From c3e7455285e6767af57fcb42b76d285af8289e3c Mon Sep 17 00:00:00 2001 From: krahets Date: Sun, 21 May 2023 04:51:32 +0800 Subject: [PATCH] Refactor the section of bianry search. --- codes/c/chapter_binary_search/binary_search.c | 8 +- .../chapter_binary_search/binary_search.cpp | 8 +- .../chapter_binary_search/binary_search.cs | 4 +- .../chapter_binary_search/binary_search.dart | 4 +- .../go/chapter_binary_search/binary_search.go | 4 +- .../chapter_binary_search/binary_search.java | 12 +- .../chapter_binary_search/binary_search.js | 4 +- .../chapter_binary_search/binary_search.py | 21 +-- .../chapter_binary_search/binary_search.rs | 18 +-- .../chapter_binary_search/binary_search.swift | 4 +- .../chapter_binary_search/binary_search.ts | 4 +- .../chapter_binary_search/binary_search.zig | 4 +- .../binary_search_ranges.png | Bin 0 -> 80317 bytes .../binary_search_step0.png | Bin 0 -> 63442 bytes .../binary_search_step1.png | Bin 54139 -> 59875 bytes .../binary_search_step2.png | Bin 54485 -> 58391 bytes .../binary_search_step3.png | Bin 65396 -> 67208 bytes .../binary_search_step4.png | Bin 54413 -> 58211 bytes .../binary_search_step5.png | Bin 65146 -> 66645 bytes .../binary_search_step6.png | Bin 54591 -> 57813 bytes .../binary_search_step7.png | Bin 64872 -> 62040 bytes docs/chapter_binary_search/binary_search.md | 142 ++++-------------- 22 files changed, 76 insertions(+), 161 deletions(-) create mode 100644 docs/chapter_binary_search/binary_search.assets/binary_search_ranges.png create mode 100644 docs/chapter_binary_search/binary_search.assets/binary_search_step0.png diff --git a/codes/c/chapter_binary_search/binary_search.c b/codes/c/chapter_binary_search/binary_search.c index f9183860a..c1407e3af 100644 --- a/codes/c/chapter_binary_search/binary_search.c +++ b/codes/c/chapter_binary_search/binary_search.c @@ -12,8 +12,8 @@ int binarySearch(int *nums, int len, int target) { int i = 0, j = len - 1; // 循环,当搜索区间为空时跳出(当 i > j 时为空) while (i <= j) { - int m = (i + j) / 2; // 计算中点索引 m - if (nums[m] < target) // 此情况说明 target 在区间 [m+1, j] 中 + int m = i + (j - i) / 2; // 计算中点索引 m + if (nums[m] < target) // 此情况说明 target 在区间 [m+1, j] 中 i = m + 1; else if (nums[m] > target) // 此情况说明 target 在区间 [i, m-1] 中 j = m - 1; @@ -30,8 +30,8 @@ int binarySearchLCRO(int *nums, int len, int target) { int i = 0, j = len; // 循环,当搜索区间为空时跳出(当 i = j 时为空) while (i < j) { - int m = (i + j) / 2; // 计算中点索引 m - if (nums[m] < target) // 此情况说明 target 在区间 [m+1, j) 中 + int m = i + (j - i) / 2; // 计算中点索引 m + if (nums[m] < target) // 此情况说明 target 在区间 [m+1, j) 中 i = m + 1; else if (nums[m] > target) // 此情况说明 target 在区间 [i, m) 中 j = m; diff --git a/codes/cpp/chapter_binary_search/binary_search.cpp b/codes/cpp/chapter_binary_search/binary_search.cpp index 60a26305c..7c56649e8 100644 --- a/codes/cpp/chapter_binary_search/binary_search.cpp +++ b/codes/cpp/chapter_binary_search/binary_search.cpp @@ -12,8 +12,8 @@ int binarySearch(vector &nums, int target) { int i = 0, j = nums.size() - 1; // 循环,当搜索区间为空时跳出(当 i > j 时为空) while (i <= j) { - int m = (i + j) / 2; // 计算中点索引 m - if (nums[m] < target) // 此情况说明 target 在区间 [m+1, j] 中 + int m = i + (j - i) / 2; // 计算中点索引 m + if (nums[m] < target) // 此情况说明 target 在区间 [m+1, j] 中 i = m + 1; else if (nums[m] > target) // 此情况说明 target 在区间 [i, m-1] 中 j = m - 1; @@ -30,8 +30,8 @@ int binarySearchLCRO(vector &nums, int target) { int i = 0, j = nums.size(); // 循环,当搜索区间为空时跳出(当 i = j 时为空) while (i < j) { - int m = (i + j) / 2; // 计算中点索引 m - if (nums[m] < target) // 此情况说明 target 在区间 [m+1, j) 中 + int m = i + (j - i) / 2; // 计算中点索引 m + if (nums[m] < target) // 此情况说明 target 在区间 [m+1, j) 中 i = m + 1; else if (nums[m] > target) // 此情况说明 target 在区间 [i, m) 中 j = m; diff --git a/codes/csharp/chapter_binary_search/binary_search.cs b/codes/csharp/chapter_binary_search/binary_search.cs index 358db04e1..ec80f29d3 100644 --- a/codes/csharp/chapter_binary_search/binary_search.cs +++ b/codes/csharp/chapter_binary_search/binary_search.cs @@ -13,7 +13,7 @@ public class binary_search { int i = 0, j = nums.Length - 1; // 循环,当搜索区间为空时跳出(当 i > j 时为空) while (i <= j) { - int m = (i + j) / 2; // 计算中点索引 m + int m = i + (j - i) / 2; // 计算中点索引 m if (nums[m] < target) // 此情况说明 target 在区间 [m+1, j] 中 i = m + 1; else if (nums[m] > target) // 此情况说明 target 在区间 [i, m-1] 中 @@ -31,7 +31,7 @@ public class binary_search { int i = 0, j = nums.Length; // 循环,当搜索区间为空时跳出(当 i = j 时为空) while (i < j) { - int m = (i + j) / 2; // 计算中点索引 m + int m = i + (j - i) / 2; // 计算中点索引 m if (nums[m] < target) // 此情况说明 target 在区间 [m+1, j) 中 i = m + 1; else if (nums[m] > target) // 此情况说明 target 在区间 [i, m) 中 diff --git a/codes/dart/chapter_binary_search/binary_search.dart b/codes/dart/chapter_binary_search/binary_search.dart index d195ae6dc..348987f80 100644 --- a/codes/dart/chapter_binary_search/binary_search.dart +++ b/codes/dart/chapter_binary_search/binary_search.dart @@ -10,7 +10,7 @@ int binarySearch(List nums, int target) { int i = 0, j = nums.length - 1; // 循环,当搜索区间为空时跳出(当 i > j 时为空) while (i <= j) { - int m = (i + j) ~/ 2; // 计算中点索引 m + int m = i + (j - i) ~/ 2; // 计算中点索引 m if (nums[m] < target) { // 此情况说明 target 在区间 [m+1, j] 中 i = m + 1; @@ -32,7 +32,7 @@ int binarySearchLCRO(List nums, int target) { int i = 0, j = nums.length; // 循环,当搜索区间为空时跳出(当 i = j 时为空) while (i < j) { - int m = (i + j) ~/ 2; // 计算中点索引 m + int m = i + (j - i) ~/ 2; // 计算中点索引 m if (nums[m] < target) { // 此情况说明 target 在区间 [m+1, j) 中 i = m + 1; diff --git a/codes/go/chapter_binary_search/binary_search.go b/codes/go/chapter_binary_search/binary_search.go index a1f61b27b..1f792c417 100644 --- a/codes/go/chapter_binary_search/binary_search.go +++ b/codes/go/chapter_binary_search/binary_search.go @@ -10,7 +10,7 @@ func binarySearch(nums []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) for i <= j { - m := (i + j) / 2 // 计算中点索引 m + m := i + (j-i)/2 // 计算中点索引 m if nums[m] < target { // 此情况说明 target 在区间 [m+1, j] 中 i = m + 1 } else if nums[m] > target { // 此情况说明 target 在区间 [i, m-1] 中 @@ -29,7 +29,7 @@ func binarySearchLCRO(nums []int, target int) int { i, j := 0, len(nums) // 循环,当搜索区间为空时跳出(当 i = j 时为空) for i < j { - m := (i + j) / 2 // 计算中点索引 m + m := i + (j-i)/2 // 计算中点索引 m if nums[m] < target { // 此情况说明 target 在区间 [m+1, j) 中 i = m + 1 } else if nums[m] > target { // 此情况说明 target 在区间 [i, m) 中 diff --git a/codes/java/chapter_binary_search/binary_search.java b/codes/java/chapter_binary_search/binary_search.java index f9608a663..b2115e445 100644 --- a/codes/java/chapter_binary_search/binary_search.java +++ b/codes/java/chapter_binary_search/binary_search.java @@ -13,12 +13,12 @@ public class binary_search { int i = 0, j = nums.length - 1; // 循环,当搜索区间为空时跳出(当 i > j 时为空) while (i <= j) { - int m = (i + j) / 2; // 计算中点索引 m - if (nums[m] < target) // 此情况说明 target 在区间 [m+1, j] 中 + int m = i + (j - i) / 2; // 计算中点索引 m + if (nums[m] < target) // 此情况说明 target 在区间 [m+1, j] 中 i = m + 1; else if (nums[m] > target) // 此情况说明 target 在区间 [i, m-1] 中 j = m - 1; - else // 找到目标元素,返回其索引 + else // 找到目标元素,返回其索引 return m; } // 未找到目标元素,返回 -1 @@ -31,12 +31,12 @@ public class binary_search { int i = 0, j = nums.length; // 循环,当搜索区间为空时跳出(当 i = j 时为空) while (i < j) { - int m = (i + j) / 2; // 计算中点索引 m - if (nums[m] < target) // 此情况说明 target 在区间 [m+1, j) 中 + int m = i + (j - i) / 2; // 计算中点索引 m + if (nums[m] < target) // 此情况说明 target 在区间 [m+1, j) 中 i = m + 1; else if (nums[m] > target) // 此情况说明 target 在区间 [i, m) 中 j = m; - else // 找到目标元素,返回其索引 + else // 找到目标元素,返回其索引 return m; } // 未找到目标元素,返回 -1 diff --git a/codes/javascript/chapter_binary_search/binary_search.js b/codes/javascript/chapter_binary_search/binary_search.js index 2e33ac0b8..8c1224548 100644 --- a/codes/javascript/chapter_binary_search/binary_search.js +++ b/codes/javascript/chapter_binary_search/binary_search.js @@ -12,7 +12,7 @@ function binarySearch(nums, target) { // 循环,当搜索区间为空时跳出(当 i > j 时为空) while (i <= j) { // 计算中点索引 m ,使用 parseInt() 向下取整 - const m = parseInt((i + j) / 2); + const m = parseInt(i + (j - i) / 2); if (nums[m] < target) // 此情况说明 target 在区间 [m+1, j] 中 i = m + 1; @@ -33,7 +33,7 @@ function binarySearchLCRO(nums, target) { // 循环,当搜索区间为空时跳出(当 i = j 时为空) while (i < j) { // 计算中点索引 m ,使用 parseInt() 向下取整 - const m = parseInt((i + j) / 2); + const m = parseInt(i + (j - i) / 2); if (nums[m] < target) // 此情况说明 target 在区间 [m+1, j) 中 i = m + 1; diff --git a/codes/python/chapter_binary_search/binary_search.py b/codes/python/chapter_binary_search/binary_search.py index 056e9baf4..92e1587d2 100644 --- a/codes/python/chapter_binary_search/binary_search.py +++ b/codes/python/chapter_binary_search/binary_search.py @@ -9,12 +9,13 @@ def binary_search(nums: list[int], target: int) -> int: """二分查找(双闭区间)""" # 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 + # 循环,当搜索区间为空时跳出(当 i > j 时为空) while i <= j: m = (i + j) // 2 # 计算中点索引 m - if nums[m] < target: # 此情况说明 target 在区间 [m+1, j] 中 - i = m + 1 - elif nums[m] > target: # 此情况说明 target 在区间 [i, m-1] 中 - j = m - 1 + 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: return m # 找到目标元素,返回其索引 return -1 # 未找到目标元素,返回 -1 @@ -27,12 +28,12 @@ def binary_search_lcro(nums: list[int], target: int) -> int: # 循环,当搜索区间为空时跳出(当 i = j 时为空) while i < j: m = (i + j) // 2 # 计算中点索引 m - if nums[m] < target: # 此情况说明 target 在区间 [m+1, j) 中 - i = m + 1 - elif nums[m] > target: # 此情况说明 target 在区间 [i, m) 中 - j = m - else: # 找到目标元素,返回其索引 - return 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 # 此情况说明 target 在区间 [i, m) 中 + else: + return m # 找到目标元素,返回其索引 return -1 # 未找到目标元素,返回 -1 diff --git a/codes/rust/chapter_binary_search/binary_search.rs b/codes/rust/chapter_binary_search/binary_search.rs index 6a74c9067..1c12bd91b 100644 --- a/codes/rust/chapter_binary_search/binary_search.rs +++ b/codes/rust/chapter_binary_search/binary_search.rs @@ -11,12 +11,12 @@ fn binary_search(nums: &[i32], target: i32) -> i32 { let mut j = nums.len() - 1; // 循环,当搜索区间为空时跳出(当 i > j 时为空) while i <= j { - let m = (i + j) / 2; // 计算中点索引 m - if nums[m] < target { // 此情况说明 target 在区间 [m+1, j] 中 + let m = i + (j - i) / 2; // 计算中点索引 m + if nums[m] < target { // 此情况说明 target 在区间 [m+1, j] 中 i = m + 1; - } else if nums[m] > target { // 此情况说明 target 在区间 [i, m-1] 中 + } else if nums[m] > target { // 此情况说明 target 在区间 [i, m-1] 中 j = m - 1; - } else { // 找到目标元素,返回其索引 + } else { // 找到目标元素,返回其索引 return m as i32; } } @@ -31,12 +31,12 @@ fn binary_search_lcro(nums: &[i32], target: i32) -> i32 { let mut j = nums.len(); // 循环,当搜索区间为空时跳出(当 i = j 时为空) while i < j { - let m = (i + j) / 2; // 计算中点索引 m - if nums[m] < target { // 此情况说明 target 在区间 [m+1, j) 中 + let m = i + (j - i) / 2; // 计算中点索引 m + if nums[m] < target { // 此情况说明 target 在区间 [m+1, j) 中 i = m + 1; - } else if nums[m] > target { // 此情况说明 target 在区间 [i, m) 中 + } else if nums[m] > target { // 此情况说明 target 在区间 [i, m) 中 j = m - 1; - } else { // 找到目标元素,返回其索引 + } else { // 找到目标元素,返回其索引 return m as i32; } } @@ -56,4 +56,4 @@ pub fn main() { // 二分查找(左闭右开) index = binary_search_lcro(&nums, target); println!("目标元素 6 的索引 = {index}"); -} \ No newline at end of file +} diff --git a/codes/swift/chapter_binary_search/binary_search.swift b/codes/swift/chapter_binary_search/binary_search.swift index 4eddcc6fa..978979b46 100644 --- a/codes/swift/chapter_binary_search/binary_search.swift +++ b/codes/swift/chapter_binary_search/binary_search.swift @@ -11,7 +11,7 @@ func binarySearch(nums: [Int], target: Int) -> Int { var j = nums.count - 1 // 循环,当搜索区间为空时跳出(当 i > j 时为空) while i <= j { - let m = (i + j) / 2 // 计算中点索引 m + let m = i + (j - i) / 2 // 计算中点索引 m if nums[m] < target { // 此情况说明 target 在区间 [m+1, j] 中 i = m + 1 } else if nums[m] > target { // 此情况说明 target 在区间 [i, m-1] 中 @@ -31,7 +31,7 @@ func binarySearchLCRO(nums: [Int], target: Int) -> Int { var j = nums.count // 循环,当搜索区间为空时跳出(当 i = j 时为空) while i < j { - let m = (i + j) / 2 // 计算中点索引 m + let m = i + (j - i) / 2 // 计算中点索引 m if nums[m] < target { // 此情况说明 target 在区间 [m+1, j) 中 i = m + 1 } else if nums[m] > target { // 此情况说明 target 在区间 [i, m) 中 diff --git a/codes/typescript/chapter_binary_search/binary_search.ts b/codes/typescript/chapter_binary_search/binary_search.ts index bc305b3c4..34b170e4c 100644 --- a/codes/typescript/chapter_binary_search/binary_search.ts +++ b/codes/typescript/chapter_binary_search/binary_search.ts @@ -12,7 +12,7 @@ function binarySearch(nums: number[], target: number): number { // 循环,当搜索区间为空时跳出(当 i > j 时为空) while (i <= j) { // 计算中点索引 m - const m = Math.floor((i + j) / 2); + const m = Math.floor(i + (j - i) / 2); if (nums[m] < target) { // 此情况说明 target 在区间 [m+1, j] 中 i = m + 1; @@ -35,7 +35,7 @@ function binarySearchLCRO(nums: number[], target: number): number { // 循环,当搜索区间为空时跳出(当 i = j 时为空) while (i < j) { // 计算中点索引 m - const m = Math.floor((i + j) / 2); + const m = Math.floor(i + (j - i) / 2); if (nums[m] < target) { // 此情况说明 target 在区间 [m+1, j) 中 i = m + 1; diff --git a/codes/zig/chapter_binary_search/binary_search.zig b/codes/zig/chapter_binary_search/binary_search.zig index ac0bf0351..15022e26a 100644 --- a/codes/zig/chapter_binary_search/binary_search.zig +++ b/codes/zig/chapter_binary_search/binary_search.zig @@ -12,7 +12,7 @@ fn binarySearch(comptime T: type, nums: std.ArrayList(T), target: T) T { var j: usize = nums.items.len - 1; // 循环,当搜索区间为空时跳出(当 i > j 时为空) while (i <= j) { - var m = (i + j) / 2; // 计算中点索引 m + var m = i + (j - i) / 2; // 计算中点索引 m if (nums.items[m] < target) { // 此情况说明 target 在区间 [m+1, j] 中 i = m + 1; } else if (nums.items[m] > target) { // 此情况说明 target 在区间 [i, m-1] 中 @@ -32,7 +32,7 @@ fn binarySearchLCRO(comptime T: type, nums: std.ArrayList(T), target: T) T { var j: usize = nums.items.len; // 循环,当搜索区间为空时跳出(当 i = j 时为空) while (i <= j) { - var m = (i + j) / 2; // 计算中点索引 m + var m = i + (j - i) / 2; // 计算中点索引 m if (nums.items[m] < target) { // 此情况说明 target 在区间 [m+1, j) 中 i = m + 1; } else if (nums.items[m] > 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b/docs/chapter_binary_search/binary_search.md @@ -1,24 +1,30 @@ # 二分查找 -「二分查找 Binary Search」利用数据的有序性,通过每轮减少一半搜索范围来定位目标元素。 +「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,实现定位目标元素。 -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。数组索引的取值范围为: +我们先来求解一个简单的二分查找问题。 -$$ -0, 1, 2, \cdots, n-1 -$$ +!!! 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。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。" -我们通常使用以下两种方法来表示这个取值范围: +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。 -1. **双闭区间 $[0, n-1]$** ,即两个边界都包含自身;在此方法下,区间 $[i, i]$ 仍包含 $1$ 个元素; -2. **左闭右开 $[0, n)$** ,即左边界包含自身、右边界不包含自身;在此方法下,区间 $[i, i)$ 不包含元素; +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: -## 双闭区间实现 +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: + 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; -首先,我们采用“双闭区间”表示法,在数组 `nums` 中查找目标元素 `target` 的对应索引。 +**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。 + +=== "<0>" + ![二分查找步骤](binary_search.assets/binary_search_step0.png) === "<1>" - ![二分查找步骤](binary_search.assets/binary_search_step1.png) + ![binary_search_step1](binary_search.assets/binary_search_step1.png) === "<2>" ![binary_search_step2](binary_search.assets/binary_search_step2.png) @@ -38,7 +44,9 @@ $$ === "<7>" ![binary_search_step7](binary_search.assets/binary_search_step7.png) -二分查找在“双闭区间”表示下的代码如下所示。 +值得注意的是,**当数组长度 $n$ 很大时,加法 $i + j$ 的结果可能会超出 `int` 类型的取值范围**。为了避免大数越界,我们通常采用公式 $m = \lfloor {i + (j - i) / 2} \rfloor$ 来计算中点。 + +有趣的是,理论上 Python 的数字可以无限大(取决于内存大小),因此无需考虑大数越界问题。 === "Java" @@ -100,96 +108,15 @@ $$ [class]{}-[func]{binarySearch} ``` -需要注意的是,**当数组长度非常大时,加法 $i + j$ 的结果可能会超出 `int` 类型的取值范围**。在这种情况下,我们需要采用一种更安全的计算中点的方法。 - -=== "Java" - - ```java title="" - // (i + j) 有可能超出 int 的取值范围 - int m = (i + j) / 2; - // 更换为此写法则不会越界 - int m = i + (j - i) / 2; - ``` - -=== "C++" - - ```cpp title="" - // (i + j) 有可能超出 int 的取值范围 - int m = (i + j) / 2; - // 更换为此写法则不会越界 - int m = i + (j - i) / 2; - ``` - -=== "Python" - - ```py title="" - # Python 中的数字理论上可以无限大(取决于内存大小) - # 因此无需考虑大数越界问题 - ``` - -=== "Go" - - ```go title="" - // (i + j) 有可能超出 int 的取值范围 - m := (i + j) / 2 - // 更换为此写法则不会越界 - m := i + (j - i) / 2 - ``` +时间复杂度为 $O(\log n)$ 。每轮缩小一半区间,因此二分循环次数为 $\log_2 n$ 。 -=== "JavaScript" +空间复杂度为 $O(1)$ 。指针 `i` , `j` 使用常数大小空间。 - ```javascript title="" - // (i + j) 有可能超出 int 的取值范围 - let m = parseInt((i + j) / 2); - // 更换为此写法则不会越界 - let m = parseInt(i + (j - i) / 2); - ``` +## 区间表示方法 -=== "TypeScript" +除了上述的双闭区间外,常见的区间表示还有“左闭右开”区间,定义为 $[0, n)$ ,即左边界包含自身,右边界不包含自身。在该表示下,区间 $[i, j]$ 在 $i = j$ 时为空。 - ```typescript title="" - // (i + j) 有可能超出 Number 的取值范围 - let m = Math.floor((i + j) / 2); - // 更换为此写法则不会越界 - let m = Math.floor(i + (j - i) / 2); - ``` - -=== "C" - - ```c title="" - // (i + j) 有可能超出 int 的取值范围 - int m = (i + j) / 2; - // 更换为此写法则不会越界 - int m = i + (j - i) / 2; - ``` - -=== "C#" - - ```csharp title="" - // (i + j) 有可能超出 int 的取值范围 - int m = (i + j) / 2; - // 更换为此写法则不会越界 - int m = i + (j - i) / 2; - ``` - -=== "Swift" - - ```swift title="" - // (i + j) 有可能超出 int 的取值范围 - let m = (i + j) / 2 - // 更换为此写法则不会越界 - let m = i + (j - 1) / 2 - ``` - -=== "Zig" - - ```zig title="" - - ``` - -## 左闭右开实现 - -我们可以采用“左闭右开”的表示法,编写具有相同功能的二分查找代码。 +我们可以基于该表示实现具有相同功能的二分查找算法。 === "Java" @@ -251,24 +178,11 @@ $$ [class]{}-[func]{binarySearchLCRO} ``` -对比这两种代码写法,我们可以发现以下不同点: - -

- -在“双闭区间”表示法中,由于对左右两边界的定义相同,因此缩小区间的 $i$ 和 $j$ 的处理方法也是对称的,这样更不容易出错。因此,**建议采用“双闭区间”的写法**。 - -## 复杂度分析 +如下图所示,在两种区间表示下,二分查找算法的初始化、循环条件和缩小区间操作皆有所不同。 -**时间复杂度 $O(\log n)$** :其中 $n$ 为数组长度;每轮排除一半的区间,因此循环轮数为 $\log_2 n$ ,使用 $O(\log n)$ 时间。 +在“双闭区间”表示法中,由于左右边界都被定义为闭区间,因此指针 $i$ 和 $j$ 缩小区间操作也是对称的。这样更不容易出错。因此,**我们通常采用“双闭区间”的写法**。 -**空间复杂度 $O(1)$** :指针 `i` , `j` 使用常数大小空间。 +![两种区间定义](binary_search.assets/binary_search_ranges.png) ## 优点与局限性