From 081b76d6209ffbfe43b16b62cfce3c64bfbdf9ac Mon Sep 17 00:00:00 2001 From: krahets Date: Mon, 22 May 2023 01:37:12 +0800 Subject: [PATCH] Polish the sections of binary search. --- docs/chapter_searching/binary_search.md | 28 +++++------- .../binary_search_left_edge_naive.png | Bin 46255 -> 47475 bytes .../binary_search_left_edge_step1.png | Bin 63633 -> 60456 bytes .../binary_search_left_right_edge.png | Bin 82664 -> 79974 bytes docs/chapter_searching/binary_search_edge.md | 41 ++++++++++-------- 5 files changed, 34 insertions(+), 35 deletions(-) diff --git a/docs/chapter_searching/binary_search.md b/docs/chapter_searching/binary_search.md index 9d512c525..9600e140b 100755 --- a/docs/chapter_searching/binary_search.md +++ b/docs/chapter_searching/binary_search.md @@ -2,25 +2,21 @@ 「二分查找 Binary Search」是一种基于分治思想的高效搜索算法。它利用数据的有序性,每轮减少一半搜索范围,直至找到目标元素或搜索区间为空为止。 -我们先来求解一个简单的二分查找问题。 - !!! question - 给定一个长度为 $n$ 的有序数组 `nums` ,元素按从小到大的顺序排列。请查找并返回元素 `target` 在该数组中的索引。若数组中不包含该元素,则返回 $-1$ 。数组中不包含重复元素。 + 给定一个长度为 $n$ 的数组 `nums` ,元素按从小到大的顺序排列,数组不包含重复元素。请查找并返回元素 `target` 在该数组中的索引。若数组不包含该元素,则返回 $-1$ 。 -该数组的索引范围可以使用区间 $[0, n - 1]$ 来表示。其中,**中括号表示“闭区间”,即包含边界值本身**。在该表示下,区间 $[i, j]$ 在 $i = j$ 时仍包含一个元素,在 $i > j$ 时为空区间。 +对于上述问题,我们先初始化指针 $i = 0$ 和 $j = n - 1$ ,分别指向数组首元素和尾元素,代表搜索区间 $[0, n - 1]$ 。其中,中括号表示“闭区间”,即包含边界值本身。 -接下来,我们基于上述区间定义实现二分查找。先初始化指针 $i = 0$ 和 $j = n - 1$ ,分别指向数组首元素和尾元素。之后循环执行以下两个步骤: +接下来,循环执行以下两个步骤: 1. 计算中点索引 $m = \lfloor {(i + j) / 2} \rfloor$ ,其中 $\lfloor \space \rfloor$ 表示向下取整操作。 -2. 根据 `nums[m]` 和 `target` 缩小搜索区间,分为三种情况: +2. 判断 `nums[m]` 和 `target` 的大小关系,分为三种情况: 1. 当 `nums[m] < target` 时,说明 `target` 在区间 $[m + 1, j]$ 中,因此执行 $i = m + 1$ ; 2. 当 `nums[m] > target` 时,说明 `target` 在区间 $[i, m - 1]$ 中,因此执行 $j = m - 1$ ; - 3. 当 `nums[m] = target` 时,说明找到目标元素,直接返回索引 $m$ 即可; - -**若数组不包含目标元素,搜索区间最终会缩小为空**,即达到 $i > j$ 。此时,终止循环并返回 $-1$ 即可。 + 3. 当 `nums[m] = target` 时,说明找到 `target` ,因此返回索引 $m$ ; -如下图所示,为了更清晰地表示区间,我们以折线图的形式表示数组。 +若数组不包含目标元素,搜索区间最终会缩小为空。此时返回 $-1$ 。 === "<0>" ![二分查找步骤](binary_search.assets/binary_search_step0.png) @@ -46,7 +42,7 @@ === "<7>" ![binary_search_step7](binary_search.assets/binary_search_step7.png) -值得注意的是,**当数组长度 $n$ 很大时,加法 $i + j$ 的结果可能会超出 `int` 类型的取值范围**。为了避免大数越界,我们通常采用公式 $m = \lfloor {i + (j - i) / 2} \rfloor$ 来计算中点。 +值得注意的是,由于 $i$ 和 $j$ 都是 `int` 类型,**因此 $i + j$ 可能会超出 `int` 类型的取值范围**。为了避免大数越界,我们通常采用公式 $m = \lfloor {i + (j - i) / 2} \rfloor$ 来计算中点。 === "Java" @@ -188,11 +184,11 @@ 二分查找在时间和空间方面都有较好的性能: -- **二分查找的时间效率高**。在大数据量下,对数阶的时间复杂度具有显著优势。例如,当数据大小 $n = 2^{20}$ 时,线性查找需要 $2^{20} = 1048576$ 轮循环,而二分查找仅需 $\log_2 2^{20} = 20$ 轮循环。 -- **二分查找无需额外空间**。相较于需要借助额外空间的搜索算法(例如哈希查找),二分查找更加节省空间。 +- 二分查找的时间效率高。在大数据量下,对数阶的时间复杂度具有显著优势。例如,当数据大小 $n = 2^{20}$ 时,线性查找需要 $2^{20} = 1048576$ 轮循环,而二分查找仅需 $\log_2 2^{20} = 20$ 轮循环。 +- 二分查找无需额外空间。相较于需要借助额外空间的搜索算法(例如哈希查找),二分查找更加节省空间。 然而,二分查找并非适用于所有情况,原因如下: -- **二分查找仅适用于有序数据**。若输入数据无序,为了使用二分查找而专门进行排序,得不偿失。因为排序算法的时间复杂度通常为 $O(n \log n)$ ,比线性查找和二分查找都更高。对于频繁插入元素的场景,为保持数组有序性,需要将元素插入到特定位置,时间复杂度为 $O(n)$ ,也是非常昂贵的。 -- **二分查找仅适用于数组**。二分查找需要跳跃式(非连续地)访问元素,而在链表中执行跳跃式访问的效率较低,因此不适合应用在链表或基于链表实现的数据结构。 -- **小数据量下,线性查找性能更佳**。在线性查找中,每轮只需要 1 次判断操作;而在二分查找中,需要 1 次加法、1 次除法、1 ~ 3 次判断操作、1 次加法(减法),共 4 ~ 6 个单元操作;因此,当数据量 $n$ 较小时,线性查找反而比二分查找更快。 +- 二分查找仅适用于有序数据。若输入数据无序,为了使用二分查找而专门进行排序,得不偿失。因为排序算法的时间复杂度通常为 $O(n \log n)$ ,比线性查找和二分查找都更高。对于频繁插入元素的场景,为保持数组有序性,需要将元素插入到特定位置,时间复杂度为 $O(n)$ ,也是非常昂贵的。 +- 二分查找仅适用于数组。二分查找需要跳跃式(非连续地)访问元素,而在链表中执行跳跃式访问的效率较低,因此不适合应用在链表或基于链表实现的数据结构。 +- 小数据量下,线性查找性能更佳。在线性查找中,每轮只需要 1 次判断操作;而在二分查找中,需要 1 次加法、1 次除法、1 ~ 3 次判断操作、1 次加法(减法),共 4 ~ 6 个单元操作;因此,当数据量 $n$ 较小时,线性查找反而比二分查找更快。 diff --git a/docs/chapter_searching/binary_search_edge.assets/binary_search_left_edge_naive.png b/docs/chapter_searching/binary_search_edge.assets/binary_search_left_edge_naive.png index ef57a7e24faee494506c7520215d3b568be14266..b5cc56fe35833ad479405d205b239385674c5f05 100644 GIT binary patch literal 47475 zcmeFZcUTi!+dc{bqzMR!DBT7KQk33Nq)C;Il%O=Jq4!WkK}A4Cq=a6j_ZCVJRHXL~ z3B808dVml(gYVw^`(DrQT<5=Yoj7-G8F|A1pxs8m8!}k 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来缩小区间,**从而使指针 `j` 向 `leftarget` 收缩靠近**。 -- 二分查找完成后,`i` 指向最左一个 `target` ,`j` 指向 `leftarget` ,因此最终返回索引 `i` 即可。 +二分查找完成后,**$i$ 指向最左边的 `target` ,$j$ 指向首个小于 `target` 的元素**,因此返回索引 $i$ 即可。 === "<1>" - ![二分查找最左元素的步骤](binary_search_edge.assets/binary_search_left_edge_step1.png) + ![二分查找最左边元素的步骤](binary_search_edge.assets/binary_search_left_edge_step1.png) === "<2>" ![binary_search_left_edge_step2](binary_search_edge.assets/binary_search_left_edge_step2.png) @@ -46,7 +50,7 @@ === "<8>" ![binary_search_left_edge_step8](binary_search_edge.assets/binary_search_left_edge_step8.png) -注意,数组可能不包含目标元素 `target` 。因此在函数返回前,我们需要先判断 `nums[i]` 与 `target` 是否相等。另外,当 `target` 大于数组中的所有元素时,索引 `i` 会越界,因此也需要额外判断。 +注意,数组可能不包含目标元素 `target` 。因此在函数返回前,我们需要先判断 `nums[i]` 与 `target` 是否相等,以及索引 $i$ 是否越界。 === "Java" @@ -108,12 +112,11 @@ [class]{}-[func]{binarySearchLeftEdge} ``` -## 查找最右一个元素 +## 查找右边界 -类似地,我们也可以二分查找最右一个元素。设首个大于 `target` 的元素为 `rightarget` 。 +类似地,我们也可以二分查找最右边的 `target` 。当 `nums[m] == target` 时,说明大于 `target` 的元素在区间 $[m + 1, j]$ 中,因此执行 `i = m + 1` ,**使得指针 $i$ 向大于 `target` 的元素靠近**。 -- 当 `nums[m] == target` 时,说明 `rightarget` 在区间 `[m + 1, j]` 中,因此执行 `i = m + 1` 将搜索区间向右收缩。 -- 完成二分后,`i` 指向 `rightarget` ,`j` 指向最右一个 `target` ,因此最终返回索引 `j` 即可。 +完成二分后,**$i$ 指向首个大于 `target` 的元素,$j$ 指向最右边的 `target`** ,因此返回索引 $j$ 即可。 === "Java" @@ -175,9 +178,9 @@ [class]{}-[func]{binarySearchRightEdge} ``` -观察下图,搜索最右元素时指针 `j` 起到了搜索最左元素时指针 `i` 的作用,反之亦然。本质上看,**搜索最左元素和最右元素的实现是镜像对称的**。 +观察下图,搜索最右边元素时指针 $j$ 的作用与搜索最左边元素时指针 $i$ 的作用一致,反之亦然。也就是说,**搜索最左边元素和最右边元素的实现是镜像对称的**。 -![二分查找最左元素和最右元素](binary_search_edge.assets/binary_search_left_right_edge.png) +![查找最左边和最右边元素的对称性](binary_search_edge.assets/binary_search_left_right_edge.png) !!! tip