From 881d57379036f9f680caa46f8db22d191cd64194 Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Mon, 17 Apr 2023 18:22:18 +0800 Subject: [PATCH] Refactor the articles related to searching algorithm. Add the chapter of binary search. Add the section of searching algorithm revisited. (#464) --- codes/c/CMakeLists.txt | 5 +- codes/c/chapter_binary_search/CMakeLists.txt | 1 + .../binary_search.c | 0 .../CMakeLists.txt | 1 - codes/c/chapter_searching/CMakeLists.txt | 5 +- .../leetcode_two_sum.c | 0 codes/cpp/CMakeLists.txt | 1 + .../cpp/chapter_binary_search/CMakeLists.txt | 1 + .../binary_search.cpp | 0 .../CMakeLists.txt | 3 +- codes/cpp/chapter_searching/CMakeLists.txt | 2 +- .../leetcode_two_sum.cpp | 0 .../binary_search.cs | 2 +- .../leetcode_two_sum.cs | 2 +- .../binary_search.go | 2 +- .../binary_search_test.go | 2 +- .../leetcode_two_sum.go | 2 +- .../leetcode_two_sum_test.go | 2 +- .../backtrack_find_constrained_paths.java | 2 + .../preorder_find_constrained_paths.java | 2 + .../preorder_find_nodes.java | 2 + .../preorder_find_paths.java | 2 + .../binary_search.java | 2 +- .../leetcode_two_sum.java | 2 +- .../binary_search.js | 0 .../leetcode_two_sum.js | 0 .../binary_search.py | 0 .../leetcode_two_sum.py | 0 .../binary_search.rs | 0 .../binary_search.swift | 0 .../leetcode_two_sum.swift | 0 .../binary_search.ts | 0 .../leetcode_two_sum.ts | 0 .../binary_search.zig | 1 - .../leetcode_two_sum.zig | 0 .../binary_search_step1.png | Bin .../binary_search_step2.png | Bin .../binary_search_step3.png | Bin .../binary_search_step4.png | Bin .../binary_search_step5.png | Bin .../binary_search_step6.png | Bin .../binary_search_step7.png | Bin .../binary_search.md | 168 +++++++++--------- .../space_complexity.md | 8 + .../summary.md | 6 +- docs/chapter_hashing/hash_map.md | 21 +-- .../hash_search_listnode.png | Bin 70105 -> 0 bytes docs/chapter_searching/hashing_search.md | 152 ---------------- docs/chapter_searching/linear_search.md | 143 --------------- .../hashing_search.png} | Bin .../linear_search.png | Bin .../replace_linear_by_hashing.md} | 30 ++-- .../searching_algorithms.png | Bin 0 -> 104675 bytes .../searching_algorithm_revisited.md | 81 +++++++++ docs/chapter_searching/summary.md | 18 +- docs/chapter_tree/binary_search_tree.md | 31 +--- mkdocs.yml | 68 +++---- 57 files changed, 265 insertions(+), 505 deletions(-) create mode 100644 codes/c/chapter_binary_search/CMakeLists.txt rename codes/c/{chapter_searching => chapter_binary_search}/binary_search.c (100%) rename codes/c/{chapter_computational_complexity => chapter_searching}/leetcode_two_sum.c (100%) create mode 100644 codes/cpp/chapter_binary_search/CMakeLists.txt rename codes/cpp/{chapter_searching => chapter_binary_search}/binary_search.cpp (100%) rename codes/cpp/{chapter_computational_complexity => chapter_searching}/leetcode_two_sum.cpp (100%) rename codes/csharp/{chapter_searching => chapter_binary_search}/binary_search.cs (98%) rename codes/csharp/{chapter_computational_complexity => chapter_searching}/leetcode_two_sum.cs (96%) rename codes/go/{chapter_searching => chapter_binary_search}/binary_search.go (97%) rename codes/go/{chapter_searching => chapter_binary_search}/binary_search_test.go (94%) rename codes/go/{chapter_computational_complexity => chapter_searching}/leetcode_two_sum.go (94%) rename codes/go/{chapter_computational_complexity => chapter_searching}/leetcode_two_sum_test.go (91%) rename codes/java/{chapter_searching => chapter_binary_search}/binary_search.java (98%) rename codes/java/{chapter_computational_complexity => chapter_searching}/leetcode_two_sum.java (97%) rename codes/javascript/{chapter_searching => chapter_binary_search}/binary_search.js (100%) rename codes/javascript/{chapter_computational_complexity => chapter_searching}/leetcode_two_sum.js (100%) rename codes/python/{chapter_searching => chapter_binary_search}/binary_search.py (100%) rename codes/python/{chapter_computational_complexity => chapter_searching}/leetcode_two_sum.py (100%) rename codes/rust/{chapter_searching => chapter_binary_search}/binary_search.rs (100%) rename codes/swift/{chapter_searching => chapter_binary_search}/binary_search.swift (100%) rename codes/swift/{chapter_computational_complexity => chapter_searching}/leetcode_two_sum.swift (100%) rename codes/typescript/{chapter_searching => chapter_binary_search}/binary_search.ts (100%) rename codes/typescript/{chapter_computational_complexity => chapter_searching}/leetcode_two_sum.ts (100%) rename codes/zig/{chapter_searching => chapter_binary_search}/binary_search.zig (99%) rename codes/zig/{chapter_computational_complexity => chapter_searching}/leetcode_two_sum.zig (100%) rename docs/{chapter_searching => chapter_binary_search}/binary_search.assets/binary_search_step1.png (100%) rename docs/{chapter_searching => chapter_binary_search}/binary_search.assets/binary_search_step2.png (100%) rename docs/{chapter_searching => chapter_binary_search}/binary_search.assets/binary_search_step3.png (100%) rename docs/{chapter_searching => chapter_binary_search}/binary_search.assets/binary_search_step4.png (100%) rename docs/{chapter_searching => chapter_binary_search}/binary_search.assets/binary_search_step5.png (100%) rename docs/{chapter_searching => chapter_binary_search}/binary_search.assets/binary_search_step6.png (100%) rename docs/{chapter_searching => chapter_binary_search}/binary_search.assets/binary_search_step7.png (100%) rename docs/{chapter_searching => chapter_binary_search}/binary_search.md (93%) delete mode 100644 docs/chapter_searching/hashing_search.assets/hash_search_listnode.png delete mode 100755 docs/chapter_searching/hashing_search.md delete mode 100755 docs/chapter_searching/linear_search.md rename docs/chapter_searching/{hashing_search.assets/hash_search_index.png => replace_linear_by_hashing.assets/hashing_search.png} (100%) rename docs/chapter_searching/{linear_search.assets => replace_linear_by_hashing.assets}/linear_search.png (100%) rename docs/{chapter_computational_complexity/space_time_tradeoff.md => chapter_searching/replace_linear_by_hashing.md} (59%) create mode 100644 docs/chapter_searching/searching_algorithm_revisited.assets/searching_algorithms.png create mode 100644 docs/chapter_searching/searching_algorithm_revisited.md diff --git a/codes/c/CMakeLists.txt b/codes/c/CMakeLists.txt index 853475b5f..588025726 100644 --- a/codes/c/CMakeLists.txt +++ b/codes/c/CMakeLists.txt @@ -8,8 +8,7 @@ include_directories(./include) add_subdirectory(include) add_subdirectory(chapter_computational_complexity) add_subdirectory(chapter_array_and_linkedlist) -add_subdirectory(chapter_sorting) -add_subdirectory(chapter_tree) add_subdirectory(chapter_stack_and_queue) +add_subdirectory(chapter_binary_search) add_subdirectory(chapter_heap) -add_subdirectory(chapter_searching) \ No newline at end of file +add_subdirectory(chapter_searching) diff --git a/codes/c/chapter_binary_search/CMakeLists.txt b/codes/c/chapter_binary_search/CMakeLists.txt new file mode 100644 index 000000000..f7ddf6761 --- /dev/null +++ b/codes/c/chapter_binary_search/CMakeLists.txt @@ -0,0 +1 @@ +add_executable(linear_search linear_search.c) \ No newline at end of file diff --git a/codes/c/chapter_searching/binary_search.c b/codes/c/chapter_binary_search/binary_search.c similarity index 100% rename from codes/c/chapter_searching/binary_search.c rename to codes/c/chapter_binary_search/binary_search.c diff --git a/codes/c/chapter_computational_complexity/CMakeLists.txt b/codes/c/chapter_computational_complexity/CMakeLists.txt index 30646d4e1..b11e25877 100644 --- a/codes/c/chapter_computational_complexity/CMakeLists.txt +++ b/codes/c/chapter_computational_complexity/CMakeLists.txt @@ -1,4 +1,3 @@ add_executable(time_complexity time_complexity.c) add_executable(worst_best_time_complexity worst_best_time_complexity.c) -add_executable(leetcode_two_sum leetcode_two_sum.c) add_executable(space_complexity space_complexity.c) \ No newline at end of file diff --git a/codes/c/chapter_searching/CMakeLists.txt b/codes/c/chapter_searching/CMakeLists.txt index d6ef4b895..58f81dd39 100644 --- a/codes/c/chapter_searching/CMakeLists.txt +++ b/codes/c/chapter_searching/CMakeLists.txt @@ -1,3 +1,2 @@ -add_executable(binary_search binary_search.c ) -add_executable(linear_search linear_search.c) -add_executable(hashing_search hashing_search.c) \ No newline at end of file +add_executable(binary_search binary_search.c) +add_executable(leetcode_two_sum leetcode_two_sum.c) \ No newline at end of file diff --git a/codes/c/chapter_computational_complexity/leetcode_two_sum.c b/codes/c/chapter_searching/leetcode_two_sum.c similarity index 100% rename from codes/c/chapter_computational_complexity/leetcode_two_sum.c rename to codes/c/chapter_searching/leetcode_two_sum.c diff --git a/codes/cpp/CMakeLists.txt b/codes/cpp/CMakeLists.txt index 9e3f5583a..164e94c81 100644 --- a/codes/cpp/CMakeLists.txt +++ b/codes/cpp/CMakeLists.txt @@ -8,6 +8,7 @@ include_directories(./include) add_subdirectory(chapter_computational_complexity) add_subdirectory(chapter_array_and_linkedlist) add_subdirectory(chapter_stack_and_queue) +add_subdirectory(chapter_binary_search) add_subdirectory(chapter_hashing) add_subdirectory(chapter_tree) add_subdirectory(chapter_heap) diff --git a/codes/cpp/chapter_binary_search/CMakeLists.txt b/codes/cpp/chapter_binary_search/CMakeLists.txt new file mode 100644 index 000000000..304737e1f --- /dev/null +++ b/codes/cpp/chapter_binary_search/CMakeLists.txt @@ -0,0 +1 @@ +add_executable(binary_search binary_search.cpp) \ No newline at end of file diff --git a/codes/cpp/chapter_searching/binary_search.cpp b/codes/cpp/chapter_binary_search/binary_search.cpp similarity index 100% rename from codes/cpp/chapter_searching/binary_search.cpp rename to codes/cpp/chapter_binary_search/binary_search.cpp diff --git a/codes/cpp/chapter_computational_complexity/CMakeLists.txt b/codes/cpp/chapter_computational_complexity/CMakeLists.txt index b484b3af2..4c1c11261 100644 --- a/codes/cpp/chapter_computational_complexity/CMakeLists.txt +++ b/codes/cpp/chapter_computational_complexity/CMakeLists.txt @@ -1,4 +1,3 @@ -add_executable(leetcode_two_sum leetcode_two_sum.cpp) -add_executable(space_complexity space_complexity.cpp ) +add_executable(space_complexity space_complexity.cpp) add_executable(time_complexity time_complexity.cpp) add_executable(worst_best_time_complexity worst_best_time_complexity.cpp) \ No newline at end of file diff --git a/codes/cpp/chapter_searching/CMakeLists.txt b/codes/cpp/chapter_searching/CMakeLists.txt index 9d2a23995..04448bf5d 100644 --- a/codes/cpp/chapter_searching/CMakeLists.txt +++ b/codes/cpp/chapter_searching/CMakeLists.txt @@ -1,3 +1,3 @@ -add_executable(binary_search binary_search.cpp) add_executable(hashing_search hashing_search.cpp) +add_executable(leetcode_two_sum leetcode_two_sum.cpp) add_executable(linear_search linear_search.cpp) \ No newline at end of file diff --git a/codes/cpp/chapter_computational_complexity/leetcode_two_sum.cpp b/codes/cpp/chapter_searching/leetcode_two_sum.cpp similarity index 100% rename from codes/cpp/chapter_computational_complexity/leetcode_two_sum.cpp rename to codes/cpp/chapter_searching/leetcode_two_sum.cpp diff --git a/codes/csharp/chapter_searching/binary_search.cs b/codes/csharp/chapter_binary_search/binary_search.cs similarity index 98% rename from codes/csharp/chapter_searching/binary_search.cs rename to codes/csharp/chapter_binary_search/binary_search.cs index 361f40723..0820399f4 100644 --- a/codes/csharp/chapter_searching/binary_search.cs +++ b/codes/csharp/chapter_binary_search/binary_search.cs @@ -6,7 +6,7 @@ using NUnit.Framework; -namespace hello_algo.chapter_searching; +namespace hello_algo.chapter_binary_search; public class binary_search { diff --git a/codes/csharp/chapter_computational_complexity/leetcode_two_sum.cs b/codes/csharp/chapter_searching/leetcode_two_sum.cs similarity index 96% rename from codes/csharp/chapter_computational_complexity/leetcode_two_sum.cs rename to codes/csharp/chapter_searching/leetcode_two_sum.cs index 4d716f971..e2557508a 100644 --- a/codes/csharp/chapter_computational_complexity/leetcode_two_sum.cs +++ b/codes/csharp/chapter_searching/leetcode_two_sum.cs @@ -6,7 +6,7 @@ using NUnit.Framework; -namespace hello_algo.chapter_computational_complexity; +namespace hello_algo.chapter_searching; public class leetcode_two_sum { diff --git a/codes/go/chapter_searching/binary_search.go b/codes/go/chapter_binary_search/binary_search.go similarity index 97% rename from codes/go/chapter_searching/binary_search.go rename to codes/go/chapter_binary_search/binary_search.go index cbaa85162..159524bd9 100644 --- a/codes/go/chapter_searching/binary_search.go +++ b/codes/go/chapter_binary_search/binary_search.go @@ -2,7 +2,7 @@ // Created Time: 2022-12-05 // Author: Slone123c (274325721@qq.com) -package chapter_searching +package chapter_binary_search /* 二分查找（双闭区间） */ func binarySearch(nums []int, target int) int { diff --git a/codes/go/chapter_searching/binary_search_test.go b/codes/go/chapter_binary_search/binary_search_test.go similarity index 94% rename from codes/go/chapter_searching/binary_search_test.go rename to codes/go/chapter_binary_search/binary_search_test.go index 3dada9593..0cfedbdcd 100644 --- a/codes/go/chapter_searching/binary_search_test.go +++ b/codes/go/chapter_binary_search/binary_search_test.go @@ -2,7 +2,7 @@ // Created Time: 2022-12-05 // Author: Slone123c (274325721@qq.com) -package chapter_searching +package chapter_binary_search import ( "fmt" diff --git a/codes/go/chapter_computational_complexity/leetcode_two_sum.go b/codes/go/chapter_searching/leetcode_two_sum.go similarity index 94% rename from codes/go/chapter_computational_complexity/leetcode_two_sum.go rename to codes/go/chapter_searching/leetcode_two_sum.go index 663020381..138ea88c5 100644 --- a/codes/go/chapter_computational_complexity/leetcode_two_sum.go +++ b/codes/go/chapter_searching/leetcode_two_sum.go @@ -2,7 +2,7 @@ // Created Time: 2022-11-25 // Author: reanon (793584285@qq.com) -package chapter_computational_complexity +package chapter_searching /* 方法一：暴力枚举 */ func twoSumBruteForce(nums []int, target int) []int { diff --git a/codes/go/chapter_computational_complexity/leetcode_two_sum_test.go b/codes/go/chapter_searching/leetcode_two_sum_test.go similarity index 91% rename from codes/go/chapter_computational_complexity/leetcode_two_sum_test.go rename to codes/go/chapter_searching/leetcode_two_sum_test.go index 6e44eef98..e45d5fb21 100644 --- a/codes/go/chapter_computational_complexity/leetcode_two_sum_test.go +++ b/codes/go/chapter_searching/leetcode_two_sum_test.go @@ -2,7 +2,7 @@ // Created Time: 2022-11-25 // Author: reanon (793584285@qq.com) -package chapter_computational_complexity +package chapter_searching import ( "fmt" diff --git a/codes/java/chapter_backtracking/backtrack_find_constrained_paths.java b/codes/java/chapter_backtracking/backtrack_find_constrained_paths.java index 420c469e0..61e555ada 100644 --- a/codes/java/chapter_backtracking/backtrack_find_constrained_paths.java +++ b/codes/java/chapter_backtracking/backtrack_find_constrained_paths.java @@ -4,6 +4,8 @@ * Author: Krahets (krahets@163.com) */ +package chapter_backtracking; + import include.*; import java.util.*; diff --git a/codes/java/chapter_backtracking/preorder_find_constrained_paths.java b/codes/java/chapter_backtracking/preorder_find_constrained_paths.java index 3653710ed..304d3dbfe 100644 --- a/codes/java/chapter_backtracking/preorder_find_constrained_paths.java +++ b/codes/java/chapter_backtracking/preorder_find_constrained_paths.java @@ -4,6 +4,8 @@ * Author: Krahets (krahets@163.com) */ +package chapter_backtracking; + import include.*; import java.util.*; diff --git a/codes/java/chapter_backtracking/preorder_find_nodes.java b/codes/java/chapter_backtracking/preorder_find_nodes.java index 085a76593..966d1b7e6 100644 --- a/codes/java/chapter_backtracking/preorder_find_nodes.java +++ b/codes/java/chapter_backtracking/preorder_find_nodes.java @@ -4,6 +4,8 @@ * Author: Krahets (krahets@163.com) */ +package chapter_backtracking; + import include.*; import java.util.*; diff --git a/codes/java/chapter_backtracking/preorder_find_paths.java b/codes/java/chapter_backtracking/preorder_find_paths.java index 39568dcda..3131673e5 100644 --- a/codes/java/chapter_backtracking/preorder_find_paths.java +++ b/codes/java/chapter_backtracking/preorder_find_paths.java @@ -4,6 +4,8 @@ * Author: Krahets (krahets@163.com) */ +package chapter_backtracking; + import include.*; import java.util.*; diff --git a/codes/java/chapter_searching/binary_search.java b/codes/java/chapter_binary_search/binary_search.java similarity index 98% rename from codes/java/chapter_searching/binary_search.java rename to codes/java/chapter_binary_search/binary_search.java index 491b47757..3e20efb56 100644 --- a/codes/java/chapter_searching/binary_search.java +++ b/codes/java/chapter_binary_search/binary_search.java @@ -4,7 +4,7 @@ * Author: Krahets (krahets@163.com) */ -package chapter_searching; +package chapter_binary_search; public class binary_search { /* 二分查找（双闭区间） */ diff --git a/codes/java/chapter_computational_complexity/leetcode_two_sum.java b/codes/java/chapter_searching/leetcode_two_sum.java similarity index 97% rename from codes/java/chapter_computational_complexity/leetcode_two_sum.java rename to codes/java/chapter_searching/leetcode_two_sum.java index 6b9784759..690e07bef 100644 --- a/codes/java/chapter_computational_complexity/leetcode_two_sum.java +++ b/codes/java/chapter_searching/leetcode_two_sum.java @@ -4,7 +4,7 @@ * Author: Krahets (krahets@163.com) */ -package chapter_computational_complexity; +package chapter_searching; import java.util.*; diff --git a/codes/javascript/chapter_searching/binary_search.js b/codes/javascript/chapter_binary_search/binary_search.js similarity index 100% rename from codes/javascript/chapter_searching/binary_search.js rename to codes/javascript/chapter_binary_search/binary_search.js diff --git a/codes/javascript/chapter_computational_complexity/leetcode_two_sum.js b/codes/javascript/chapter_searching/leetcode_two_sum.js similarity index 100% rename from codes/javascript/chapter_computational_complexity/leetcode_two_sum.js rename to codes/javascript/chapter_searching/leetcode_two_sum.js diff --git a/codes/python/chapter_searching/binary_search.py b/codes/python/chapter_binary_search/binary_search.py similarity index 100% rename from codes/python/chapter_searching/binary_search.py rename to codes/python/chapter_binary_search/binary_search.py diff --git a/codes/python/chapter_computational_complexity/leetcode_two_sum.py b/codes/python/chapter_searching/leetcode_two_sum.py similarity index 100% rename from codes/python/chapter_computational_complexity/leetcode_two_sum.py rename to codes/python/chapter_searching/leetcode_two_sum.py diff --git a/codes/rust/chapter_searching/binary_search.rs b/codes/rust/chapter_binary_search/binary_search.rs similarity index 100% rename from codes/rust/chapter_searching/binary_search.rs rename to codes/rust/chapter_binary_search/binary_search.rs diff --git a/codes/swift/chapter_searching/binary_search.swift b/codes/swift/chapter_binary_search/binary_search.swift similarity index 100% rename from codes/swift/chapter_searching/binary_search.swift rename to codes/swift/chapter_binary_search/binary_search.swift diff --git a/codes/swift/chapter_computational_complexity/leetcode_two_sum.swift b/codes/swift/chapter_searching/leetcode_two_sum.swift similarity index 100% rename from codes/swift/chapter_computational_complexity/leetcode_two_sum.swift rename to codes/swift/chapter_searching/leetcode_two_sum.swift diff --git a/codes/typescript/chapter_searching/binary_search.ts b/codes/typescript/chapter_binary_search/binary_search.ts similarity index 100% rename from codes/typescript/chapter_searching/binary_search.ts rename to codes/typescript/chapter_binary_search/binary_search.ts diff --git a/codes/typescript/chapter_computational_complexity/leetcode_two_sum.ts b/codes/typescript/chapter_searching/leetcode_two_sum.ts similarity index 100% rename from codes/typescript/chapter_computational_complexity/leetcode_two_sum.ts rename to codes/typescript/chapter_searching/leetcode_two_sum.ts diff --git a/codes/zig/chapter_searching/binary_search.zig b/codes/zig/chapter_binary_search/binary_search.zig similarity index 99% rename from codes/zig/chapter_searching/binary_search.zig rename to codes/zig/chapter_binary_search/binary_search.zig index 0a2f754f4..300b7c491 100644 --- a/codes/zig/chapter_searching/binary_search.zig +++ b/codes/zig/chapter_binary_search/binary_search.zig @@ -62,4 +62,3 @@ pub fn main() !void { _ = try std.io.getStdIn().reader().readByte(); } - diff --git a/codes/zig/chapter_computational_complexity/leetcode_two_sum.zig b/codes/zig/chapter_searching/leetcode_two_sum.zig similarity index 100% rename from codes/zig/chapter_computational_complexity/leetcode_two_sum.zig rename to codes/zig/chapter_searching/leetcode_two_sum.zig diff --git a/docs/chapter_searching/binary_search.assets/binary_search_step1.png b/docs/chapter_binary_search/binary_search.assets/binary_search_step1.png similarity index 100% rename from docs/chapter_searching/binary_search.assets/binary_search_step1.png rename to docs/chapter_binary_search/binary_search.assets/binary_search_step1.png diff --git a/docs/chapter_searching/binary_search.assets/binary_search_step2.png b/docs/chapter_binary_search/binary_search.assets/binary_search_step2.png similarity index 100% rename from docs/chapter_searching/binary_search.assets/binary_search_step2.png rename to docs/chapter_binary_search/binary_search.assets/binary_search_step2.png diff --git a/docs/chapter_searching/binary_search.assets/binary_search_step3.png b/docs/chapter_binary_search/binary_search.assets/binary_search_step3.png similarity index 100% rename from docs/chapter_searching/binary_search.assets/binary_search_step3.png rename to docs/chapter_binary_search/binary_search.assets/binary_search_step3.png diff --git a/docs/chapter_searching/binary_search.assets/binary_search_step4.png b/docs/chapter_binary_search/binary_search.assets/binary_search_step4.png similarity index 100% rename from docs/chapter_searching/binary_search.assets/binary_search_step4.png rename to docs/chapter_binary_search/binary_search.assets/binary_search_step4.png diff --git a/docs/chapter_searching/binary_search.assets/binary_search_step5.png b/docs/chapter_binary_search/binary_search.assets/binary_search_step5.png similarity index 100% rename from docs/chapter_searching/binary_search.assets/binary_search_step5.png rename to docs/chapter_binary_search/binary_search.assets/binary_search_step5.png diff --git a/docs/chapter_searching/binary_search.assets/binary_search_step6.png b/docs/chapter_binary_search/binary_search.assets/binary_search_step6.png similarity index 100% rename from docs/chapter_searching/binary_search.assets/binary_search_step6.png rename to docs/chapter_binary_search/binary_search.assets/binary_search_step6.png diff --git a/docs/chapter_searching/binary_search.assets/binary_search_step7.png b/docs/chapter_binary_search/binary_search.assets/binary_search_step7.png similarity index 100% rename from docs/chapter_searching/binary_search.assets/binary_search_step7.png rename to docs/chapter_binary_search/binary_search.assets/binary_search_step7.png diff --git a/docs/chapter_searching/binary_search.md b/docs/chapter_binary_search/binary_search.md similarity index 93% rename from docs/chapter_searching/binary_search.md rename to docs/chapter_binary_search/binary_search.md index 126b685c5..23d762a20 100755 --- a/docs/chapter_searching/binary_search.md +++ b/docs/chapter_binary_search/binary_search.md @@ -2,8 +2,6 @@ 「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。 -## 算法实现 - 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： $$ @@ -12,10 +10,10 @@ $$ 我们通常使用以下两种方法来表示这个取值范围： -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 $[0, 0]$ 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 $[0, 0)$ 不包含元素； +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； -### “双闭区间”实现 +## 双闭区间实现 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。 @@ -102,88 +100,7 @@ $$ [class]{}-[func]{binarySearch} ``` -### “左闭右开”实现 - -此外，我们也可以采用“左闭右开”的表示法，编写具有相同功能的二分查找代码。 - -=== "Java" - - ```java title="binary_search.java" - [class]{binary_search}-[func]{binarySearch1} - ``` - -=== "C++" - - ```cpp title="binary_search.cpp" - [class]{}-[func]{binarySearch1} - ``` - -=== "Python" - - ```python title="binary_search.py" - [class]{}-[func]{binary_search1} - ``` - -=== "Go" - - ```go title="binary_search.go" - [class]{}-[func]{binarySearch1} - ``` - -=== "JavaScript" - - ```javascript title="binary_search.js" - [class]{}-[func]{binarySearch1} - ``` - -=== "TypeScript" - - ```typescript title="binary_search.ts" - [class]{}-[func]{binarySearch1} - ``` - -=== "C" - - ```c title="binary_search.c" - [class]{}-[func]{binarySearch1} - ``` - -=== "C#" - - ```csharp title="binary_search.cs" - [class]{binary_search}-[func]{binarySearch1} - ``` - -=== "Swift" - - ```swift title="binary_search.swift" - [class]{}-[func]{binarySearch1} - ``` - -=== "Zig" - - ```zig title="binary_search.zig" - [class]{}-[func]{binarySearch1} - ``` - -### 两种表示对比 - -对比这两种代码写法，我们可以发现以下不同点： - -
- -| 表示方法 | 初始化指针 | 缩小区间 | 循环终止条件 | -| ------------------- | ------------------- | ------------------------- | ------------ | -| 双闭区间 $[0, n-1]$ | $i = 0$ , $j = n-1$ | $i = m + 1$ , $j = m - 1$ | $i > j$ | -| 左闭右开 $[0, n)$ | $i = 0$ , $j = n$ | $i = m + 1$ , $j = m$ | $i = j$ | - -
- -在“双闭区间”表示法中，由于对左右两边界的定义相同，因此缩小区间的 $i$ 和 $j$ 的处理方法也是对称的，这样更不容易出错。因此，**建议采用“双闭区间”的写法**。 - -### 大数越界处理 - -当数组长度非常大时，加法 $i + j$ 的结果可能会超出 `int` 类型的取值范围。在这种情况下，我们需要采用一种更安全的计算中点的方法。 +需要注意的是，**当数组长度非常大时，加法 $i + j$ 的结果可能会超出 `int` 类型的取值范围**。在这种情况下，我们需要采用一种更安全的计算中点的方法。 === "Java" @@ -267,6 +184,83 @@ $$ ``` +## 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。 + +=== "Java" + + ```java title="binary_search.java" + [class]{binary_search}-[func]{binarySearch1} + ``` + +=== "C++" + + ```cpp title="binary_search.cpp" + [class]{}-[func]{binarySearch1} + ``` + +=== "Python" + + ```python title="binary_search.py" + [class]{}-[func]{binary_search1} + ``` + +=== "Go" + + ```go title="binary_search.go" + [class]{}-[func]{binarySearch1} + ``` + +=== "JavaScript" + + ```javascript title="binary_search.js" + [class]{}-[func]{binarySearch1} + ``` + +=== "TypeScript" + + ```typescript title="binary_search.ts" + [class]{}-[func]{binarySearch1} + ``` + +=== "C" + + ```c title="binary_search.c" + [class]{}-[func]{binarySearch1} + ``` + +=== "C#" + + ```csharp title="binary_search.cs" + [class]{binary_search}-[func]{binarySearch1} + ``` + +=== "Swift" + + ```swift title="binary_search.swift" + [class]{}-[func]{binarySearch1} + ``` + +=== "Zig" + + ```zig title="binary_search.zig" + [class]{}-[func]{binarySearch1} + ``` + +对比这两种代码写法，我们可以发现以下不同点： + +
+ +| 表示方法 | 初始化指针 | 缩小区间 | 循环终止条件 | +| ------------------- | ------------------- | ------------------------- | ------------ | +| 双闭区间 $[0, n-1]$ | $i = 0$ , $j = n-1$ | $i = m + 1$ , $j = m - 1$ | $i > j$ | +| 左闭右开 $[0, n)$ | $i = 0$ , $j = n$ | $i = m + 1$ , $j = m$ | $i = j$ | + +
+ +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 **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。 diff --git a/docs/chapter_computational_complexity/space_complexity.md b/docs/chapter_computational_complexity/space_complexity.md index 8fa733858..ad42dbefa 100755 --- a/docs/chapter_computational_complexity/space_complexity.md +++ b/docs/chapter_computational_complexity/space_complexity.md @@ -961,3 +961,11 @@ $$ 例如“归并排序”算法，输入长度为 $n$ 的数组，每轮递归将数组从中点划分为两半，形成高度为 $\log n$ 的递归树，使用 $O(\log n)$ 栈帧空间。 再例如“数字转化为字符串”，输入任意正整数 $n$ ，它的位数为 $\log_{10} n$ ，即对应字符串长度为 $\log_{10} n$ ，因此空间复杂度为 $O(\log_{10} n) = O(\log n)$ 。 + +## 权衡时间与空间 + +理想情况下，我们希望算法的时间复杂度和空间复杂度都能达到最优。然而在实际情况中，同时优化时间复杂度和空间复杂度通常是非常困难的。 + +**降低时间复杂度通常需要以提升空间复杂度为代价，反之亦然**。我们将牺牲内存空间来提升算法运行速度的思路称为“以空间换时间”；反之，则称为“以时间换空间”。 + +选择哪种思路取决于我们更看重哪个方面。在大多数情况下，时间比空间更宝贵，因此以空间换时间通常是更常用的策略。当然，在数据量很大的情况下，控制空间复杂度也是非常重要的。 diff --git a/docs/chapter_computational_complexity/summary.md b/docs/chapter_computational_complexity/summary.md index 973fac6cd..ad2e3889f 100644 --- a/docs/chapter_computational_complexity/summary.md +++ b/docs/chapter_computational_complexity/summary.md @@ -1,12 +1,12 @@ # 小结 -### 算法效率评估 +**算法效率评估** - 时间效率和空间效率是评价算法性能的两个关键维度。 - 我们可以通过实际测试来评估算法效率，但难以消除测试环境的影响，且会耗费大量计算资源。 - 复杂度分析可以克服实际测试的弊端，分析结果适用于所有运行平台，并且能够揭示算法在不同数据规模下的效率。 -### 时间复杂度 +**时间复杂度** - 时间复杂度用于衡量算法运行时间随数据量增长的趋势，可以有效评估算法效率，但在某些情况下可能失效，如在输入数据量较小或时间复杂度相同时，无法精确对比算法效率的优劣。 - 最差时间复杂度使用大 $O$ 符号表示，即函数渐近上界，反映当 $n$ 趋向正无穷时，$T(n)$ 的增长级别。 @@ -15,7 +15,7 @@ - 某些算法的时间复杂度非固定，而是与输入数据的分布有关。时间复杂度分为最差、最佳、平均时间复杂度，最佳时间复杂度几乎不用，因为输入数据一般需要满足严格条件才能达到最佳情况。 - 平均时间复杂度反映算法在随机数据输入下的运行效率，最接近实际应用中的算法性能。计算平均时间复杂度需要统计输入数据分布以及综合后的数学期望。 -### 空间复杂度 +**空间复杂度** - 类似于时间复杂度，空间复杂度用于衡量算法占用空间随数据量增长的趋势。 - 算法运行过程中的相关内存空间可分为输入空间、暂存空间、输出空间。通常情况下，输入空间不计入空间复杂度计算。暂存空间可分为指令空间、数据空间、栈帧空间，其中栈帧空间通常仅在递归函数中影响空间复杂度。 diff --git a/docs/chapter_hashing/hash_map.md b/docs/chapter_hashing/hash_map.md index a5a0a9abb..35acbd227 100755 --- a/docs/chapter_hashing/hash_map.md +++ b/docs/chapter_hashing/hash_map.md @@ -6,24 +6,17 @@ ![哈希表的抽象表示](hash_map.assets/hash_map.png) -## 哈希表效率 +除哈希表外，我们还可以使用数组或链表实现查询功能，各项操作的时间复杂度如下表所示。 -除哈希表外，还可以使用以下数据结构来实现上述查询功能： - -1. **无序数组**：每个元素为 `[学号, 姓名]` ； -2. **有序数组**：将 `1.` 中的数组按照学号从小到大排序； -3. **链表**：每个节点的值为 `[学号, 姓名]` ； -4. **二叉搜索树**：每个节点的值为 `[学号, 姓名]` ，根据学号大小来构建树； - -各项操作的时间复杂度如下表所示（详解可见[二叉搜索树章节](https://www.hello-algo.com/chapter_tree/binary_search_tree/)）。无论是查找元素还是增删元素，哈希表的时间复杂度都是 $O(1)$ ，全面胜出！ +在哈希表中增删查改的时间复杂度都是 $O(1)$ ，全面胜出！因此，哈希表常用于对查找效率要求较高的场景。
-| | 无序数组 | 有序数组 | 链表 | 二叉搜索树 | 哈希表 | -| -------- | -------- | ----------- | ------ | ----------- | ------ | -| 查找元素 | $O(n)$ | $O(\log n)$ | $O(n)$ | $O(\log n)$ | $O(1)$ | -| 插入元素 | $O(1)$ | $O(n)$ | $O(1)$ | $O(\log n)$ | $O(1)$ | -| 删除元素 | $O(n)$ | $O(n)$ | $O(n)$ | $O(\log n)$ | $O(1)$ | +| | 数组 | 链表 | 哈希表 | +| -------- | ------ | ------ | ------ | +| 查找元素 | $O(n)$ | $O(n)$ | $O(1)$ | +| 插入元素 | $O(1)$ | $O(1)$ | $O(1)$ | +| 删除元素 | $O(n)$ | $O(n)$ | $O(1)$ |
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b/docs/chapter_searching/hashing_search.md deleted file mode 100755 index 09e420cbe..000000000 --- a/docs/chapter_searching/hashing_search.md +++ /dev/null @@ -1,152 +0,0 @@ -# 哈希查找 - -「哈希查找 Hash Searching」通过使用哈希表来存储所需的键值对，从而可在 $O(1)$ 时间内完成“键 $\rightarrow$ 值”的查找操作。 - -与线性查找相比，哈希查找通过利用额外空间来提高效率，体现了“以空间换时间”的算法思想。 - -## 算法实现 - -例如，若我们想要在给定数组中找到目标元素 `target` 的索引，则可以使用哈希查找来实现。 - -![哈希查找数组索引](hashing_search.assets/hash_search_index.png) - -=== "Java" - - ```java title="hashing_search.java" - [class]{hashing_search}-[func]{hashingSearchArray} - ``` - -=== "C++" - - ```cpp title="hashing_search.cpp" - [class]{}-[func]{hashingSearchArray} - ``` - -=== "Python" - - ```python title="hashing_search.py" - [class]{}-[func]{hashing_search_array} - ``` - -=== "Go" - - ```go title="hashing_search.go" - [class]{}-[func]{hashingSearchArray} - ``` - -=== "JavaScript" - - ```javascript title="hashing_search.js" - [class]{}-[func]{hashingSearchArray} - ``` - -=== "TypeScript" - - ```typescript title="hashing_search.ts" - [class]{}-[func]{hashingSearchArray} - ``` - -=== "C" - - ```c title="hashing_search.c" - [class]{}-[func]{hashingSearchArray} - ``` - -=== "C#" - - ```csharp title="hashing_search.cs" - [class]{hashing_search}-[func]{hashingSearchArray} - ``` - -=== "Swift" - - ```swift title="hashing_search.swift" - [class]{}-[func]{hashingSearchArray} - ``` - -=== "Zig" - - ```zig title="hashing_search.zig" - [class]{}-[func]{hashingSearchArray} - ``` - -同样，若要根据目标节点值 target 查找对应的链表节点对象，也可以采用哈希查找方法。 - -![哈希查找链表节点](hashing_search.assets/hash_search_listnode.png) - -=== "Java" - - ```java title="hashing_search.java" - [class]{hashing_search}-[func]{hashingSearchLinkedList} - ``` - -=== "C++" - - ```cpp title="hashing_search.cpp" - [class]{}-[func]{hashingSearchLinkedList} - ``` - -=== "Python" - - ```python title="hashing_search.py" - [class]{}-[func]{hashing_search_linkedlist} - ``` - -=== "Go" - - ```go title="hashing_search.go" - [class]{}-[func]{hashingSearchLinkedList} - ``` - -=== "JavaScript" - - ```javascript title="hashing_search.js" - [class]{}-[func]{hashingSearchLinkedList} - ``` - -=== "TypeScript" - - ```typescript title="hashing_search.ts" - [class]{}-[func]{hashingSearchLinkedList} - ``` - -=== "C" - - ```c title="hashing_search.c" - [class]{}-[func]{hashingSearchLinkedList} - ``` - -=== "C#" - - ```csharp title="hashing_search.cs" - [class]{hashing_search}-[func]{hashingSearchLinkedList} - ``` - -=== "Swift" - - ```swift title="hashing_search.swift" - [class]{}-[func]{hashingSearchLinkedList} - ``` - -=== "Zig" - - ```zig title="hashing_search.zig" - [class]{}-[func]{hashingSearchLinkedList} - ``` - -## 复杂度分析 - -**时间复杂度 $O(1)$** ：哈希表的查找操作使用 $O(1)$ 时间。 - -**空间复杂度 $O(n)$** ：其中 $n$ 是数组或链表的长度。 - -## 优点与局限性 - -哈希查找的性能表现相当优秀，查找、插入、删除操作的平均时间复杂度均为 $O(1)$ 。尽管如此，哈希查找仍然存在一些问题： - -- 辅助哈希表需要占用 $O(n)$ 的额外空间，意味着需要预留更多的计算机内存； -- 构建和维护哈希表需要时间，因此哈希查找不适用于高频增删、低频查找的场景； -- 当哈希冲突严重时，哈希表可能退化为链表，导致时间复杂度劣化至 $O(n)$ ； -- 当数据量较小时，线性查找可能比哈希查找更快。这是因为计算哈希函数可能比遍历一个小型数组更慢； - -因此，在实际应用中，我们需要根据具体情况灵活选择解决方案。 diff --git a/docs/chapter_searching/linear_search.md b/docs/chapter_searching/linear_search.md deleted file mode 100755 index 4e4011df5..000000000 --- a/docs/chapter_searching/linear_search.md +++ /dev/null @@ -1,143 +0,0 @@ -# 线性查找 - -「线性查找 Linear Search」是一种简单的查找方法，其从数据结构的一端开始，逐个访问每个元素，直至另一端为止。 - -## 算法实现 - -例如，若我们想要在数组 `nums` 中查找目标元素 `target` 的对应索引，可以采用线性查找方法。 - -![在数组中线性查找元素](linear_search.assets/linear_search.png) - -=== "Java" - - ```java title="linear_search.java" - [class]{linear_search}-[func]{linearSearchArray} - ``` - -=== "C++" - - ```cpp title="linear_search.cpp" - [class]{}-[func]{linearSearchArray} - ``` - -=== "Python" - - ```python title="linear_search.py" - [class]{}-[func]{linear_search_array} - ``` - -=== "Go" - - ```go title="linear_search.go" - [class]{}-[func]{linearSearchArray} - ``` - -=== "JavaScript" - - ```javascript title="linear_search.js" - [class]{}-[func]{linearSearchArray} - ``` - -=== "TypeScript" - - ```typescript title="linear_search.ts" - [class]{}-[func]{linearSearchArray} - ``` - -=== "C" - - ```c title="linear_search.c" - [class]{}-[func]{linearSearchArray} - ``` - -=== "C#" - - ```csharp title="linear_search.cs" - [class]{linear_search}-[func]{linearSearchArray} - ``` - -=== "Swift" - - ```swift title="linear_search.swift" - [class]{}-[func]{linearSearchArray} - ``` - -=== "Zig" - - ```zig title="linear_search.zig" - [class]{}-[func]{linearSearchArray} - ``` - -另一个例子，若需要在链表中查找给定目标节点值 `target` 并返回该节点对象，同样可以使用线性查找。 - -=== "Java" - - ```java title="linear_search.java" - [class]{linear_search}-[func]{linearSearchLinkedList} - ``` - -=== "C++" - - ```cpp title="linear_search.cpp" - [class]{}-[func]{linearSearchLinkedList} - ``` - -=== "Python" - - ```python title="linear_search.py" - [class]{}-[func]{linear_search_linkedlist} - ``` - -=== "Go" - - ```go title="linear_search.go" - [class]{}-[func]{linearSearchLinkedList} - ``` - -=== "JavaScript" - - ```javascript title="linear_search.js" - [class]{}-[func]{linearSearchLinkedList} - ``` - -=== "TypeScript" - - ```typescript title="linear_search.ts" - [class]{}-[func]{linearSearchLinkedList} - ``` - -=== "C" - - ```c title="linear_search.c" - [class]{}-[func]{linearSearchLinkedList} - ``` - -=== "C#" - - ```csharp title="linear_search.cs" - [class]{linear_search}-[func]{linearSearchLinkedList} - ``` - -=== "Swift" - - ```swift title="linear_search.swift" - [class]{}-[func]{linearSearchLinkedList} - ``` - -=== "Zig" - - ```zig title="linear_search.zig" - [class]{}-[func]{linearSearchLinkedList} - ``` - -## 复杂度分析 - -**时间复杂度 $O(n)$** ：其中 $n$ 代表数组或链表的长度。 - -**空间复杂度 $O(1)$** ：无需借助额外的存储空间。 - -## 优点与局限性 - -**线性查找具有极佳的通用性**。由于线性查找是逐个访问元素的，没有跳跃式访问，因此适用于数组和链表的查找。 - -**线性查找的时间复杂度较高**。当数据量 $n$ 较大时，线性查找的效率较低。 diff --git a/docs/chapter_searching/hashing_search.assets/hash_search_index.png b/docs/chapter_searching/replace_linear_by_hashing.assets/hashing_search.png similarity index 100% rename from docs/chapter_searching/hashing_search.assets/hash_search_index.png rename to docs/chapter_searching/replace_linear_by_hashing.assets/hashing_search.png diff --git a/docs/chapter_searching/linear_search.assets/linear_search.png b/docs/chapter_searching/replace_linear_by_hashing.assets/linear_search.png similarity index 100% rename from docs/chapter_searching/linear_search.assets/linear_search.png rename to docs/chapter_searching/replace_linear_by_hashing.assets/linear_search.png diff --git a/docs/chapter_computational_complexity/space_time_tradeoff.md b/docs/chapter_searching/replace_linear_by_hashing.md similarity index 59% rename from docs/chapter_computational_complexity/space_time_tradeoff.md rename to docs/chapter_searching/replace_linear_by_hashing.md index 409f6b59e..fcd87f311 100755 --- a/docs/chapter_computational_complexity/space_time_tradeoff.md +++ b/docs/chapter_searching/replace_linear_by_hashing.md @@ -1,14 +1,6 @@ -# 权衡时间与空间 +# 哈希优化策略 -理想情况下，我们希望算法的时间复杂度和空间复杂度都能达到最优。然而在实际情况中，同时优化时间复杂度和空间复杂度通常是非常困难的。 - -**降低时间复杂度通常需要以提升空间复杂度为代价，反之亦然**。我们将牺牲内存空间来提升算法运行速度的思路称为「以空间换时间」；反之，则称之为「以时间换空间」。 - -选择哪种思路取决于我们更看重哪个方面。在大多数情况下，时间比空间更宝贵，因此以空间换时间通常是更常用的策略。当然，在数据量很大的情况下，控制空间复杂度也是非常重要的。 - -## 示例题目 * - -以 LeetCode 全站第一题 [两数之和](https://leetcode.cn/problems/two-sum/) 为例。 +在算法题中，**我们时常通过将线性查找替换为哈希查找来降低算法的时间复杂度**。以 LeetCode 全站第一题 [两数之和](https://leetcode.cn/problems/two-sum/) 为例。 !!! question "两数之和" @@ -18,11 +10,11 @@ 你可以按任意顺序返回答案。 -「暴力枚举」和「辅助哈希表」分别对应“空间最优”和“时间最优”的两种解法。遵循时间比空间更宝贵的原则，后者是本题的最佳解法。 +## 线性查找：以时间换空间 -### 方法一：暴力枚举 +考虑直接遍历所有可能的组合。开启一个两层循环，在每轮中判断两个整数的和是否为 `target` ，若是，则返回它们的索引。 -考虑直接遍历所有可能的组合。通过开启一个两层循环，判断两个整数的和是否为 `target` ，若是，则返回它们的索引（即下标）。 +（图） === "Java" @@ -84,15 +76,17 @@ [class]{}-[func]{twoSumBruteForce} ``` -该方法的时间复杂度为 $O(n^2)$ ，空间复杂度为 $O(1)$ ，**属于以时间换空间**。此方法时间复杂度太高，在大数据量下非常耗时。 +此方法的时间复杂度为 $O(n^2)$ ，空间复杂度为 $O(1)$ ，在大数据量下非常耗时。 -### 方法二：辅助哈希表 +## 哈希查找：以空间换时间 -考虑借助一个哈希表，key-value 分别为数组元素和元素索引。循环遍历数组中的每个元素 num，并执行： +考虑借助一个哈希表，将数组元素和元素索引构建为键值对。循环遍历数组中的每个元素 `num` 并执行： 1. 判断数字 `target - num` 是否在哈希表中，若是则直接返回该两个元素的索引； 2. 将元素 `num` 和其索引添加进哈希表； +（图） + === "Java" ```java title="leetcode_two_sum.java" @@ -153,4 +147,6 @@ [class]{}-[func]{twoSumHashTable} ``` -该方法的时间复杂度为 $O(N)$ ，空间复杂度为 $O(N)$ ，**体现了以空间换时间**。尽管此方法引入了额外的空间使用，但在时间和空间的整体效率更为均衡，因此它是本题的最优解法。 +此方法通过哈希查找将时间复杂度从 $O(n^2)$ 降低至 $O(n)$ ，大幅提升运行效率。 + +由于需要维护一个额外的哈希表，因此空间复杂度为 $O(n)$ 。**尽管如此，该方法的整体时空效率更为均衡，因此它是本题的最优解法**。 diff --git a/docs/chapter_searching/searching_algorithm_revisited.assets/searching_algorithms.png b/docs/chapter_searching/searching_algorithm_revisited.assets/searching_algorithms.png new file mode 100644 index 0000000000000000000000000000000000000000..4dc6d762723b72177a25f339800c948db399475b GIT binary patch literal 104675 zcmeFZWms0*_BTv-BaL)PH_}LVN_R_lcXvxmcY`2^bc1vVNZfQcNY}I2anAl9_m}tk z3k%k|FXoyxdW_#BLQ!4<837*w3=9leN>WrA3=A50gqVPX0B%CW(%`@i+(}tN7_54X za1ZzpY^ot;CMO3*3p~StL4#v}LB2ErzQMrp|9JW36=`sS|2!*$)BM#2=rha$4C;^e 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z9tNujG)orbbI%I%h5oW*$OEz@$*bEb^(?&L29( z$%Swor5oJ-+lcyE0wCMM`F13aUuZp`i@4?7e~<)U^jzIk73AjPqLePkPqSJ}rI@av zhq%Gc5a*xt7iA4A;YY{PJxqgYI}>f{IrCMoOwPsodoMVq%6viw8O8_QPjsBE?cu=*5R9Ez#q=NfLx(22VO} z^rloetVly9ERkw&bKx2qyEC`RxJXK$Ff{+g+cDrg^e!6nFx3Jj$^=o_g=qnKcOa#R z!i>QYzzZMPO*(+Y3@#y5M(?`%W(Y@%uTB7{51>pLP5>2qGvE*P zNF@uzZheMl_YiBz&Pf*k#xVacfbxrhGtxwB>m)bT1xM{>-^u|sj;u#OmeRrgem@Z- zqTw(wpQ#LJN--^Q5pqg`{rv{Ue%6T~&5fyKqV#X(a7GBq|3>cPfjnc^1^5xz0f4m& z^aMdHn!o_B!V9nk=z=!3lk7FHA)gcIg158&`wkJ50n|QO%9XZp-{tN?K(rD?d%W^v zc;IX!ZvAFoU>2L#5O`Au+fFxVg%XXOiy#^!Oc(>}o7dkWjTkl{`$_y{1b{lG;X4yY zS)(U?`Om_Ans+yfW{f2uh`E6l0z5crf_PeX=}Q7K@WCnEMQA`|lj3|3=D zgD#vY{L8MDKM9j;31l;qyaeSwIexqN5iswjD&P(*iY-_PyHe$hII_$Ca(0M9wo<5S zYlMJTl^OAr`*ok?C~g6N+uMZ7-&P3%K%{1oX7PVLSR)4f0LN3$-Z({xQx5|E + +| | 线性搜索 | 二分查找 | 树查找 | 哈希查找 | +| ------------ | -------- | ------------------ | ------------------ | --------------- | +| 查找元素 | $O(n)$ | $O(\log n)$ | $O(\log n)$ | $O(1)$ | +| 插入元素 | $O(1)$ | $O(n)$ | $O(\log n)$ | $O(1)$ | +| 删除元素 | $O(n)$ | $O(n)$ | $O(\log n)$ | $O(1)$ | +| 额外空间 | $O(1)$ | $O(1)$ | $O(n)$ | $O(n)$ | +| 数据预处理 | / | 排序 $O(n \log n)$ | 建树 $O(n \log n)$ | 建哈希表 $O(n)$ | +| 数据是否有序 | 无序 | 有序 | 有序 | 无序 | + + + +除了以上表格内容，搜索算法的选择还取决于数据体量、搜索性能要求、数据查询与更新频率等。 + +**线性搜索** + +- 通用性较好，无需任何数据预处理操作。加入我们仅需查询一次数据，那么其他三种方法的数据预处理的时间比线性搜索的时间还要更长。 +- 适用于体量较小的数据，此情况下时间复杂度对效率影响较小。 +- 适用于数据更新频率较高的场景，因为该方法不需要对数据进行任何额外维护。 + +**二分查找** + +- 适用于大数据量的情况，效率表现稳定，最差时间复杂度为 $O(\log n)$ 。 +- 数据量不能过大，因为存储数组需要连续的内存空间。 +- 不适用于高频增删数据的场景，因为维护有序数组的开销较大。 + +**哈希查找** + +- 适合对查询性能要求很高的场景，平均时间复杂度为 $O(1)$ 。 +- 不适合需要有序数据或范围查找的场景，因为哈希表无法维护数据的有序性。 +- 对哈希函数和哈希冲突处理策略的依赖性较高，具有较大的性能劣化风险。 +- 不适合数据量过大的情况，因为哈希表需要额外空间来最大程度地减少冲突，从而提供良好的查询性能。 + +**树查找** + +- 适用于海量数据，因为树节点在内存中是离散存储的。 +- 适合需要维护有序数据或范围查找的场景。 +- 在持续增删节点的过程中，二叉搜索树可能产生倾斜，时间复杂度劣化至 $O(n)$ 。 +- 若使用 AVL 树或红黑树，则各项操作可在 $O(\log n)$ 效率下稳定运行，但维护树平衡的操作会增加额外开销。 diff --git a/docs/chapter_searching/summary.md b/docs/chapter_searching/summary.md index dbee447c5..31798c02c 100644 --- a/docs/chapter_searching/summary.md +++ b/docs/chapter_searching/summary.md @@ -1,16 +1,8 @@ # 小结 -- 线性查找通过遍历数据结构并进行条件判断来完成查找任务。 - 二分查找依赖于数据的有序性，通过循环逐步缩减一半搜索区间来实现查找。它要求输入数据有序，且仅适用于数组或基于数组实现的数据结构。 -- 哈希查找利用哈希表实现常数阶时间复杂度的查找操作，体现了空间换时间的算法思维。 -- 下表概括并对比了三种查找算法的特性和时间复杂度。 - -
- -| | 线性查找 | 二分查找 | 哈希查找 | -| ------------------------------------- | ------------------------ | ----------------------------- | ------------------------ | -| 适用数据结构 | 数组、链表 | 有序数组 | 数组、链表 | -| 时间复杂度
（查找，插入，删除） | $O(n)$ , $O(1)$ , $O(n)$ | $O(\log n)$ , $O(n)$ , $O(n)$ | $O(1)$ , $O(1)$ , $O(1)$ | -| 空间复杂度 | $O(1)$ | $O(1)$ | $O(n)$ | - -
+- 暴力搜索通过遍历数据结构来定位数据。线性搜索适用于数组和链表，广度优先搜索和深度优先搜索适用于图和树。此类算法通用性好，无需对数据预处理，但时间复杂度 $O(n)$ 较高。 +- 哈希查找、树查找和二分查找属于高效搜索方法，可在特定数据结构中快速定位目标元素。此类算法效率高，时间复杂度可达 $O(\log n)$ 甚至 $O(1)$ ，但通常需要借助额外数据结构。 +- 实际中，我们需要对数据体量、搜索性能要求、数据查询和更新频率等因素进行具体分析，从而选择合适的搜索方法。 +- 线性搜索适用于小型或频繁更新的数据；二分查找适用于大型、排序的数据；哈希查找适合对查询效率要求较高且无需范围查询的数据；树查找适用于需要维护顺序和支持范围查询的大型动态数据。 +- 用哈希查找替换线性查找是一种常用的优化运行时间的策略，可将时间复杂度从 $O(n)$ 降低至 $O(1)$ 。 diff --git a/docs/chapter_tree/binary_search_tree.md b/docs/chapter_tree/binary_search_tree.md index dd38bdf6c..731883abe 100755 --- a/docs/chapter_tree/binary_search_tree.md +++ b/docs/chapter_tree/binary_search_tree.md @@ -268,36 +268,21 @@ ## 二叉搜索树的效率 -假设给定 $n$ 个数字，最常见的存储方式是「数组」。对于这串乱序的数字，常见操作的效率如下： +给定一组数据，我们考虑使用数组或二叉搜索树存储。 -- **查找元素**：由于数组是无序的，因此需要遍历数组来确定，使用 $O(n)$ 时间； -- **插入元素**：只需将元素添加至数组尾部即可，使用 $O(1)$ 时间； -- **删除元素**：先查找元素，使用 $O(n)$ 时间，再在数组中删除该元素，使用 $O(n)$ 时间； -- **获取最小 / 最大元素**：需要遍历数组来确定，使用 $O(n)$ 时间； - -为了获得先验信息，我们可以预先将数组元素进行排序，得到一个「排序数组」。此时操作效率如下： - -- **查找元素**：由于数组已排序，可以使用二分查找，平均使用 $O(\log n)$ 时间； -- **插入元素**：先查找插入位置，使用 $O(\log n)$ 时间，再插入到指定位置，使用 $O(n)$ 时间； -- **删除元素**：先查找元素，使用 $O(\log n)$ 时间，再在数组中删除该元素，使用 $O(n)$ 时间； -- **获取最小 / 最大元素**：数组头部和尾部元素即是最小和最大元素，使用 $O(1)$ 时间； - -观察可知，无序数组和有序数组中的各项操作的时间复杂度呈现“偏科”的特点，即有的快有的慢。**然而，二叉搜索树的各项操作的时间复杂度都是对数阶，在数据量 $n$ 较大时具有显著优势**。 +观察可知，二叉搜索树的各项操作的时间复杂度都是对数阶，具有稳定且高效的性能表现。只有在高频添加、低频查找删除的数据适用场景下，数组比二叉搜索树的效率更高。
-| | 无序数组 | 有序数组 | 二叉搜索树 | -| ------------------- | -------- | ----------- | ----------- | -| 查找指定元素 | $O(n)$ | $O(\log n)$ | $O(\log n)$ | -| 插入元素 | $O(1)$ | $O(n)$ | $O(\log n)$ | -| 删除元素 | $O(n)$ | $O(n)$ | $O(\log n)$ | -| 获取最小 / 最大元素 | $O(n)$ | $O(1)$ | $O(\log n)$ | +| | 无序数组 | 二叉搜索树 | +| -------- | -------- | ----------- | +| 查找元素 | $O(n)$ | $O(\log n)$ | +| 插入元素 | $O(1)$ | $O(\log n)$ | +| 删除元素 | $O(n)$ | $O(\log n)$ |
-## 二叉搜索树的退化 - -在理想情况下，我们希望二叉搜索树是“平衡”的，这样就可以在 $\log n$ 轮循环内查找任意节点。 +在理想情况下，二叉搜索树是“平衡”的，这样就可以在 $\log n$ 轮循环内查找任意节点。 然而，如果我们在二叉搜索树中不断地插入和删除节点，可能导致二叉树退化为链表，这时各种操作的时间复杂度也会退化为 $O(n)$ 。 diff --git a/mkdocs.yml b/mkdocs.yml index 805ff23ce..ec619f982 100644 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -136,8 +136,7 @@ nav: - 2.1.   算法效率评估: chapter_computational_complexity/performance_evaluation.md - 2.2.   时间复杂度: chapter_computational_complexity/time_complexity.md - 2.3.   空间复杂度: chapter_computational_complexity/space_complexity.md - - 2.4.   权衡时间与空间: chapter_computational_complexity/space_time_tradeoff.md - - 2.5.   小结: chapter_computational_complexity/summary.md + - 2.4.   小结: chapter_computational_complexity/summary.md - 3.     数据结构简介: - 3.1.   数据与内存: chapter_data_structure/data_and_memory.md - 3.2.   数据结构分类: chapter_data_structure/classification_of_data_structure.md @@ -152,44 +151,45 @@ nav: - 5.2.   队列: chapter_stack_and_queue/queue.md - 5.3.   双向队列: chapter_stack_and_queue/deque.md - 5.4.   小结: chapter_stack_and_queue/summary.md - - 6.     散列表: - - 6.1.   哈希表: chapter_hashing/hash_map.md - - 6.2.   哈希冲突处理: chapter_hashing/hash_collision.md - - 6.3.   小结: chapter_hashing/summary.md - - 7.     树: - - 7.1.   二叉树: chapter_tree/binary_tree.md - - 7.2.   二叉树遍历: chapter_tree/binary_tree_traversal.md - - 7.3.   二叉搜索树: chapter_tree/binary_search_tree.md - - 7.4.   AVL 树 *: chapter_tree/avl_tree.md - - 7.5.   小结: chapter_tree/summary.md - - 8.     堆: - - 8.1.   堆: chapter_heap/heap.md - - 8.2.   建堆操作 *: chapter_heap/build_heap.md - - 8.3.   小结: chapter_heap/summary.md - - 9.     图: - - 9.1.   图: chapter_graph/graph.md - - 9.2.   图基础操作: chapter_graph/graph_operations.md - - 9.3.   图的遍历: chapter_graph/graph_traversal.md - - 9.4.   小结: chapter_graph/summary.md - - 10.     查找算法: - - 10.1.   线性查找: chapter_searching/linear_search.md - - 10.2.   二分查找: chapter_searching/binary_search.md - - 10.3.   哈希查找: chapter_searching/hashing_search.md - - 10.4.   小结: chapter_searching/summary.md + - 6.     二分查找: + - 6.1.   二分查找: chapter_binary_search/binary_search.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 + - 7.3.   小结: chapter_hashing/summary.md + - 8.     树: + - 8.1.   二叉树: chapter_tree/binary_tree.md + - 8.2.   二叉树遍历: chapter_tree/binary_tree_traversal.md + - 8.3.   二叉搜索树: chapter_tree/binary_search_tree.md + - 8.4.   AVL 树 *: chapter_tree/avl_tree.md + - 8.5.   小结: chapter_tree/summary.md + - 9.     堆: + - 9.1.   堆: chapter_heap/heap.md + - 9.2.   建堆操作 *: chapter_heap/build_heap.md + - 9.3.   小结: chapter_heap/summary.md + - 10.     图: + - 10.1.   图: chapter_graph/graph.md + - 10.2.   图基础操作: chapter_graph/graph_operations.md + - 10.3.   图的遍历: chapter_graph/graph_traversal.md + - 10.4.   小结: chapter_graph/summary.md - 11.     排序算法: - 11.1.   排序算法: chapter_sorting/sorting_algorithm.md - 11.2.   冒泡排序: chapter_sorting/bubble_sort.md - 11.3.   插入排序: chapter_sorting/insertion_sort.md - 11.4.   快速排序: chapter_sorting/quick_sort.md - 11.5.   归并排序: chapter_sorting/merge_sort.md - - 11.6.   桶排序（New）: chapter_sorting/bucket_sort.md - - 11.7.   计数排序（New）: chapter_sorting/counting_sort.md - - 11.8.   基数排序（New）: chapter_sorting/radix_sort.md + - 11.6.   桶排序: chapter_sorting/bucket_sort.md + - 11.7.   计数排序: chapter_sorting/counting_sort.md + - 11.8.   基数排序: chapter_sorting/radix_sort.md - 11.9.   小结: chapter_sorting/summary.md - - 12.     回溯算法: - - 12.1.   回溯算法（New）: chapter_backtracking/backtracking_algorithm.md - - 13.     附录: - - 13.1.   编程环境安装: chapter_appendix/installation.md - - 13.2.   一起参与创作: chapter_appendix/contribution.md + - 12.     搜索算法: + - 12.1.   搜索算法（New）: chapter_searching/searching_algorithm_revisited.md + - 12.2.   哈希优化策略: chapter_searching/replace_linear_by_hashing.md + - 12.3.   小结: chapter_searching/summary.md + - 13.     回溯算法: + - 13.1.   回溯算法（New）: chapter_backtracking/backtracking_algorithm.md + - 14.     附录: + - 14.1.   编程环境安装: chapter_appendix/installation.md + - 14.2.   一起参与创作: chapter_appendix/contribution.md - 参考文献: - chapter_reference/index.md