中小于 8 的元素有 4 个。 +再比如说 `nums = [2,3,5,7], target = 8`,算法会返回 4,含义是:`nums` 中小于 8 的元素有 4 个。 -综上可以看出,函数的返回值(即 left 变量的值)取值区间是闭区间 [0, nums.length],所以我们简单添加两行代码就能在正确的时候 return -1: +综上可以看出,函数的返回值(即 `left` 变量的值)取值区间是闭区间 `[0, nums.length]`,所以我们简单添加两行代码就能在正确的时候 return -1: ```java while (left < right) { @@ -168,29 +172,99 @@ if (left == nums.length) return -1; return nums[left] == target ? left : -1; ``` -1、为什么 left = mid + 1,right = mid ?和之前的算法不一样? +**3、为什么 `left = mid + 1`,`right = mid` ?和之前的算法不一样**? -答:这个很好解释,因为我们的「搜索区间」是 [left, right) 左闭右开,所以当 nums[mid] 被检测之后,下一步的搜索区间应该去掉 mid 分割成两个区间,即 [left, mid) 或 [mid + 1, right)。 +答:这个很好解释,因为我们的「搜索区间」是 `[left, right)` 左闭右开,所以当 `nums[mid]` 被检测之后,下一步的搜索区间应该去掉 `mid` 分割成两个区间,即 `[left, mid)` 或 `[mid + 1, right)`。 -4、为什么该算法能够搜索左侧边界? +**4、为什么该算法能够搜索左侧边界**? -答:关键在于对于 nums[mid] == target 这种情况的处理: +答:关键在于对于 `nums[mid] == target` 这种情况的处理: ```java if (nums[mid] == target) right = mid; ``` -可见,找到 target 时不要立即返回,而是缩小「搜索区间」的上界 right,在区间 [left, mid) 中继续搜索,即不断向左收缩,达到锁定左侧边界的目的。 +可见,找到 target 时不要立即返回,而是缩小「搜索区间」的上界 `right`,在区间 `[left, mid)` 中继续搜索,即不断向左收缩,达到锁定左侧边界的目的。 + +**5、为什么返回 `left` 而不是 `right`**? + +答:都是一样的,因为 while 终止的条件是 `left == right`。 + +**6、能不能想办法把 `right` 变成 `nums.length - 1`,也就是继续使用两边都闭的「搜索区间」?这样就可以和第一种二分搜索在某种程度上统一起来了**。 + +答:当然可以,只要你明白了「搜索区间」这个概念,就能有效避免漏掉元素,随便你怎么改都行。下面我们严格根据逻辑来修改: + +因为你非要让搜索区间两端都闭,所以 `right` 应该初始化为 `nums.length - 1`,while 的终止条件应该是 `left == right + 1`,也就是其中应该用 `<=`: + +```java +int left_bound(int[] nums, int target) { + // 搜索区间为 [left, right] + int left = 0, right = nums.length - 1; + while (left <= right) { + int mid = left + (right - left) / 2; + // if else ... + } +``` + +因为搜索区间是两端都闭的,且现在是搜索左侧边界,所以 `left` 和 `right` 的更新逻辑如下: -5、为什么返回 left 而不是 right? +```java +if (nums[mid] < target) { + // 搜索区间变为 [mid+1, right] + left = mid + 1; +} else if (nums[mid] > target) { + // 搜索区间变为 [left, mid-1] + right = mid - 1; +} else if (nums[mid] == target) { + // 收缩右侧边界 + right = mid - 1; +} +``` -答:都是一样的,因为 while 终止的条件是 left == right。 +由于 while 的退出条件是 `left == right + 1`,所以当 `target` 比 `nums` 中所有元素都大时,会存在以下情况使得索引越界: +![](../pictures/%E4%BA%8C%E5%88%86%E6%9F%A5%E6%89%BE/2.jpg) + +因此,最后返回结果的代码应该检查越界情况: + +```java +if (left >= nums.length || nums[left] != target) + return -1; +return left; +``` + +至此,整个算法就写完了,完整代码如下: + +```java +int left_bound(int[] nums, int target) { + int left = 0, right = nums.length - 1; + // 搜索区间为 [left, right] + while (left <= right) { + int mid = left + (right - left) / 2; + if (nums[mid] < target) { + // 搜索区间变为 [mid+1, right] + left = mid + 1; + } else if (nums[mid] > target) { + // 搜索区间变为 [left, mid-1] + right = mid - 1; + } else if (nums[mid] == target) { + // 收缩右侧边界 + right = mid - 1; + } + } + // 检查出界情况 + if (left >= nums.length || nums[left] != target) + return -1; + return left; +} +``` + +这样就和第一种二分搜索算法统一了,都是两端都闭的「搜索区间」,而且最后返回的也是 `left` 变量的值。只要把住二分搜索的逻辑,两种形式大家看自己喜欢哪种记哪种吧。 ### 三、寻找右侧边界的二分查找 -寻找右侧边界和寻找左侧边界的代码差不多,只有两处不同,已标注: +类似寻找左侧边界的算法,这里也会提供两种写法,还是先写常见的左闭右开的写法,只有两处和搜索左侧边界不同,已标注: ```java int right_bound(int[] nums, int target) { @@ -211,38 +285,38 @@ int right_bound(int[] nums, int target) { } ``` -1、为什么这个算法能够找到右侧边界? +**1、为什么这个算法能够找到右侧边界**? 答:类似地,关键点还是这里: ```java - if (nums[mid] == target) { - left = mid + 1; +if (nums[mid] == target) { + left = mid + 1; ``` -当 nums[mid] == target 时,不要立即返回,而是增大「搜索区间」的下界 left,使得区间不断向右收缩,达到锁定右侧边界的目的。 +当 `nums[mid] == target` 时,不要立即返回,而是增大「搜索区间」的下界 `left`,使得区间不断向右收缩,达到锁定右侧边界的目的。 -2、为什么最后返回 left - 1 而不像左侧边界的函数,返回 left?而且我觉得这里既然是搜索右侧边界,应该返回 right 才对。 +**2、为什么最后返回 `left - 1` 而不像左侧边界的函数,返回 `left`?而且我觉得这里既然是搜索右侧边界,应该返回 `right` 才对**。 -答:首先,while 循环的终止条件是 left == right,所以 left 和 right 是一样的,你非要体现右侧的特点,返回 right - 1 好了。 +答:首先,while 循环的终止条件是 `left == right`,所以 `left` 和 `right` 是一样的,你非要体现右侧的特点,返回 `right - 1` 好了。 至于为什么要减一,这是搜索右侧边界的一个特殊点,关键在这个条件判断: ```java - if (nums[mid] == target) { - left = mid + 1; - // 这样想: mid = left - 1 +if (nums[mid] == target) { + left = mid + 1; + // 这样想: mid = left - 1 ``` -![](../pictures/%E4%BA%8C%E5%88%86%E6%9F%A5%E6%89%BE/binarySearch2.png) +![](../pictures/%E4%BA%8C%E5%88%86%E6%9F%A5%E6%89%BE/3.jpg) -因为我们对 left 的更新必须是 left = mid + 1,就是说 while 循环结束时,nums[left] 一定不等于 target 了,而 nums[left-1] 可能是 target。 +因为我们对 `left` 的更新必须是 `left = mid + 1`,就是说 while 循环结束时,`nums[left]` 一定不等于 `target` 了,而 `nums[left-1]` 可能是 `target`。 -至于为什么 left 的更新必须是 left = mid + 1,同左侧边界搜索,就不再赘述。 +至于为什么 `left` 的更新必须是 `left = mid + 1`,同左侧边界搜索,就不再赘述。 -1、为什么没有返回 -1 的操作?如果 nums 中不存在 target 这个值,怎么办? +**3、为什么没有返回 -1 的操作?如果 `nums` 中不存在 `target` 这个值,怎么办**? -答:类似之前的左侧边界搜索,因为 while 的终止条件是 left == right,就是说 left 的取值范围是 [0, nums.length],所以可以添加两行代码,正确地返回 -1: +答:类似之前的左侧边界搜索,因为 while 的终止条件是 `left == right`,就是说 `left` 的取值范围是 `[0, nums.length]`,所以可以添加两行代码,正确地返回 -1: ```java while (left < right) { @@ -252,11 +326,42 @@ if (left == 0) return -1; return nums[left-1] == target ? (left-1) : -1; ``` -### 四、最后总结 +**4、是否也可以把这个算法的「搜索区间」也统一成两端都闭的形式呢?这样这三个写法就完全统一了,以后就可以闭着眼睛写出来了**。 + +答:当然可以,类似搜索左侧边界的统一写法,其实只要改两个地方就行了: + +```java +int right_bound(int[] nums, int target) { + int left = 0, right = nums.length - 1; + while (left <= right) { + int mid = left + (right - left) / 2; + if (nums[mid] < target) { + left = mid + 1; + } else if (nums[mid] > target) { + right = mid - 1; + } else if (nums[mid] == target) { + // 这里改成收缩左侧边界即可 + left = mid + 1; + } + } + // 这里改为检查 right 越界的情况,见下图 + if (right < 0 || nums[right] != target) + return -1; + return right; +} +``` + +当 `target` 比所有元素都小时,`right` 会被减到 -1,所以需要在最后防止越界: + +![](../pictures/%E4%BA%8C%E5%88%86%E6%9F%A5%E6%89%BE/4.jpg) + +至此,搜索右侧边界的二分查找的两种写法也完成了,其实将「搜索区间」统一成两端都闭反而更容易记忆,你说是吧? + +### 四、逻辑统一 来梳理一下这些细节差异的因果逻辑: -第一个,最基本的二分查找算法: +**第一个,最基本的二分查找算法**: ```python 因为我们初始化 right = nums.length - 1 @@ -268,7 +373,7 @@ return nums[left-1] == target ? (left-1) : -1; 所以当 nums[mid] == target 时可以立即返回 ``` -第二个,寻找左侧边界的二分查找: +**第二个,寻找左侧边界的二分查找**: ```python 因为我们初始化 right = nums.length @@ -281,7 +386,7 @@ return nums[left-1] == target ? (left-1) : -1; 而要收紧右侧边界以锁定左侧边界 ``` -第三个,寻找右侧边界的二分查找: +**第三个,寻找右侧边界的二分查找**: ```python 因为我们初始化 right = nums.length @@ -297,6 +402,66 @@ return nums[left-1] == target ? (left-1) : -1; 所以最后无论返回 left 还是 right,必须减一 ``` +对于寻找左右边界的二分搜索,常见的手法是使用左闭右开的「搜索区间」,**我们还根据逻辑将「搜索区间」全都统一成了两端都闭,便于记忆,只要修改两处即可变化出三种写法**: + +```java +int binary_search(int[] nums, int target) { + int left = 0, right = nums.length - 1; + while(left <= right) { + int mid = left + (right - left) / 2; + if (nums[mid] < target) { + left = mid + 1; + } else if (nums[mid] > target) { + right = mid - 1; + } else if(nums[mid] == target) { + // 直接返回 + return mid; + } + } + // 直接返回 + return -1; +} + +int left_bound(int[] nums, int target) { + int left = 0, right = nums.length - 1; + while (left <= right) { + int mid = left + (right - left) / 2; + if (nums[mid] < target) { + left = mid + 1; + } else if (nums[mid] > target) { + right = mid - 1; + } else if (nums[mid] == target) { + // 别返回,锁定左侧边界 + right = mid - 1; + } + } + // 最后要检查 left 越界的情况 + if (left >= nums.length || nums[left] != target) + return -1; + return left; +} + + +int right_bound(int[] nums, int target) { + int left = 0, right = nums.length - 1; + while (left <= right) { + int mid = left + (right - left) / 2; + if (nums[mid] < target) { + left = mid + 1; + } else if (nums[mid] > target) { + right = mid - 1; + } else if (nums[mid] == target) { + // 别返回,锁定右侧边界 + left = mid + 1; + } + } + // 最后要检查 right 越界的情况 + if (right < 0 || nums[right] != target) + return -1; + return right; +} +``` + 如果以上内容你都能理解,那么恭喜你,二分查找算法的细节不过如此。 通过本文,你学会了: @@ -305,8 +470,9 @@ return nums[left-1] == target ? (left-1) : -1; 2、注意「搜索区间」和 while 的终止条件,如果存在漏掉的元素,记得在最后检查。 -3、如需要搜索左右边界,只要在 nums[mid] == target 时做修改即可。搜索右侧时需要减一。 +3、如需定义左闭右开的「搜索区间」搜索左右边界,只要在 `nums[mid] == target` 时做修改即可,搜索右侧时需要减一。 +4、如果将「搜索区间」全都统一成两端都闭,好记,只要稍改 `nums[mid] == target` 条件处的代码和返回的逻辑即可,**推荐拿小本本记下,作为二分搜索模板**。 呵呵,此文对二分查找的问题无敌好吧!**致力于把算法讲清楚!欢迎关注我的微信公众号 labuladong,查看更多通俗易懂的文章**: -- GitLab