From b5eb9ca2713e197d78e0f48336b7e5475c1f936b Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Sun, 21 May 2023 19:04:21 +0800 Subject: [PATCH] feat: Add the section of binary search edge. (#508) * Add the section of binary search edge. * Delete binary_search_rotation.py --- .../binary_search_edge.cpp | 57 ++++++ .../binary_search_edge.java | 56 ++++++ .../chapter_binary_search/binary_search.py | 1 + .../binary_search_edge.py | 53 +++++ docs/chapter_binary_search/binary_search.md | 14 +- .../binary_search_left_edge_naive.png | Bin 0 -> 46255 bytes .../binary_search_left_edge_step1.png | Bin 0 -> 63633 bytes .../binary_search_left_edge_step2.png | Bin 0 -> 56203 bytes .../binary_search_left_edge_step3.png | Bin 0 -> 65305 bytes .../binary_search_left_edge_step4.png | Bin 0 -> 55958 bytes .../binary_search_left_edge_step5.png | Bin 0 -> 64626 bytes .../binary_search_left_edge_step6.png | Bin 0 -> 55894 bytes .../binary_search_left_edge_step7.png | Bin 0 -> 61121 bytes .../binary_search_left_edge_step8.png | Bin 0 -> 54297 bytes .../binary_search_left_right_edge.png | Bin 0 -> 82664 bytes .../binary_search_edge.md | 182 ++++++++++++++++++ mkdocs.yml | 1 + 17 files changed, 356 insertions(+), 8 deletions(-) create mode 100644 codes/cpp/chapter_binary_search/binary_search_edge.cpp create mode 100644 codes/java/chapter_binary_search/binary_search_edge.java create mode 100644 codes/python/chapter_binary_search/binary_search_edge.py create mode 100644 docs/chapter_binary_search/binary_search_edge.assets/binary_search_left_edge_naive.png create mode 100644 docs/chapter_binary_search/binary_search_edge.assets/binary_search_left_edge_step1.png create mode 100644 docs/chapter_binary_search/binary_search_edge.assets/binary_search_left_edge_step2.png create mode 100644 docs/chapter_binary_search/binary_search_edge.assets/binary_search_left_edge_step3.png create mode 100644 docs/chapter_binary_search/binary_search_edge.assets/binary_search_left_edge_step4.png create mode 100644 docs/chapter_binary_search/binary_search_edge.assets/binary_search_left_edge_step5.png create mode 100644 docs/chapter_binary_search/binary_search_edge.assets/binary_search_left_edge_step6.png create mode 100644 docs/chapter_binary_search/binary_search_edge.assets/binary_search_left_edge_step7.png create mode 100644 docs/chapter_binary_search/binary_search_edge.assets/binary_search_left_edge_step8.png create mode 100644 docs/chapter_binary_search/binary_search_edge.assets/binary_search_left_right_edge.png create mode 100644 docs/chapter_binary_search/binary_search_edge.md diff --git a/codes/cpp/chapter_binary_search/binary_search_edge.cpp b/codes/cpp/chapter_binary_search/binary_search_edge.cpp new file mode 100644 index 00000000..332fe6c1 --- /dev/null +++ b/codes/cpp/chapter_binary_search/binary_search_edge.cpp @@ -0,0 +1,57 @@ +/** + * File: binary_search_edge.cpp + * Created Time: 2023-05-21 + * Author: Krahets (krahets@163.com) + */ + +#include "../utils/common.hpp" + +/* 二分查找最左一个元素 */ +int binarySearchLeftEdge(vector &nums, int target) { + int i = 0, j = nums.size() - 1; // 初始化双闭区间 [0, n-1] + while (i <= j) { + int m = i + (j - i) / 2; // 计算中点索引 m + if (nums[m] < target) + i = m + 1; // target 在区间 [m+1, j] 中 + else if (nums[m] > target) + j = m - 1; // target 在区间 [i, m-1] 中 + else + j = m - 1; // 首个小于 target 的元素在区间 [i, m-1] 中 + } + if (i == nums.size() || nums[i] != target) + return -1; // 未找到目标元素,返回 -1 + return i; +} + +/* 二分查找最右一个元素 */ +int binarySearchRightEdge(vector &nums, int target) { + int i = 0, j = nums.size() - 1; // 初始化双闭区间 [0, n-1] + while (i <= j) { + int m = i + (j - i) / 2; // 计算中点索引 m + if (nums[m] < target) + i = m + 1; // target 在区间 [m+1, j] 中 + else if (nums[m] > target) + j = m - 1; // target 在区间 [i, m-1] 中 + else + i = m + 1; // 首个大于 target 的元素在区间 [m+1, j] 中 + } + if (j < 0 || nums[j] != target) + return -1; // 未找到目标元素,返回 -1 + return j; +} + +/* Driver Code */ +int main() { + int target = 6; + vector nums = {1, 3, 6, 6, 6, 6, 6, 10, 12, 15}; + + // 二分查找最左一个元素 + int indexLeft = binarySearchLeftEdge(nums, target); + cout << "数组中最左一个元素 6 的索引 = " << indexLeft << endl; + + // 二分查找最右一个元素 + int indexRight = binarySearchRightEdge(nums, target); + cout << "数组中最右一个元素 6 的索引 = " << indexRight << endl; + + return 0; +} diff --git a/codes/java/chapter_binary_search/binary_search_edge.java b/codes/java/chapter_binary_search/binary_search_edge.java new file mode 100644 index 00000000..d63a8f65 --- /dev/null +++ b/codes/java/chapter_binary_search/binary_search_edge.java @@ -0,0 +1,56 @@ +/** + * File: binary_search_edge.java + * Created Time: 2023-05-21 + * Author: Krahets (krahets@163.com) + */ + +package chapter_binary_search; + +public class binary_search_edge { + /* 二分查找最左一个元素 */ + static int binarySearchLeftEdge(int[] nums, int target) { + int i = 0, j = nums.length - 1; // 初始化双闭区间 [0, n-1] + while (i <= j) { + int m = i + (j - i) / 2; // 计算中点索引 m + if (nums[m] < target) + i = m + 1; // target 在区间 [m+1, j] 中 + else if (nums[m] > target) + j = m - 1; // target 在区间 [i, m-1] 中 + else + j = m - 1; // 首个小于 target 的元素在区间 [i, m-1] 中 + } + if (i == nums.length || nums[i] != target) + return -1; // 未找到目标元素,返回 -1 + return i; + } + + /* 二分查找最右一个元素 */ + static int binarySearchRightEdge(int[] nums, int target) { + int i = 0, j = nums.length - 1; // 初始化双闭区间 [0, n-1] + while (i <= j) { + int m = i + (j - i) / 2; // 计算中点索引 m + if (nums[m] < target) + i = m + 1; // target 在区间 [m+1, j] 中 + else if (nums[m] > target) + j = m - 1; // target 在区间 [i, m-1] 中 + else + i = m + 1; // 首个大于 target 的元素在区间 [m+1, j] 中 + } + if (j < 0 || nums[j] != target) + return -1; // 未找到目标元素,返回 -1 + return j; + } + + public static void main(String[] args) { + int target = 6; + int[] nums = { 1, 3, 6, 6, 6, 6, 6, 10, 12, 15 }; + + // 二分查找最左一个元素 + int indexLeft = binarySearchLeftEdge(nums, target); + System.out.println("数组中最左一个元素 6 的索引 = " + indexLeft); + + // 二分查找最右一个元素 + int indexRight = binarySearchRightEdge(nums, target); + System.out.println("数组中最右一个元素 6 的索引 = " + indexRight); + } +} diff --git a/codes/python/chapter_binary_search/binary_search.py b/codes/python/chapter_binary_search/binary_search.py index 92e1587d..23bb71ce 100644 --- a/codes/python/chapter_binary_search/binary_search.py +++ b/codes/python/chapter_binary_search/binary_search.py @@ -11,6 +11,7 @@ def binary_search(nums: list[int], target: int) -> int: i, j = 0, len(nums) - 1 # 循环,当搜索区间为空时跳出(当 i > j 时为空) while i <= j: + # 理论上 Python 的数字可以无限大(取决于内存大小),无需考虑大数越界问题 m = (i + j) // 2 # 计算中点索引 m if nums[m] < target: i = m + 1 # 此情况说明 target 在区间 [m+1, j] 中 diff --git a/codes/python/chapter_binary_search/binary_search_edge.py b/codes/python/chapter_binary_search/binary_search_edge.py new file mode 100644 index 00000000..c14d8347 --- /dev/null +++ b/codes/python/chapter_binary_search/binary_search_edge.py @@ -0,0 +1,53 @@ +""" +File: binary_search_edge.py +Created Time: 2023-05-18 +Author: Krahets (krahets@163.com) +""" + + +def binary_search_left_edge(nums: list[int], target: int) -> int: + """二分查找最左一个元素""" + # 初始化双闭区间 [0, n-1] ,即 i, j 分别指向数组首元素、尾元素 + i, j = 0, len(nums) - 1 + while i <= j: + m = (i + j) // 2 # 计算中点索引 m + if nums[m] < target: + i = m + 1 # 此情况说明 target 在区间 [m+1, j] 中 + elif nums[m] > target: + j = m - 1 # 此情况说明 target 在区间 [i, m-1] 中 + else: + j = m - 1 # 此情况说明首个小于 target 的元素在区间 [i, m-1] 中 + if i == len(nums) or nums[i] != target: + return -1 # 未找到目标元素,返回 -1 + return i + + +def binary_search_right_edge(nums: list[int], target: int) -> int: + """二分查找最右一个元素""" + # 初始化双闭区间 [0, n-1] ,即 i, j 分别指向数组首元素、尾元素 + i, j = 0, len(nums) - 1 + while i <= j: + m = (i + j) // 2 # 计算中点索引 m + if nums[m] < target: + i = m + 1 # target 在区间 [m+1, j] 中 + elif nums[m] > target: + j = m - 1 # target 在区间 [i, m-1] 中 + else: + i = m + 1 # 首个大于 target 的元素在区间 [m+1, j] 中 + if j == len(nums) or nums[j] != target: + return -1 # 未找到目标元素,返回 -1 + return j + + +"""Driver Code""" +if __name__ == "__main__": + target: int = 6 + nums: list[int] = [1, 3, 6, 6, 6, 6, 6, 10, 12, 15] + + # 二分查找最左一个元素 + index_left = binary_search_left_edge(nums, target) + print("数组中最左一个元素 6 的索引 = ", index_left) + + # 二分查找最右一个元素 + index_right = binary_search_right_edge(nums, target) + print("数组中最右一个元素 6 的索引 = ", index_right) diff --git a/docs/chapter_binary_search/binary_search.md b/docs/chapter_binary_search/binary_search.md index e7bd4caa..4a81aa69 100755 --- a/docs/chapter_binary_search/binary_search.md +++ b/docs/chapter_binary_search/binary_search.md @@ -1,6 +1,6 @@ # 二分查找 -「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,实现定位目标元素。 +「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。 @@ -18,7 +18,7 @@ **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。 -为了更清晰地表示区间,我们在下图中以折线图的形式表示数组。 +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。 === "<0>" ![二分查找步骤](binary_search.assets/binary_search_step0.png) @@ -46,8 +46,6 @@ 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。 -有趣的是,理论上 Python 的数字可以无限大(取决于内存大小),因此无需考虑大数越界问题。 - === "Java" ```java title="binary_search.java" @@ -186,12 +184,12 @@ ## 优点与局限性 -二分查找效率很高,主要体现在: +二分查找在时间和空间方面都有较好的性能: -- **二分查找的时间复杂度较低**。对数阶在大数据量情况下具有显著优势。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**。与哈希查找相比,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,并非所有情况下都可使用二分查找,原因如下: +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。 diff --git a/docs/chapter_binary_search/binary_search_edge.assets/binary_search_left_edge_naive.png b/docs/chapter_binary_search/binary_search_edge.assets/binary_search_left_edge_naive.png new file mode 100644 index 0000000000000000000000000000000000000000..ef57a7e24faee494506c7520215d3b568be14266 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z+Rapik1Yv{eMb=x6`l9Zj;*q73!K(*^!wL36ee()Hjgc>8V(EgG?qNln6yV$QBY{- zb-+`2TE485FX+j*Sy=Rn8T+%f_upvW-V;Bh~;PciqkF?YMnpo6f0ziQ)DP4&O^^8+ zgTMelxYM3QN8iggmFC2-7450QsD1jWDV7)J7+s`~DEnhP1HF}3CCk6eAPGCQ? zpbiq{p<>`dO8{8vr{mV6b4`RtL;#&psNbqFLqIcqoemO06^Bzg1_=rIHB?$cD<8+p zt5>h=U%OikT4doEV}KxEjS0g;_4XwY{4+8$H|k%b4`wynY`d9`GFcc1MTAEpCE<9P zjTQdke&Upb<--%BPsXe7OujT>2xi5%Ipgz2Cv7wAuxkRxW`&1e<>fjhd1H?^ypn|@ z$ZM;j+gk&@tKu~^B@W3^|rjO_V#wb z5r~x?X+SGiujeS~Wi7el2>pvz^~-pGF9a8d+aBb_wyca`n#R9hX&Fd)569&zffW{H zWd&2mm0!t$ufatMqN@r=nNVib{yHw^mm*fB1P|5DIuZI$qasI+@iQ`_ug$z*bJRzg z;i!;tfkL6azJl{ZRCyvJ!Ik8Qyx?ZopTRDW#0(WzQ@2H9$~0VV>q%u06%}P=WraW> s33jX_ILK7yQMZ`>7zqK?TmHiZ25)e#q~_H(%t+u5Dx)G@Drx%QKb!N(8vp target` 时,说明还没有找到 `target` ,因此采取与上节代码相同的缩小区间操作。 +- 当 `nums[m] == target` 时,说明找到了一个目标元素,此时应该如何缩小区间? + +对于该情况,**我们可以将查找目标想象为 `leftarget`**,其中 `leftarget` 表示从右到左首个小于 `target` 的元素。具体来说: + +- 当 `nums[m] == target` 时,说明 `leftarget` 在区间 `[i, m - 1]` 中,因此采用 `j = m - 1` 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。 + +=== "<1>" + ![二分查找最左元素的步骤](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) + +=== "<3>" + ![binary_search_left_edge_step3](binary_search_edge.assets/binary_search_left_edge_step3.png) + +=== "<4>" + ![binary_search_left_edge_step4](binary_search_edge.assets/binary_search_left_edge_step4.png) + +=== "<5>" + ![binary_search_left_edge_step5](binary_search_edge.assets/binary_search_left_edge_step5.png) + +=== "<6>" + ![binary_search_left_edge_step6](binary_search_edge.assets/binary_search_left_edge_step6.png) + +=== "<7>" + ![binary_search_left_edge_step7](binary_search_edge.assets/binary_search_left_edge_step7.png) + +=== "<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。 + +=== "Java" + + ```java title="binary_search_edge.java" + [class]{binary_search_edge}-[func]{binarySearchLeftEdge} + ``` + +=== "C++" + + ```cpp title="binary_search_edge.cpp" + [class]{}-[func]{binarySearchLeftEdge} + ``` + +=== "Python" + + ```python title="binary_search_edge.py" + [class]{}-[func]{binary_search_left_edge} + ``` + +=== "Go" + + ```go title="binary_search_edge.go" + [class]{}-[func]{binarySearchLeftEdge} + ``` + +=== "JavaScript" + + ```javascript title="binary_search_edge.js" + [class]{}-[func]{binarySearchLeftEdge} + ``` + +=== "TypeScript" + + ```typescript title="binary_search_edge.ts" + [class]{}-[func]{binarySearchLeftEdge} + ``` + +=== "C" + + ```c title="binary_search_edge.c" + [class]{}-[func]{binarySearchLeftEdge} + ``` + +=== "C#" + + ```csharp title="binary_search_edge.cs" + [class]{binary_search_edge}-[func]{binarySearchLeftEdge} + ``` + +=== "Swift" + + ```swift title="binary_search_edge.swift" + [class]{}-[func]{binarySearchLeftEdge} + ``` + +=== "Zig" + + ```zig title="binary_search_edge.zig" + [class]{}-[func]{binarySearchLeftEdge} + ``` + +## 查找最右一个元素 + +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` 。 + +- 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。 + +=== "Java" + + ```java title="binary_search_edge.java" + [class]{binary_search_edge}-[func]{binarySearchRightEdge} + ``` + +=== "C++" + + ```cpp title="binary_search_edge.cpp" + [class]{}-[func]{binarySearchRightEdge} + ``` + +=== "Python" + + ```python title="binary_search_edge.py" + [class]{}-[func]{binary_search_right_edge} + ``` + +=== "Go" + + ```go title="binary_search_edge.go" + [class]{}-[func]{binarySearchRightEdge} + ``` + +=== "JavaScript" + + ```javascript title="binary_search_edge.js" + [class]{}-[func]{binarySearchRightEdge} + ``` + +=== "TypeScript" + + ```typescript title="binary_search_edge.ts" + [class]{}-[func]{binarySearchRightEdge} + ``` + +=== "C" + + ```c title="binary_search_edge.c" + [class]{}-[func]{binarySearchRightEdge} + ``` + +=== "C#" + + ```csharp title="binary_search_edge.cs" + [class]{binary_search_edge}-[func]{binarySearchRightEdge} + ``` + +=== "Swift" + + ```swift title="binary_search_edge.swift" + [class]{}-[func]{binarySearchRightEdge} + ``` + +=== "Zig" + + ```zig title="binary_search_edge.zig" + [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**。 + +![二分查找最左元素和最右元素](binary_search_edge.assets/binary_search_left_right_edge.png) + +!!! tip + + 以上代码采取的都是“双闭区间”写法。有兴趣的读者可以自行实现“左闭右开”写法。 diff --git a/mkdocs.yml b/mkdocs.yml index f4753c45..fcfc369d 100644 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -155,6 +155,7 @@ nav: - 5.4.   小结: chapter_stack_and_queue/summary.md - 6.     二分查找: - 6.1.   二分查找: chapter_binary_search/binary_search.md + - 6.2.   二分查找边界: chapter_binary_search/binary_search_edge.md - 7.     散列表: - 7.1.   哈希表: chapter_hashing/hash_map.md - 7.2.   哈希冲突处理: chapter_hashing/hash_collision.md