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66
.gitignore vendored
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@ -1,14 +1,62 @@
# Ignore all
*
# Unignore all with extensions
!*.*
# Unignore all dirs
!*/
# ---> CMake
CMakeLists.txt.user
CMakeCache.txt
CMakeFiles
CMakeScripts
Testing
Makefile
cmake_install.cmake
install_manifest.txt
compile_commands.json
CTestTestfile.cmake
_deps
*.dSYM/
# ---> VisualStudioCode
.vscode/*
!.vscode/settings.json
!.vscode/tasks.json
!.vscode/launch.json
!.vscode/extensions.json
!.vscode/*.code-snippets
build/
# Local History for Visual Studio Code
.history/
# Built Visual Studio Code Extensions
*.vsix
# ---> C++
# Prerequisites
*.d
# Compiled Object files
*.slo
*.lo
*.o
*.obj
# Precompiled Headers
*.gch
*.pch
# Compiled Dynamic libraries
*.so
*.dylib
*.dll
# Fortran module files
*.mod
*.smod
# Compiled Static libraries
*.lai
*.la
*.a
*.lib
# Executables
*.exe
*.out
*.app
*/output/
build

20
CMakeLists.txt
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@ -1,20 +0,0 @@
cmake_minimum_required(VERSION 3.10)
project(hello_algo CXX)
set(CMAKE_CXX_STANDARD 11)
include_directories(./include)
add_subdirectory(chapter_computational_complexity)
add_subdirectory(chapter_array_and_linkedlist)
add_subdirectory(chapter_stack_and_queue)
add_subdirectory(chapter_hashing)
add_subdirectory(chapter_tree)
add_subdirectory(chapter_heap)
add_subdirectory(chapter_graph)
add_subdirectory(chapter_searching)
add_subdirectory(chapter_sorting)
add_subdirectory(chapter_divide_and_conquer)
add_subdirectory(chapter_backtracking)
add_subdirectory(chapter_dynamic_programming)
add_subdirectory(chapter_greedy)

4
chapter_array_and_linkedlist/CMakeLists.txt
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@ -1,4 +0,0 @@
add_executable(array array.cpp)
add_executable(linked_list linked_list.cpp)
add_executable(list list.cpp)
add_executable(my_list my_list.cpp)

113
chapter_array_and_linkedlist/array.cpp
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@ -1,113 +0,0 @@
/**
* File: array.cpp
* Created Time: 2022-11-25
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 随机访问元素 */
int randomAccess(int *nums, int size) {
// 在区间 [0, size) 中随机抽取一个数字
int randomIndex = rand() % size;
// 获取并返回随机元素
int randomNum = nums[randomIndex];
return randomNum;
}
/* 扩展数组长度 */
int *extend(int *nums, int size, int enlarge) {
// 初始化一个扩展长度后的数组
int *res = new int[size + enlarge];
// 将原数组中的所有元素复制到新数组
for (int i = 0; i < size; i++) {
res[i] = nums[i];
}
// 释放内存
delete[] nums;
// 返回扩展后的新数组
return res;
}
/* 在数组的索引 index 处插入元素 num */
void insert(int *nums, int size, int num, int index) {
// 把索引 index 以及之后的所有元素向后移动一位
for (int i = size - 1; i > index; i--) {
nums[i] = nums[i - 1];
}
// 将 num 赋给 index 处的元素
nums[index] = num;
}
/* 删除索引 index 处的元素 */
void remove(int *nums, int size, int index) {
// 把索引 index 之后的所有元素向前移动一位
for (int i = index; i < size - 1; i++) {
nums[i] = nums[i + 1];
}
}
/* 遍历数组 */
void traverse(int *nums, int size) {
int count = 0;
// 通过索引遍历数组
for (int i = 0; i < size; i++) {
count += nums[i];
}
}
/* 在数组中查找指定元素 */
int find(int *nums, int size, int target) {
for (int i = 0; i < size; i++) {
if (nums[i] == target)
return i;
}
return -1;
}
/* Driver Code */
int main() {
/* 初始化数组 */
int size = 5;
int *arr = new int[size];
cout << "数组 arr = ";
printArray(arr, size);
int *nums = new int[size]{1, 3, 2, 5, 4};
cout << "数组 nums = ";
printArray(nums, size);
/* 随机访问 */
int randomNum = randomAccess(nums, size);
cout << "在 nums 中获取随机元素 " << randomNum << endl;
/* 长度扩展 */
int enlarge = 3;
nums = extend(nums, size, enlarge);
size += enlarge;
cout << "将数组长度扩展至 8 ，得到 nums = ";
printArray(nums, size);
/* 插入元素 */
insert(nums, size, 6, 3);
cout << "在索引 3 处插入数字 6 ，得到 nums = ";
printArray(nums, size);
/* 删除元素 */
remove(nums, size, 2);
cout << "删除索引 2 处的元素，得到 nums = ";
printArray(nums, size);
/* 遍历数组 */
traverse(nums, size);
/* 查找元素 */
int index = find(nums, size, 3);
cout << "在 nums 中查找元素 3 ，得到索引 = " << index << endl;
// 释放内存
delete[] arr;
delete[] nums;
return 0;
}

89
chapter_array_and_linkedlist/linked_list.cpp
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@ -1,89 +0,0 @@
/**
* File: linked_list.cpp
* Created Time: 2022-11-25
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 在链表的节点 n0 之后插入节点 P */
void insert(ListNode *n0, ListNode *P) {
ListNode *n1 = n0->next;
P->next = n1;
n0->next = P;
}
/* 删除链表的节点 n0 之后的首个节点 */
void remove(ListNode *n0) {
if (n0->next == nullptr)
return;
// n0 -> P -> n1
ListNode *P = n0->next;
ListNode *n1 = P->next;
n0->next = n1;
// 释放内存
delete P;
}
/* 访问链表中索引为 index 的节点 */
ListNode *access(ListNode *head, int index) {
for (int i = 0; i < index; i++) {
if (head == nullptr)
return nullptr;
head = head->next;
}
return head;
}
/* 在链表中查找值为 target 的首个节点 */
int find(ListNode *head, int target) {
int index = 0;
while (head != nullptr) {
if (head->val == target)
return index;
head = head->next;
index++;
}
return -1;
}
/* Driver Code */
int main() {
/* 初始化链表 */
// 初始化各个节点
ListNode *n0 = new ListNode(1);
ListNode *n1 = new ListNode(3);
ListNode *n2 = new ListNode(2);
ListNode *n3 = new ListNode(5);
ListNode *n4 = new ListNode(4);
// 构建节点之间的引用
n0->next = n1;
n1->next = n2;
n2->next = n3;
n3->next = n4;
cout << "初始化的链表为" << endl;
printLinkedList(n0);
/* 插入节点 */
insert(n0, new ListNode(0));
cout << "插入节点后的链表为" << endl;
printLinkedList(n0);
/* 删除节点 */
remove(n0);
cout << "删除节点后的链表为" << endl;
printLinkedList(n0);
/* 访问节点 */
ListNode *node = access(n0, 3);
cout << "链表中索引 3 处的节点的值 = " << node->val << endl;
/* 查找节点 */
int index = find(n0, 2);
cout << "链表中值为 2 的节点的索引 = " << index << endl;
// 释放内存
freeMemoryLinkedList(n0);
return 0;
}

72
chapter_array_and_linkedlist/list.cpp
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/**
* File: list.cpp
* Created Time: 2022-11-25
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* Driver Code */
int main() {
/* 初始化列表 */
vector<int> nums = {1, 3, 2, 5, 4};
cout << "列表 nums = ";
printVector(nums);
/* 访问元素 */
int num = nums[1];
cout << "访问索引 1 处的元素，得到 num = " << num << endl;
/* 更新元素 */
nums[1] = 0;
cout << "将索引 1 处的元素更新为 0 ，得到 nums = ";
printVector(nums);
/* 清空列表 */
nums.clear();
cout << "清空列表后 nums = ";
printVector(nums);
/* 在尾部添加元素 */
nums.push_back(1);
nums.push_back(3);
nums.push_back(2);
nums.push_back(5);
nums.push_back(4);
cout << "添加元素后 nums = ";
printVector(nums);
/* 在中间插入元素 */
nums.insert(nums.begin() + 3, 6);
cout << "在索引 3 处插入数字 6 ，得到 nums = ";
printVector(nums);
/* 删除元素 */
nums.erase(nums.begin() + 3);
cout << "删除索引 3 处的元素，得到 nums = ";
printVector(nums);
/* 通过索引遍历列表 */
int count = 0;
for (int i = 0; i < nums.size(); i++) {
count += nums[i];
}
/* 直接遍历列表元素 */
count = 0;
for (int x : nums) {
count += x;
}
/* 拼接两个列表 */
vector<int> nums1 = {6, 8, 7, 10, 9};
nums.insert(nums.end(), nums1.begin(), nums1.end());
cout << "将列表 nums1 拼接到 nums 之后，得到 nums = ";
printVector(nums);
/* 排序列表 */
sort(nums.begin(), nums.end());
cout << "排序列表后 nums = ";
printVector(nums);
return 0;
}

171
chapter_array_and_linkedlist/my_list.cpp
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/**
* File: my_list.cpp
* Created Time: 2022-11-25
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 列表类 */
class MyList {
private:
int *arr; // 数组（存储列表元素）
int arrCapacity = 10; // 列表容量
int arrSize = 0; // 列表长度（当前元素数量）
int extendRatio = 2; // 每次列表扩容的倍数
public:
/* 构造方法 */
MyList() {
arr = new int[arrCapacity];
}
/* 析构方法 */
~MyList() {
delete[] arr;
}
/* 获取列表长度（当前元素数量）*/
int size() {
return arrSize;
}
/* 获取列表容量 */
int capacity() {
return arrCapacity;
}
/* 访问元素 */
int get(int index) {
// 索引如果越界，则抛出异常，下同
if (index < 0 || index >= size())
throw out_of_range("索引越界");
return arr[index];
}
/* 更新元素 */
void set(int index, int num) {
if (index < 0 || index >= size())
throw out_of_range("索引越界");
arr[index] = num;
}
/* 在尾部添加元素 */
void add(int num) {
// 元素数量超出容量时，触发扩容机制
if (size() == capacity())
extendCapacity();
arr[size()] = num;
// 更新元素数量
arrSize++;
}
/* 在中间插入元素 */
void insert(int index, int num) {
if (index < 0 || index >= size())
throw out_of_range("索引越界");
// 元素数量超出容量时，触发扩容机制
if (size() == capacity())
extendCapacity();
// 将索引 index 以及之后的元素都向后移动一位
for (int j = size() - 1; j >= index; j--) {
arr[j + 1] = arr[j];
}
arr[index] = num;
// 更新元素数量
arrSize++;
}
/* 删除元素 */
int remove(int index) {
if (index < 0 || index >= size())
throw out_of_range("索引越界");
int num = arr[index];
// 将索引 index 之后的元素都向前移动一位
for (int j = index; j < size() - 1; j++) {
arr[j] = arr[j + 1];
}
// 更新元素数量
arrSize--;
// 返回被删除的元素
return num;
}
/* 列表扩容 */
void extendCapacity() {
// 新建一个长度为原数组 extendRatio 倍的新数组
int newCapacity = capacity() * extendRatio;
int *tmp = arr;
arr = new int[newCapacity];
// 将原数组中的所有元素复制到新数组
for (int i = 0; i < size(); i++) {
arr[i] = tmp[i];
}
// 释放内存
delete[] tmp;
arrCapacity = newCapacity;
}
/* 将列表转换为 Vector 用于打印 */
vector<int> toVector() {
// 仅转换有效长度范围内的列表元素
vector<int> vec(size());
for (int i = 0; i < size(); i++) {
vec[i] = arr[i];
}
return vec;
}
};
/* Driver Code */
int main() {
/* 初始化列表 */
MyList *nums = new MyList();
/* 在尾部添加元素 */
nums->add(1);
nums->add(3);
nums->add(2);
nums->add(5);
nums->add(4);
cout << "列表 nums = ";
vector<int> vec = nums->toVector();
printVector(vec);
cout << "容量 = " << nums->capacity() << " ，长度 = " << nums->size() << endl;
/* 在中间插入元素 */
nums->insert(3, 6);
cout << "在索引 3 处插入数字 6 ，得到 nums = ";
vec = nums->toVector();
printVector(vec);
/* 删除元素 */
nums->remove(3);
cout << "删除索引 3 处的元素，得到 nums = ";
vec = nums->toVector();
printVector(vec);
/* 访问元素 */
int num = nums->get(1);
cout << "访问索引 1 处的元素，得到 num = " << num << endl;
/* 更新元素 */
nums->set(1, 0);
cout << "将索引 1 处的元素更新为 0 ，得到 nums = ";
vec = nums->toVector();
printVector(vec);
/* 测试扩容机制 */
for (int i = 0; i < 10; i++) {
// 在 i = 5 时，列表长度将超出列表容量，此时触发扩容机制
nums->add(i);
}
cout << "扩容后的列表 nums = ";
vec = nums->toVector();
printVector(vec);
cout << "容量 = " << nums->capacity() << " ，长度 = " << nums->size() << endl;
// 释放内存
delete nums;
return 0;
}

10
chapter_backtracking/CMakeLists.txt
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add_executable(preorder_traversal_i_compact preorder_traversal_i_compact.cpp)
add_executable(preorder_traversal_ii_compact preorder_traversal_ii_compact.cpp)
add_executable(preorder_traversal_iii_compact preorder_traversal_iii_compact.cpp)
add_executable(preorder_traversal_iii_template preorder_traversal_iii_template.cpp)
add_executable(permutations_i permutations_i.cpp)
add_executable(permutations_ii permutations_ii.cpp)
add_executable(n_queens n_queens.cpp)
add_executable(subset_sum_i_naive subset_sum_i_naive.cpp)
add_executable(subset_sum_i subset_sum_i.cpp)
add_executable(subset_sum_ii subset_sum_ii.cpp)

65
chapter_backtracking/n_queens.cpp
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/**
* File: n_queens.cpp
* Created Time: 2023-05-04
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 回溯算法：n 皇后 */
void backtrack(int row, int n, vector<vector<string>> &state, vector<vector<vector<string>>> &res, vector<bool> &cols,
vector<bool> &diags1, vector<bool> &diags2) {
// 当放置完所有行时，记录解
if (row == n) {
res.push_back(state);
return;
}
// 遍历所有列
for (int col = 0; col < n; col++) {
// 计算该格子对应的主对角线和次对角线
int diag1 = row - col + n - 1;
int diag2 = row + col;
// 剪枝：不允许该格子所在列、主对角线、次对角线上存在皇后
if (!cols[col] && !diags1[diag1] && !diags2[diag2]) {
// 尝试：将皇后放置在该格子
state[row][col] = "Q";
cols[col] = diags1[diag1] = diags2[diag2] = true;
// 放置下一行
backtrack(row + 1, n, state, res, cols, diags1, diags2);
// 回退：将该格子恢复为空位
state[row][col] = "#";
cols[col] = diags1[diag1] = diags2[diag2] = false;
}
}
}
/* 求解 n 皇后 */
vector<vector<vector<string>>> nQueens(int n) {
// 初始化 n*n 大小的棋盘，其中 'Q' 代表皇后，'#' 代表空位
vector<vector<string>> state(n, vector<string>(n, "#"));
vector<bool> cols(n, false); // 记录列是否有皇后
vector<bool> diags1(2 * n - 1, false); // 记录主对角线上是否有皇后
vector<bool> diags2(2 * n - 1, false); // 记录次对角线上是否有皇后
vector<vector<vector<string>>> res;
backtrack(0, n, state, res, cols, diags1, diags2);
return res;
}
/* Driver Code */
int main() {
int n = 4;
vector<vector<vector<string>>> res = nQueens(n);
cout << "输入棋盘长宽为 " << n << endl;
cout << "皇后放置方案共有 " << res.size() << " 种" << endl;
for (const vector<vector<string>> &state : res) {
cout << "--------------------" << endl;
for (const vector<string> &row : state) {
printVector(row);
}
}
return 0;
}

54
chapter_backtracking/permutations_i.cpp
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/**
* File: permutations_i.cpp
* Created Time: 2023-04-24
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 回溯算法：全排列 I */
void backtrack(vector<int> &state, const vector<int> &choices, vector<bool> &selected, vector<vector<int>> &res) {
// 当状态长度等于元素数量时，记录解
if (state.size() == choices.size()) {
res.push_back(state);
return;
}
// 遍历所有选择
for (int i = 0; i < choices.size(); i++) {
int choice = choices[i];
// 剪枝：不允许重复选择元素
if (!selected[i]) {
// 尝试：做出选择，更新状态
selected[i] = true;
state.push_back(choice);
// 进行下一轮选择
backtrack(state, choices, selected, res);
// 回退：撤销选择，恢复到之前的状态
selected[i] = false;
state.pop_back();
}
}
}
/* 全排列 I */
vector<vector<int>> permutationsI(vector<int> nums) {
vector<int> state;
vector<bool> selected(nums.size(), false);
vector<vector<int>> res;
backtrack(state, nums, selected, res);
return res;
}
/* Driver Code */
int main() {
vector<int> nums = {1, 2, 3};
vector<vector<int>> res = permutationsI(nums);
cout << "输入数组 nums = ";
printVector(nums);
cout << "所有排列 res = ";
printVectorMatrix(res);
return 0;
}

56
chapter_backtracking/permutations_ii.cpp
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/**
* File: permutations_ii.cpp
* Created Time: 2023-04-24
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 回溯算法：全排列 II */
void backtrack(vector<int> &state, const vector<int> &choices, vector<bool> &selected, vector<vector<int>> &res) {
// 当状态长度等于元素数量时，记录解
if (state.size() == choices.size()) {
res.push_back(state);
return;
}
// 遍历所有选择
unordered_set<int> duplicated;
for (int i = 0; i < choices.size(); i++) {
int choice = choices[i];
// 剪枝：不允许重复选择元素 且 不允许重复选择相等元素
if (!selected[i] && duplicated.find(choice) == duplicated.end()) {
// 尝试：做出选择，更新状态
duplicated.emplace(choice); // 记录选择过的元素值
selected[i] = true;
state.push_back(choice);
// 进行下一轮选择
backtrack(state, choices, selected, res);
// 回退：撤销选择，恢复到之前的状态
selected[i] = false;
state.pop_back();
}
}
}
/* 全排列 II */
vector<vector<int>> permutationsII(vector<int> nums) {
vector<int> state;
vector<bool> selected(nums.size(), false);
vector<vector<int>> res;
backtrack(state, nums, selected, res);
return res;
}
/* Driver Code */
int main() {
vector<int> nums = {1, 1, 2};
vector<vector<int>> res = permutationsII(nums);
cout << "输入数组 nums = ";
printVector(nums);
cout << "所有排列 res = ";
printVectorMatrix(res);
return 0;
}

39
chapter_backtracking/preorder_traversal_i_compact.cpp
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/**
* File: preorder_traversal_i_compact.cpp
* Created Time: 2023-04-16
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
vector<TreeNode *> res;
/* 前序遍历：例题一 */
void preOrder(TreeNode *root) {
if (root == nullptr) {
return;
}
if (root->val == 7) {
// 记录解
res.push_back(root);
}
preOrder(root->left);
preOrder(root->right);
}
/* Driver Code */
int main() {
TreeNode *root = vectorToTree(vector<int>{1, 7, 3, 4, 5, 6, 7});
cout << "\n初始化二叉树" << endl;
printTree(root);
// 前序遍历
preOrder(root);
cout << "\n输出所有值为 7 的节点" << endl;
vector<int> vals;
for (TreeNode *node : res) {
vals.push_back(node->val);
}
printVector(vals);
}

46
chapter_backtracking/preorder_traversal_ii_compact.cpp
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/**
* File: preorder_traversal_ii_compact.cpp
* Created Time: 2023-04-16
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
vector<TreeNode *> path;
vector<vector<TreeNode *>> res;
/* 前序遍历：例题二 */
void preOrder(TreeNode *root) {
if (root == nullptr) {
return;
}
// 尝试
path.push_back(root);
if (root->val == 7) {
// 记录解
res.push_back(path);
}
preOrder(root->left);
preOrder(root->right);
// 回退
path.pop_back();
}
/* Driver Code */
int main() {
TreeNode *root = vectorToTree(vector<int>{1, 7, 3, 4, 5, 6, 7});
cout << "\n初始化二叉树" << endl;
printTree(root);
// 前序遍历
preOrder(root);
cout << "\n输出所有根节点到节点 7 的路径" << endl;
for (vector<TreeNode *> &path : res) {
vector<int> vals;
for (TreeNode *node : path) {
vals.push_back(node->val);
}
printVector(vals);
}
}

47
chapter_backtracking/preorder_traversal_iii_compact.cpp
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/**
* File: preorder_traversal_iii_compact.cpp
* Created Time: 2023-04-16
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
vector<TreeNode *> path;
vector<vector<TreeNode *>> res;
/* 前序遍历：例题三 */
void preOrder(TreeNode *root) {
// 剪枝
if (root == nullptr || root->val == 3) {
return;
}
// 尝试
path.push_back(root);
if (root->val == 7) {
// 记录解
res.push_back(path);
}
preOrder(root->left);
preOrder(root->right);
// 回退
path.pop_back();
}
/* Driver Code */
int main() {
TreeNode *root = vectorToTree(vector<int>{1, 7, 3, 4, 5, 6, 7});
cout << "\n初始化二叉树" << endl;
printTree(root);
// 前序遍历
preOrder(root);
cout << "\n输出所有根节点到节点 7 的路径，要求路径中不包含值为 3 的节点" << endl;
for (vector<TreeNode *> &path : res) {
vector<int> vals;
for (TreeNode *node : path) {
vals.push_back(node->val);
}
printVector(vals);
}
}

76
chapter_backtracking/preorder_traversal_iii_template.cpp
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/**
* File: preorder_traversal_iii_template.cpp
* Created Time: 2023-04-16
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 判断当前状态是否为解 */
bool isSolution(vector<TreeNode *> &state) {
return !state.empty() && state.back()->val == 7;
}
/* 记录解 */
void recordSolution(vector<TreeNode *> &state, vector<vector<TreeNode *>> &res) {
res.push_back(state);
}
/* 判断在当前状态下，该选择是否合法 */
bool isValid(vector<TreeNode *> &state, TreeNode *choice) {
return choice != nullptr && choice->val != 3;
}
/* 更新状态 */
void makeChoice(vector<TreeNode *> &state, TreeNode *choice) {
state.push_back(choice);
}
/* 恢复状态 */
void undoChoice(vector<TreeNode *> &state, TreeNode *choice) {
state.pop_back();
}
/* 回溯算法：例题三 */
void backtrack(vector<TreeNode *> &state, vector<TreeNode *> &choices, vector<vector<TreeNode *>> &res) {
// 检查是否为解
if (isSolution(state)) {
// 记录解
recordSolution(state, res);
}
// 遍历所有选择
for (TreeNode *choice : choices) {
// 剪枝：检查选择是否合法
if (isValid(state, choice)) {
// 尝试：做出选择，更新状态
makeChoice(state, choice);
// 进行下一轮选择
vector<TreeNode *> nextChoices{choice->left, choice->right};
backtrack(state, nextChoices, res);
// 回退：撤销选择，恢复到之前的状态
undoChoice(state, choice);
}
}
}
/* Driver Code */
int main() {
TreeNode *root = vectorToTree(vector<int>{1, 7, 3, 4, 5, 6, 7});
cout << "\n初始化二叉树" << endl;
printTree(root);
// 回溯算法
vector<TreeNode *> state;
vector<TreeNode *> choices = {root};
vector<vector<TreeNode *>> res;
backtrack(state, choices, res);
cout << "\n输出所有根节点到节点 7 的路径，要求路径中不包含值为 3 的节点" << endl;
for (vector<TreeNode *> &path : res) {
vector<int> vals;
for (TreeNode *node : path) {
vals.push_back(node->val);
}
printVector(vals);
}
}

57
chapter_backtracking/subset_sum_i.cpp
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/**
* File: subset_sum_i.cpp
* Created Time: 2023-06-21
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 回溯算法：子集和 I */
void backtrack(vector<int> &state, int target, vector<int> &choices, int start, vector<vector<int>> &res) {
// 子集和等于 target 时，记录解
if (target == 0) {
res.push_back(state);
return;
}
// 遍历所有选择
// 剪枝二：从 start 开始遍历，避免生成重复子集
for (int i = start; i < choices.size(); i++) {
// 剪枝一：若子集和超过 target ，则直接结束循环
// 这是因为数组已排序，后边元素更大，子集和一定超过 target
if (target - choices[i] < 0) {
break;
}
// 尝试：做出选择，更新 target, start
state.push_back(choices[i]);
// 进行下一轮选择
backtrack(state, target - choices[i], choices, i, res);
// 回退：撤销选择，恢复到之前的状态
state.pop_back();
}
}
/* 求解子集和 I */
vector<vector<int>> subsetSumI(vector<int> &nums, int target) {
vector<int> state; // 状态（子集）
sort(nums.begin(), nums.end()); // 对 nums 进行排序
int start = 0; // 遍历起始点
vector<vector<int>> res; // 结果列表（子集列表）
backtrack(state, target, nums, start, res);
return res;
}
/* Driver Code */
int main() {
vector<int> nums = {3, 4, 5};
int target = 9;
vector<vector<int>> res = subsetSumI(nums, target);
cout << "输入数组 nums = ";
printVector(nums);
cout << "target = " << target << endl;
cout << "所有和等于 " << target << " 的子集 res = " << endl;
printVectorMatrix(res);
return 0;
}

54
chapter_backtracking/subset_sum_i_naive.cpp
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/**
* File: subset_sum_i_naive.cpp
* Created Time: 2023-06-21
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 回溯算法：子集和 I */
void backtrack(vector<int> &state, int target, int total, vector<int> &choices, vector<vector<int>> &res) {
// 子集和等于 target 时，记录解
if (total == target) {
res.push_back(state);
return;
}
// 遍历所有选择
for (size_t i = 0; i < choices.size(); i++) {
// 剪枝：若子集和超过 target ，则跳过该选择
if (total + choices[i] > target) {
continue;
}
// 尝试：做出选择，更新元素和 total
state.push_back(choices[i]);
// 进行下一轮选择
backtrack(state, target, total + choices[i], choices, res);
// 回退：撤销选择，恢复到之前的状态
state.pop_back();
}
}
/* 求解子集和 I（包含重复子集） */
vector<vector<int>> subsetSumINaive(vector<int> &nums, int target) {
vector<int> state; // 状态（子集）
int total = 0; // 子集和
vector<vector<int>> res; // 结果列表（子集列表）
backtrack(state, target, total, nums, res);
return res;
}
/* Driver Code */
int main() {
vector<int> nums = {3, 4, 5};
int target = 9;
vector<vector<int>> res = subsetSumINaive(nums, target);
cout << "输入数组 nums = ";
printVector(nums);
cout << "target = " << target << endl;
cout << "所有和等于 " << target << " 的子集 res = " << endl;
printVectorMatrix(res);
return 0;
}

62
chapter_backtracking/subset_sum_ii.cpp
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/**
* File: subset_sum_ii.cpp
* Created Time: 2023-06-21
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 回溯算法：子集和 II */
void backtrack(vector<int> &state, int target, vector<int> &choices, int start, vector<vector<int>> &res) {
// 子集和等于 target 时，记录解
if (target == 0) {
res.push_back(state);
return;
}
// 遍历所有选择
// 剪枝二：从 start 开始遍历，避免生成重复子集
// 剪枝三：从 start 开始遍历，避免重复选择同一元素
for (int i = start; i < choices.size(); i++) {
// 剪枝一：若子集和超过 target ，则直接结束循环
// 这是因为数组已排序，后边元素更大，子集和一定超过 target
if (target - choices[i] < 0) {
break;
}
// 剪枝四：如果该元素与左边元素相等，说明该搜索分支重复，直接跳过
if (i > start && choices[i] == choices[i - 1]) {
continue;
}
// 尝试：做出选择，更新 target, start
state.push_back(choices[i]);
// 进行下一轮选择
backtrack(state, target - choices[i], choices, i + 1, res);
// 回退：撤销选择，恢复到之前的状态
state.pop_back();
}
}
/* 求解子集和 II */
vector<vector<int>> subsetSumII(vector<int> &nums, int target) {
vector<int> state; // 状态（子集）
sort(nums.begin(), nums.end()); // 对 nums 进行排序
int start = 0; // 遍历起始点
vector<vector<int>> res; // 结果列表（子集列表）
backtrack(state, target, nums, start, res);
return res;
}
/* Driver Code */
int main() {
vector<int> nums = {4, 4, 5};
int target = 9;
vector<vector<int>> res = subsetSumII(nums, target);
cout << "输入数组 nums = ";
printVector(nums);
cout << "target = " << target << endl;
cout << "所有和等于 " << target << " 的子集 res = " << endl;
printVectorMatrix(res);
return 0;
}

5
chapter_computational_complexity/CMakeLists.txt
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add_executable(iteration iteration.cpp)
add_executable(recursion recursion.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)

76
chapter_computational_complexity/iteration.cpp
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/**
* File: iteration.cpp
* Created Time: 2023-08-24
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* for 循环 */
int forLoop(int n) {
int res = 0;
// 循环求和 1, 2, ..., n-1, n
for (int i = 1; i <= n; ++i) {
res += i;
}
return res;
}
/* while 循环 */
int whileLoop(int n) {
int res = 0;
int i = 1; // 初始化条件变量
// 循环求和 1, 2, ..., n-1, n
while (i <= n) {
res += i;
i++; // 更新条件变量
}
return res;
}
/* while 循环（两次更新） */
int whileLoopII(int n) {
int res = 0;
int i = 1; // 初始化条件变量
// 循环求和 1, 4, 10, ...
while (i <= n) {
res += i;
// 更新条件变量
i++;
i *= 2;
}
return res;
}
/* 双层 for 循环 */
string nestedForLoop(int n) {
ostringstream res;
// 循环 i = 1, 2, ..., n-1, n
for (int i = 1; i <= n; ++i) {
// 循环 j = 1, 2, ..., n-1, n
for (int j = 1; j <= n; ++j) {
res << "(" << i << ", " << j << "), ";
}
}
return res.str();
}
/* Driver Code */
int main() {
int n = 5;
int res;
res = forLoop(n);
cout << "\nfor 循环的求和结果 res = " << res << endl;
res = whileLoop(n);
cout << "\nwhile 循环的求和结果 res = " << res << endl;
res = whileLoopII(n);
cout << "\nwhile 循环（两次更新）求和结果 res = " << res << endl;
string resStr = nestedForLoop(n);
cout << "\n双层 for 循环的遍历结果 " << resStr << endl;
return 0;
}

78
chapter_computational_complexity/recursion.cpp
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/**
* File: recursion.cpp
* Created Time: 2023-08-24
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 递归 */
int recur(int n) {
// 终止条件
if (n == 1)
return 1;
// 递：递归调用
int res = recur(n - 1);
// 归：返回结果
return n + res;
}
/* 使用迭代模拟递归 */
int forLoopRecur(int n) {
// 使用一个显式的栈来模拟系统调用栈
stack<int> stack;
int res = 0;
// 递：递归调用
for (int i = n; i > 0; i--) {
// 通过“入栈操作”模拟“递”
stack.push(i);
}
// 归：返回结果
while (!stack.empty()) {
// 通过“出栈操作”模拟“归”
res += stack.top();
stack.pop();
}
// res = 1+2+3+...+n
return res;
}
/* 尾递归 */
int tailRecur(int n, int res) {
// 终止条件
if (n == 0)
return res;
// 尾递归调用
return tailRecur(n - 1, res + n);
}
/* 斐波那契数列：递归 */
int fib(int n) {
// 终止条件 f(1) = 0, f(2) = 1
if (n == 1 || n == 2)
return n - 1;
// 递归调用 f(n) = f(n-1) + f(n-2)
int res = fib(n - 1) + fib(n - 2);
// 返回结果 f(n)
return res;
}
/* Driver Code */
int main() {
int n = 5;
int res;
res = recur(n);
cout << "\n递归函数的求和结果 res = " << res << endl;
res = forLoopRecur(n);
cout << "\n使用迭代模拟递归求和结果 res = " << res << endl;
res = tailRecur(n, 0);
cout << "\n尾递归函数的求和结果 res = " << res << endl;
res = fib(n);
cout << "\n斐波那契数列的第 " << n << " 项为 " << res << endl;
return 0;
}

107
chapter_computational_complexity/space_complexity.cpp
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/**
* File: space_complexity.cpp
* Created Time: 2022-11-25
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 函数 */
int func() {
// 执行某些操作
return 0;
}
/* 常数阶 */
void constant(int n) {
// 常量、变量、对象占用 O(1) 空间
const int a = 0;
int b = 0;
vector<int> nums(10000);
ListNode node(0);
// 循环中的变量占用 O(1) 空间
for (int i = 0; i < n; i++) {
int c = 0;
}
// 循环中的函数占用 O(1) 空间
for (int i = 0; i < n; i++) {
func();
}
}
/* 线性阶 */
void linear(int n) {
// 长度为 n 的数组占用 O(n) 空间
vector<int> nums(n);
// 长度为 n 的列表占用 O(n) 空间
vector<ListNode> nodes;
for (int i = 0; i < n; i++) {
nodes.push_back(ListNode(i));
}
// 长度为 n 的哈希表占用 O(n) 空间
unordered_map<int, string> map;
for (int i = 0; i < n; i++) {
map[i] = to_string(i);
}
}
/* 线性阶（递归实现） */
void linearRecur(int n) {
cout << "递归 n = " << n << endl;
if (n == 1)
return;
linearRecur(n - 1);
}
/* 平方阶 */
void quadratic(int n) {
// 二维列表占用 O(n^2) 空间
vector<vector<int>> numMatrix;
for (int i = 0; i < n; i++) {
vector<int> tmp;
for (int j = 0; j < n; j++) {
tmp.push_back(0);
}
numMatrix.push_back(tmp);
}
}
/* 平方阶（递归实现） */
int quadraticRecur(int n) {
if (n <= 0)
return 0;
vector<int> nums(n);
cout << "递归 n = " << n << " 中的 nums 长度 = " << nums.size() << endl;
return quadraticRecur(n - 1);
}
/* 指数阶（建立满二叉树） */
TreeNode *buildTree(int n) {
if (n == 0)
return nullptr;
TreeNode *root = new TreeNode(0);
root->left = buildTree(n - 1);
root->right = buildTree(n - 1);
return root;
}
/* Driver Code */
int main() {
int n = 5;
// 常数阶
constant(n);
// 线性阶
linear(n);
linearRecur(n);
// 平方阶
quadratic(n);
quadraticRecur(n);
// 指数阶
TreeNode *root = buildTree(n);
printTree(root);
// 释放内存
freeMemoryTree(root);
return 0;
}

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chapter_computational_complexity/time_complexity.cpp
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/**
* File: time_complexity.cpp
* Created Time: 2022-11-25
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 常数阶 */
int constant(int n) {
int count = 0;
int size = 100000;
for (int i = 0; i < size; i++)
count++;
return count;
}
/* 线性阶 */
int linear(int n) {
int count = 0;
for (int i = 0; i < n; i++)
count++;
return count;
}
/* 线性阶（遍历数组） */
int arrayTraversal(vector<int> &nums) {
int count = 0;
// 循环次数与数组长度成正比
for (int num : nums) {
count++;
}
return count;
}
/* 平方阶 */
int quadratic(int n) {
int count = 0;
// 循环次数与数组长度成平方关系
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
count++;
}
}
return count;
}
/* 平方阶（冒泡排序） */
int bubbleSort(vector<int> &nums) {
int count = 0; // 计数器
// 外循环：未排序区间为 [0, i]
for (int i = nums.size() - 1; i > 0; i--) {
// 内循环：将未排序区间 [0, i] 中的最大元素交换至该区间的最右端
for (int j = 0; j < i; j++) {
if (nums[j] > nums[j + 1]) {
// 交换 nums[j] 与 nums[j + 1]
int tmp = nums[j];
nums[j] = nums[j + 1];
nums[j + 1] = tmp;
count += 3; // 元素交换包含 3 个单元操作
}
}
}
return count;
}
/* 指数阶（循环实现） */
int exponential(int n) {
int count = 0, base = 1;
// 细胞每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)
for (int i = 0; i < n; i++) {
for (int j = 0; j < base; j++) {
count++;
}
base *= 2;
}
// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1
return count;
}
/* 指数阶（递归实现） */
int expRecur(int n) {
if (n == 1)
return 1;
return expRecur(n - 1) + expRecur(n - 1) + 1;
}
/* 对数阶（循环实现） */
int logarithmic(float n) {
int count = 0;
while (n > 1) {
n = n / 2;
count++;
}
return count;
}
/* 对数阶（递归实现） */
int logRecur(float n) {
if (n <= 1)
return 0;
return logRecur(n / 2) + 1;
}
/* 线性对数阶 */
int linearLogRecur(float n) {
if (n <= 1)
return 1;
int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);
for (int i = 0; i < n; i++) {
count++;
}
return count;
}
/* 阶乘阶（递归实现） */
int factorialRecur(int n) {
if (n == 0)
return 1;
int count = 0;
// 从 1 个分裂出 n 个
for (int i = 0; i < n; i++) {
count += factorialRecur(n - 1);
}
return count;
}
/* Driver Code */
int main() {
// 可以修改 n 运行，体会一下各种复杂度的操作数量变化趋势
int n = 8;
cout << "输入数据大小 n = " << n << endl;
int count = constant(n);
cout << "常数阶的操作数量 = " << count << endl;
count = linear(n);
cout << "线性阶的操作数量 = " << count << endl;
vector<int> arr(n);
count = arrayTraversal(arr);
cout << "线性阶（遍历数组）的操作数量 = " << count << endl;
count = quadratic(n);
cout << "平方阶的操作数量 = " << count << endl;
vector<int> nums(n);
for (int i = 0; i < n; i++)
nums[i] = n - i; // [n,n-1,...,2,1]
count = bubbleSort(nums);
cout << "平方阶（冒泡排序）的操作数量 = " << count << endl;
count = exponential(n);
cout << "指数阶（循环实现）的操作数量 = " << count << endl;
count = expRecur(n);
cout << "指数阶（递归实现）的操作数量 = " << count << endl;
count = logarithmic((float)n);
cout << "对数阶（循环实现）的操作数量 = " << count << endl;
count = logRecur((float)n);
cout << "对数阶（递归实现）的操作数量 = " << count << endl;
count = linearLogRecur((float)n);
cout << "线性对数阶（递归实现）的操作数量 = " << count << endl;
count = factorialRecur(n);
cout << "阶乘阶（递归实现）的操作数量 = " << count << endl;
return 0;
}

45
chapter_computational_complexity/worst_best_time_complexity.cpp
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/**
* File: worst_best_time_complexity.cpp
* Created Time: 2022-11-25
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 生成一个数组，元素为 { 1, 2, ..., n }，顺序被打乱 */
vector<int> randomNumbers(int n) {
vector<int> nums(n);
// 生成数组 nums = { 1, 2, 3, ..., n }
for (int i = 0; i < n; i++) {
nums[i] = i + 1;
}
// 使用系统时间生成随机种子
unsigned seed = chrono::system_clock::now().time_since_epoch().count();
// 随机打乱数组元素
shuffle(nums.begin(), nums.end(), default_random_engine(seed));
return nums;
}
/* 查找数组 nums 中数字 1 所在索引 */
int findOne(vector<int> &nums) {
for (int i = 0; i < nums.size(); i++) {
// 当元素 1 在数组头部时，达到最佳时间复杂度 O(1)
// 当元素 1 在数组尾部时，达到最差时间复杂度 O(n)
if (nums[i] == 1)
return i;
}
return -1;
}
/* Driver Code */
int main() {
for (int i = 0; i < 1000; i++) {
int n = 100;
vector<int> nums = randomNumbers(n);
int index = findOne(nums);
cout << "\n数组 [ 1, 2, ..., n ] 被打乱后 = ";
printVector(nums);
cout << "数字 1 的索引为 " << index << endl;
}
return 0;
}

3
chapter_divide_and_conquer/CMakeLists.txt
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add_executable(binary_search_recur binary_search_recur.cpp)
add_executable(build_tree build_tree.cpp)
add_executable(hanota hanota.cpp)

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chapter_divide_and_conquer/binary_search_recur.cpp
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/**
* File: binary_search_recur.cpp
* Created Time: 2023-07-17
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 二分查找：问题 f(i, j) */
int dfs(vector<int> &nums, int target, int i, int j) {
// 若区间为空，代表无目标元素，则返回 -1
if (i > j) {
return -1;
}
// 计算中点索引 m
int m = (i + j) / 2;
if (nums[m] < target) {
// 递归子问题 f(m+1, j)
return dfs(nums, target, m + 1, j);
} else if (nums[m] > target) {
// 递归子问题 f(i, m-1)
return dfs(nums, target, i, m - 1);
} else {
// 找到目标元素，返回其索引
return m;
}
}
/* 二分查找 */
int binarySearch(vector<int> &nums, int target) {
int n = nums.size();
// 求解问题 f(0, n-1)
return dfs(nums, target, 0, n - 1);
}
/* Driver Code */
int main() {
int target = 6;
vector<int> nums = {1, 3, 6, 8, 12, 15, 23, 26, 31, 35};
// 二分查找（双闭区间）
int index = binarySearch(nums, target);
cout << "目标元素 6 的索引 = " << index << endl;
return 0;
}

51
chapter_divide_and_conquer/build_tree.cpp
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/**
* File: build_tree.cpp
* Created Time: 2023-07-17
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 构建二叉树：分治 */
TreeNode *dfs(vector<int> &preorder, unordered_map<int, int> &inorderMap, int i, int l, int r) {
// 子树区间为空时终止
if (r - l < 0)
return NULL;
// 初始化根节点
TreeNode *root = new TreeNode(preorder[i]);
// 查询 m ，从而划分左右子树
int m = inorderMap[preorder[i]];
// 子问题：构建左子树
root->left = dfs(preorder, inorderMap, i + 1, l, m - 1);
// 子问题：构建右子树
root->right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);
// 返回根节点
return root;
}
/* 构建二叉树 */
TreeNode *buildTree(vector<int> &preorder, vector<int> &inorder) {
// 初始化哈希表，存储 inorder 元素到索引的映射
unordered_map<int, int> inorderMap;
for (int i = 0; i < inorder.size(); i++) {
inorderMap[inorder[i]] = i;
}
TreeNode *root = dfs(preorder, inorderMap, 0, 0, inorder.size() - 1);
return root;
}
/* Driver Code */
int main() {
vector<int> preorder = {3, 9, 2, 1, 7};
vector<int> inorder = {9, 3, 1, 2, 7};
cout << "前序遍历 = ";
printVector(preorder);
cout << "中序遍历 = ";
printVector(inorder);
TreeNode *root = buildTree(preorder, inorder);
cout << "构建的二叉树为：\n";
printTree(root);
return 0;
}

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chapter_divide_and_conquer/hanota.cpp
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/**
* File: hanota.cpp
* Created Time: 2023-07-17
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 移动一个圆盘 */
void move(vector<int> &src, vector<int> &tar) {
// 从 src 顶部拿出一个圆盘
int pan = src.back();
src.pop_back();
// 将圆盘放入 tar 顶部
tar.push_back(pan);
}
/* 求解汉诺塔问题 f(i) */
void dfs(int i, vector<int> &src, vector<int> &buf, vector<int> &tar) {
// 若 src 只剩下一个圆盘，则直接将其移到 tar
if (i == 1) {
move(src, tar);
return;
}
// 子问题 f(i-1) ：将 src 顶部 i-1 个圆盘借助 tar 移到 buf
dfs(i - 1, src, tar, buf);
// 子问题 f(1) ：将 src 剩余一个圆盘移到 tar
move(src, tar);
// 子问题 f(i-1) ：将 buf 顶部 i-1 个圆盘借助 src 移到 tar
dfs(i - 1, buf, src, tar);
}
/* 求解汉诺塔问题 */
void solveHanota(vector<int> &A, vector<int> &B, vector<int> &C) {
int n = A.size();
// 将 A 顶部 n 个圆盘借助 B 移到 C
dfs(n, A, B, C);
}
/* Driver Code */
int main() {
// 列表尾部是柱子顶部
vector<int> A = {5, 4, 3, 2, 1};
vector<int> B = {};
vector<int> C = {};
cout << "初始状态下：\n";
cout << "A =";
printVector(A);
cout << "B =";
printVector(B);
cout << "C =";
printVector(C);
solveHanota(A, B, C);
cout << "圆盘移动完成后：\n";
cout << "A =";
printVector(A);
cout << "B =";
printVector(B);
cout << "C =";
printVector(C);
return 0;
}

10
chapter_dynamic_programming/CMakeLists.txt
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add_executable(climbing_stairs_backtrack climbing_stairs_backtrack.cpp)
add_executable(climbing_stairs_dfs climbing_stairs_dfs.cpp)
add_executable(climbing_stairs_dfs_mem climbing_stairs_dfs_mem.cpp)
add_executable(climbing_stairs_dp climbing_stairs_dp.cpp)
add_executable(min_cost_climbing_stairs_dp min_cost_climbing_stairs_dp.cpp)
add_executable(min_path_sum min_path_sum.cpp)
add_executable(unbounded_knapsack unbounded_knapsack.cpp)
add_executable(coin_change coin_change.cpp)
add_executable(coin_change_ii coin_change_ii.cpp)
add_executable(edit_distance edit_distance.cpp)

43
chapter_dynamic_programming/climbing_stairs_backtrack.cpp
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/**
* File: climbing_stairs_backtrack.cpp
* Created Time: 2023-06-30
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 回溯 */
void backtrack(vector<int> &choices, int state, int n, vector<int> &res) {
// 当爬到第 n 阶时，方案数量加 1
if (state == n)
res[0]++;
// 遍历所有选择
for (auto &choice : choices) {
// 剪枝：不允许越过第 n 阶
if (state + choice > n)
continue;
// 尝试：做出选择，更新状态
backtrack(choices, state + choice, n, res);
// 回退
}
}
/* 爬楼梯：回溯 */
int climbingStairsBacktrack(int n) {
vector<int> choices = {1, 2}; // 可选择向上爬 1 阶或 2 阶
int state = 0; // 从第 0 阶开始爬
vector<int> res = {0}; // 使用 res[0] 记录方案数量
backtrack(choices, state, n, res);
return res[0];
}
/* Driver Code */
int main() {
int n = 9;
int res = climbingStairsBacktrack(n);
cout << "爬 " << n << " 阶楼梯共有 " << res << " 种方案" << endl;
return 0;
}

37
chapter_dynamic_programming/climbing_stairs_constraint_dp.cpp
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/**
* File: climbing_stairs_constraint_dp.cpp
* Created Time: 2023-07-01
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 带约束爬楼梯：动态规划 */
int climbingStairsConstraintDP(int n) {
if (n == 1 || n == 2) {
return 1;
}
// 初始化 dp 表，用于存储子问题的解
vector<vector<int>> dp(n + 1, vector<int>(3, 0));
// 初始状态：预设最小子问题的解
dp[1][1] = 1;
dp[1][2] = 0;
dp[2][1] = 0;
dp[2][2] = 1;
// 状态转移：从较小子问题逐步求解较大子问题
for (int i = 3; i <= n; i++) {
dp[i][1] = dp[i - 1][2];
dp[i][2] = dp[i - 2][1] + dp[i - 2][2];
}
return dp[n][1] + dp[n][2];
}
/* Driver Code */
int main() {
int n = 9;
int res = climbingStairsConstraintDP(n);
cout << "爬 " << n << " 阶楼梯共有 " << res << " 种方案" << endl;
return 0;
}

32
chapter_dynamic_programming/climbing_stairs_dfs.cpp
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/**
* File: climbing_stairs_dfs.cpp
* Created Time: 2023-06-30
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 搜索 */
int dfs(int i) {
// 已知 dp[1] 和 dp[2] ，返回之
if (i == 1 || i == 2)
return i;
// dp[i] = dp[i-1] + dp[i-2]
int count = dfs(i - 1) + dfs(i - 2);
return count;
}
/* 爬楼梯：搜索 */
int climbingStairsDFS(int n) {
return dfs(n);
}
/* Driver Code */
int main() {
int n = 9;
int res = climbingStairsDFS(n);
cout << "爬 " << n << " 阶楼梯共有 " << res << " 种方案" << endl;
return 0;
}

39
chapter_dynamic_programming/climbing_stairs_dfs_mem.cpp
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/**
* File: climbing_stairs_dfs_mem.cpp
* Created Time: 2023-06-30
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 记忆化搜索 */
int dfs(int i, vector<int> &mem) {
// 已知 dp[1] 和 dp[2] ，返回之
if (i == 1 || i == 2)
return i;
// 若存在记录 dp[i] ，则直接返回之
if (mem[i] != -1)
return mem[i];
// dp[i] = dp[i-1] + dp[i-2]
int count = dfs(i - 1, mem) + dfs(i - 2, mem);
// 记录 dp[i]
mem[i] = count;
return count;
}
/* 爬楼梯：记忆化搜索 */
int climbingStairsDFSMem(int n) {
// mem[i] 记录爬到第 i 阶的方案总数，-1 代表无记录
vector<int> mem(n + 1, -1);
return dfs(n, mem);
}
/* Driver Code */
int main() {
int n = 9;
int res = climbingStairsDFSMem(n);
cout << "爬 " << n << " 阶楼梯共有 " << res << " 种方案" << endl;
return 0;
}

49
chapter_dynamic_programming/climbing_stairs_dp.cpp
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/**
* File: climbing_stairs_dp.cpp
* Created Time: 2023-06-30
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 爬楼梯：动态规划 */
int climbingStairsDP(int n) {
if (n == 1 || n == 2)
return n;
// 初始化 dp 表，用于存储子问题的解
vector<int> dp(n + 1);
// 初始状态：预设最小子问题的解
dp[1] = 1;
dp[2] = 2;
// 状态转移：从较小子问题逐步求解较大子问题
for (int i = 3; i <= n; i++) {
dp[i] = dp[i - 1] + dp[i - 2];
}
return dp[n];
}
/* 爬楼梯：空间优化后的动态规划 */
int climbingStairsDPComp(int n) {
if (n == 1 || n == 2)
return n;
int a = 1, b = 2;
for (int i = 3; i <= n; i++) {
int tmp = b;
b = a + b;
a = tmp;
}
return b;
}
/* Driver Code */
int main() {
int n = 9;
int res = climbingStairsDP(n);
cout << "爬 " << n << " 阶楼梯共有 " << res << " 种方案" << endl;
res = climbingStairsDPComp(n);
cout << "爬 " << n << " 阶楼梯共有 " << res << " 种方案" << endl;
return 0;
}

70
chapter_dynamic_programming/coin_change.cpp
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/**
* File: coin_change.cpp
* Created Time: 2023-07-11
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 零钱兑换：动态规划 */
int coinChangeDP(vector<int> &coins, int amt) {
int n = coins.size();
int MAX = amt + 1;
// 初始化 dp 表
vector<vector<int>> dp(n + 1, vector<int>(amt + 1, 0));
// 状态转移：首行首列
for (int a = 1; a <= amt; a++) {
dp[0][a] = MAX;
}
// 状态转移：其余行和列
for (int i = 1; i <= n; i++) {
for (int a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// 若超过目标金额，则不选硬币 i
dp[i][a] = dp[i - 1][a];
} else {
// 不选和选硬币 i 这两种方案的较小值
dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);
}
}
}
return dp[n][amt] != MAX ? dp[n][amt] : -1;
}
/* 零钱兑换：空间优化后的动态规划 */
int coinChangeDPComp(vector<int> &coins, int amt) {
int n = coins.size();
int MAX = amt + 1;
// 初始化 dp 表
vector<int> dp(amt + 1, MAX);
dp[0] = 0;
// 状态转移
for (int i = 1; i <= n; i++) {
for (int a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// 若超过目标金额，则不选硬币 i
dp[a] = dp[a];
} else {
// 不选和选硬币 i 这两种方案的较小值
dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);
}
}
}
return dp[amt] != MAX ? dp[amt] : -1;
}
/* Driver code */
int main() {
vector<int> coins = {1, 2, 5};
int amt = 4;
// 动态规划
int res = coinChangeDP(coins, amt);
cout << "凑到目标金额所需的最少硬币数量为 " << res << endl;
// 空间优化后的动态规划
res = coinChangeDPComp(coins, amt);
cout << "凑到目标金额所需的最少硬币数量为 " << res << endl;
return 0;
}

68
chapter_dynamic_programming/coin_change_ii.cpp
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/**
* File: coin_change_ii.cpp
* Created Time: 2023-07-11
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 零钱兑换 II：动态规划 */
int coinChangeIIDP(vector<int> &coins, int amt) {
int n = coins.size();
// 初始化 dp 表
vector<vector<int>> dp(n + 1, vector<int>(amt + 1, 0));
// 初始化首列
for (int i = 0; i <= n; i++) {
dp[i][0] = 1;
}
// 状态转移
for (int i = 1; i <= n; i++) {
for (int a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// 若超过目标金额，则不选硬币 i
dp[i][a] = dp[i - 1][a];
} else {
// 不选和选硬币 i 这两种方案之和
dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];
}
}
}
return dp[n][amt];
}
/* 零钱兑换 II：空间优化后的动态规划 */
int coinChangeIIDPComp(vector<int> &coins, int amt) {
int n = coins.size();
// 初始化 dp 表
vector<int> dp(amt + 1, 0);
dp[0] = 1;
// 状态转移
for (int i = 1; i <= n; i++) {
for (int a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// 若超过目标金额，则不选硬币 i
dp[a] = dp[a];
} else {
// 不选和选硬币 i 这两种方案之和
dp[a] = dp[a] + dp[a - coins[i - 1]];
}
}
}
return dp[amt];
}
/* Driver code */
int main() {
vector<int> coins = {1, 2, 5};
int amt = 5;
// 动态规划
int res = coinChangeIIDP(coins, amt);
cout << "凑出目标金额的硬币组合数量为 " << res << endl;
// 空间优化后的动态规划
res = coinChangeIIDPComp(coins, amt);
cout << "凑出目标金额的硬币组合数量为 " << res << endl;
return 0;
}

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chapter_dynamic_programming/edit_distance.cpp
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/**
* File: edit_distance.cpp
* Created Time: 2023-07-13
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 编辑距离：暴力搜索 */
int editDistanceDFS(string s, string t, int i, int j) {
// 若 s 和 t 都为空，则返回 0
if (i == 0 && j == 0)
return 0;
// 若 s 为空，则返回 t 长度
if (i == 0)
return j;
// 若 t 为空，则返回 s 长度
if (j == 0)
return i;
// 若两字符相等，则直接跳过此两字符
if (s[i - 1] == t[j - 1])
return editDistanceDFS(s, t, i - 1, j - 1);
// 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1
int insert = editDistanceDFS(s, t, i, j - 1);
int del = editDistanceDFS(s, t, i - 1, j);
int replace = editDistanceDFS(s, t, i - 1, j - 1);
// 返回最少编辑步数
return min(min(insert, del), replace) + 1;
}
/* 编辑距离：记忆化搜索 */
int editDistanceDFSMem(string s, string t, vector<vector<int>> &mem, int i, int j) {
// 若 s 和 t 都为空，则返回 0
if (i == 0 && j == 0)
return 0;
// 若 s 为空，则返回 t 长度
if (i == 0)
return j;
// 若 t 为空，则返回 s 长度
if (j == 0)
return i;
// 若已有记录，则直接返回之
if (mem[i][j] != -1)
return mem[i][j];
// 若两字符相等，则直接跳过此两字符
if (s[i - 1] == t[j - 1])
return editDistanceDFSMem(s, t, mem, i - 1, j - 1);
// 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1
int insert = editDistanceDFSMem(s, t, mem, i, j - 1);
int del = editDistanceDFSMem(s, t, mem, i - 1, j);
int replace = editDistanceDFSMem(s, t, mem, i - 1, j - 1);
// 记录并返回最少编辑步数
mem[i][j] = min(min(insert, del), replace) + 1;
return mem[i][j];
}
/* 编辑距离：动态规划 */
int editDistanceDP(string s, string t) {
int n = s.length(), m = t.length();
vector<vector<int>> dp(n + 1, vector<int>(m + 1, 0));
// 状态转移：首行首列
for (int i = 1; i <= n; i++) {
dp[i][0] = i;
}
for (int j = 1; j <= m; j++) {
dp[0][j] = j;
}
// 状态转移：其余行和列
for (int i = 1; i <= n; i++) {
for (int j = 1; j <= m; j++) {
if (s[i - 1] == t[j - 1]) {
// 若两字符相等，则直接跳过此两字符
dp[i][j] = dp[i - 1][j - 1];
} else {
// 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1
dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;
}
}
}
return dp[n][m];
}
/* 编辑距离：空间优化后的动态规划 */
int editDistanceDPComp(string s, string t) {
int n = s.length(), m = t.length();
vector<int> dp(m + 1, 0);
// 状态转移：首行
for (int j = 1; j <= m; j++) {
dp[j] = j;
}
// 状态转移：其余行
for (int i = 1; i <= n; i++) {
// 状态转移：首列
int leftup = dp[0]; // 暂存 dp[i-1, j-1]
dp[0] = i;
// 状态转移：其余列
for (int j = 1; j <= m; j++) {
int temp = dp[j];
if (s[i - 1] == t[j - 1]) {
// 若两字符相等，则直接跳过此两字符
dp[j] = leftup;
} else {
// 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1
dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1;
}
leftup = temp; // 更新为下一轮的 dp[i-1, j-1]
}
}
return dp[m];
}
/* Driver Code */
int main() {
string s = "bag";
string t = "pack";
int n = s.length(), m = t.length();
// 暴力搜索
int res = editDistanceDFS(s, t, n, m);
cout << "将 " << s << " 更改为 " << t << " 最少需要编辑 " << res << " 步\n";
// 记忆化搜索
vector<vector<int>> mem(n + 1, vector<int>(m + 1, -1));
res = editDistanceDFSMem(s, t, mem, n, m);
cout << "将 " << s << " 更改为 " << t << " 最少需要编辑 " << res << " 步\n";
// 动态规划
res = editDistanceDP(s, t);
cout << "将 " << s << " 更改为 " << t << " 最少需要编辑 " << res << " 步\n";
// 空间优化后的动态规划
res = editDistanceDPComp(s, t);
cout << "将 " << s << " 更改为 " << t << " 最少需要编辑 " << res << " 步\n";
return 0;
}

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chapter_dynamic_programming/knapsack.cpp
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#include <algorithm>
#include <iostream>
#include <vector>
using namespace std;
/* 0-1 背包：暴力搜索 */
int knapsackDFS(vector<int> &wgt, vector<int> &val, int i, int c) {
// 若已选完所有物品或背包无剩余容量，则返回价值 0
if (i == 0 || c == 0) {
return 0;
}
// 若超过背包容量，则只能选择不放入背包
if (wgt[i - 1] > c) {
return knapsackDFS(wgt, val, i - 1, c);
}
// 计算不放入和放入物品 i 的最大价值
int no = knapsackDFS(wgt, val, i - 1, c);
int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];
// 返回两种方案中价值更大的那一个
return max(no, yes);
}
/* 0-1 背包：记忆化搜索 */
int knapsackDFSMem(vector<int> &wgt, vector<int> &val, vector<vector<int>> &mem, int i, int c) {
// 若已选完所有物品或背包无剩余容量，则返回价值 0
if (i == 0 || c == 0) {
return 0;
}
// 若已有记录，则直接返回
if (mem[i][c] != -1) {
return mem[i][c];
}
// 若超过背包容量，则只能选择不放入背包
if (wgt[i - 1] > c) {
return knapsackDFSMem(wgt, val, mem, i - 1, c);
}
// 计算不放入和放入物品 i 的最大价值
int no = knapsackDFSMem(wgt, val, mem, i - 1, c);
int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];
// 记录并返回两种方案中价值更大的那一个
mem[i][c] = max(no, yes);
return mem[i][c];
}
/* 0-1 背包：动态规划 */
int knapsackDP(vector<int> &wgt, vector<int> &val, int cap) {
int n = wgt.size();
// 初始化 dp 表
vector<vector<int>> dp(n + 1, vector<int>(cap + 1, 0));
// 状态转移
for (int i = 1; i <= n; i++) {
for (int c = 1; c <= cap; c++) {
if (wgt[i - 1] > c) {
// 若超过背包容量，则不选物品 i
dp[i][c] = dp[i - 1][c];
} else {
// 不选和选物品 i 这两种方案的较大值
dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);
}
}
}
return dp[n][cap];
}
/* 0-1 背包：空间优化后的动态规划 */
int knapsackDPComp(vector<int> &wgt, vector<int> &val, int cap) {
int n = wgt.size();
// 初始化 dp 表
vector<int> dp(cap + 1, 0);
// 状态转移
for (int i = 1; i <= n; i++) {
// 倒序遍历
for (int c = cap; c >= 1; c--) {
if (wgt[i - 1] <= c) {
// 不选和选物品 i 这两种方案的较大值
dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);
}
}
}
return dp[cap];
}
/* Driver Code */
int main() {
vector<int> wgt = {10, 20, 30, 40, 50};
vector<int> val = {50, 120, 150, 210, 240};
int cap = 50;
int n = wgt.size();
// 暴力搜索
int res = knapsackDFS(wgt, val, n, cap);
cout << "不超过背包容量的最大物品价值为 " << res << endl;
// 记忆化搜索
vector<vector<int>> mem(n + 1, vector<int>(cap + 1, -1));
res = knapsackDFSMem(wgt, val, mem, n, cap);
cout << "不超过背包容量的最大物品价值为 " << res << endl;
// 动态规划
res = knapsackDP(wgt, val, cap);
cout << "不超过背包容量的最大物品价值为 " << res << endl;
// 空间优化后的动态规划
res = knapsackDPComp(wgt, val, cap);
cout << "不超过背包容量的最大物品价值为 " << res << endl;
return 0;
}

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chapter_dynamic_programming/min_cost_climbing_stairs_dp.cpp
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/**
* File: min_cost_climbing_stairs_dp.cpp
* Created Time: 2023-06-30
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 爬楼梯最小代价：动态规划 */
int minCostClimbingStairsDP(vector<int> &cost) {
int n = cost.size() - 1;
if (n == 1 || n == 2)
return cost[n];
// 初始化 dp 表，用于存储子问题的解
vector<int> dp(n + 1);
// 初始状态：预设最小子问题的解
dp[1] = cost[1];
dp[2] = cost[2];
// 状态转移：从较小子问题逐步求解较大子问题
for (int i = 3; i <= n; i++) {
dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];
}
return dp[n];
}
/* 爬楼梯最小代价：空间优化后的动态规划 */
int minCostClimbingStairsDPComp(vector<int> &cost) {
int n = cost.size() - 1;
if (n == 1 || n == 2)
return cost[n];
int a = cost[1], b = cost[2];
for (int i = 3; i <= n; i++) {
int tmp = b;
b = min(a, tmp) + cost[i];
a = tmp;
}
return b;
}
/* Driver Code */
int main() {
vector<int> cost = {0, 1, 10, 1, 1, 1, 10, 1, 1, 10, 1};
cout << "输入楼梯的代价列表为 ";
printVector(cost);
int res = minCostClimbingStairsDP(cost);
cout << "爬完楼梯的最低代价为 " << res << endl;
res = minCostClimbingStairsDPComp(cost);
cout << "爬完楼梯的最低代价为 " << res << endl;
return 0;
}

116
chapter_dynamic_programming/min_path_sum.cpp
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/**
* File: min_path_sum.cpp
* Created Time: 2023-07-10
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 最小路径和：暴力搜索 */
int minPathSumDFS(vector<vector<int>> &grid, int i, int j) {
// 若为左上角单元格，则终止搜索
if (i == 0 && j == 0) {
return grid[0][0];
}
// 若行列索引越界，则返回 +∞ 代价
if (i < 0 || j < 0) {
return INT_MAX;
}
// 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价
int up = minPathSumDFS(grid, i - 1, j);
int left = minPathSumDFS(grid, i, j - 1);
// 返回从左上角到 (i, j) 的最小路径代价
return min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;
}
/* 最小路径和：记忆化搜索 */
int minPathSumDFSMem(vector<vector<int>> &grid, vector<vector<int>> &mem, int i, int j) {
// 若为左上角单元格，则终止搜索
if (i == 0 && j == 0) {
return grid[0][0];
}
// 若行列索引越界，则返回 +∞ 代价
if (i < 0 || j < 0) {
return INT_MAX;
}
// 若已有记录，则直接返回
if (mem[i][j] != -1) {
return mem[i][j];
}
// 左边和上边单元格的最小路径代价
int up = minPathSumDFSMem(grid, mem, i - 1, j);
int left = minPathSumDFSMem(grid, mem, i, j - 1);
// 记录并返回左上角到 (i, j) 的最小路径代价
mem[i][j] = min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;
return mem[i][j];
}
/* 最小路径和：动态规划 */
int minPathSumDP(vector<vector<int>> &grid) {
int n = grid.size(), m = grid[0].size();
// 初始化 dp 表
vector<vector<int>> dp(n, vector<int>(m));
dp[0][0] = grid[0][0];
// 状态转移：首行
for (int j = 1; j < m; j++) {
dp[0][j] = dp[0][j - 1] + grid[0][j];
}
// 状态转移：首列
for (int i = 1; i < n; i++) {
dp[i][0] = dp[i - 1][0] + grid[i][0];
}
// 状态转移：其余行和列
for (int i = 1; i < n; i++) {
for (int j = 1; j < m; j++) {
dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];
}
}
return dp[n - 1][m - 1];
}
/* 最小路径和：空间优化后的动态规划 */
int minPathSumDPComp(vector<vector<int>> &grid) {
int n = grid.size(), m = grid[0].size();
// 初始化 dp 表
vector<int> dp(m);
// 状态转移：首行
dp[0] = grid[0][0];
for (int j = 1; j < m; j++) {
dp[j] = dp[j - 1] + grid[0][j];
}
// 状态转移：其余行
for (int i = 1; i < n; i++) {
// 状态转移：首列
dp[0] = dp[0] + grid[i][0];
// 状态转移：其余列
for (int j = 1; j < m; j++) {
dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];
}
}
return dp[m - 1];
}
/* Driver Code */
int main() {
vector<vector<int>> grid = {{1, 3, 1, 5}, {2, 2, 4, 2}, {5, 3, 2, 1}, {4, 3, 5, 2}};
int n = grid.size(), m = grid[0].size();
// 暴力搜索
int res = minPathSumDFS(grid, n - 1, m - 1);
cout << "从左上角到右下角的最小路径和为 " << res << endl;
// 记忆化搜索
vector<vector<int>> mem(n, vector<int>(m, -1));
res = minPathSumDFSMem(grid, mem, n - 1, m - 1);
cout << "从左上角到右下角的最小路径和为 " << res << endl;
// 动态规划
res = minPathSumDP(grid);
cout << "从左上角到右下角的最小路径和为 " << res << endl;
// 空间优化后的动态规划
res = minPathSumDPComp(grid);
cout << "从左上角到右下角的最小路径和为 " << res << endl;
return 0;
}

64
chapter_dynamic_programming/unbounded_knapsack.cpp
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/**
* File: unbounded_knapsack.cpp
* Created Time: 2023-07-11
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 完全背包：动态规划 */
int unboundedKnapsackDP(vector<int> &wgt, vector<int> &val, int cap) {
int n = wgt.size();
// 初始化 dp 表
vector<vector<int>> dp(n + 1, vector<int>(cap + 1, 0));
// 状态转移
for (int i = 1; i <= n; i++) {
for (int c = 1; c <= cap; c++) {
if (wgt[i - 1] > c) {
// 若超过背包容量，则不选物品 i
dp[i][c] = dp[i - 1][c];
} else {
// 不选和选物品 i 这两种方案的较大值
dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);
}
}
}
return dp[n][cap];
}
/* 完全背包：空间优化后的动态规划 */
int unboundedKnapsackDPComp(vector<int> &wgt, vector<int> &val, int cap) {
int n = wgt.size();
// 初始化 dp 表
vector<int> dp(cap + 1, 0);
// 状态转移
for (int i = 1; i <= n; i++) {
for (int c = 1; c <= cap; c++) {
if (wgt[i - 1] > c) {
// 若超过背包容量，则不选物品 i
dp[c] = dp[c];
} else {
// 不选和选物品 i 这两种方案的较大值
dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);
}
}
}
return dp[cap];
}
/* Driver code */
int main() {
vector<int> wgt = {1, 2, 3};
vector<int> val = {5, 11, 15};
int cap = 4;
// 动态规划
int res = unboundedKnapsackDP(wgt, val, cap);
cout << "不超过背包容量的最大物品价值为 " << res << endl;
// 空间优化后的动态规划
res = unboundedKnapsackDPComp(wgt, val, cap);
cout << "不超过背包容量的最大物品价值为 " << res << endl;
return 0;
}

5
chapter_graph/CMakeLists.txt
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add_executable(graph_bfs graph_bfs.cpp)
add_executable(graph_dfs graph_dfs.cpp)
# add_executable(graph_adjacency_list graph_adjacency_list.cpp)
add_executable(graph_adjacency_list_test graph_adjacency_list_test.cpp)
add_executable(graph_adjacency_matrix graph_adjacency_matrix.cpp)

90
chapter_graph/graph_adjacency_list.cpp
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/**
* File: graph_adjacency_list.cpp
* Created Time: 2023-02-09
* Author: what-is-me (whatisme@outlook.jp), krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 基于邻接表实现的无向图类 */
class GraphAdjList {
public:
// 邻接表，key：顶点，value：该顶点的所有邻接顶点
unordered_map<Vertex *, vector<Vertex *>> adjList;
/* 在 vector 中删除指定节点 */
void remove(vector<Vertex *> &vec, Vertex *vet) {
for (int i = 0; i < vec.size(); i++) {
if (vec[i] == vet) {
vec.erase(vec.begin() + i);
break;
}
}
}
/* 构造方法 */
GraphAdjList(const vector<vector<Vertex *>> &edges) {
// 添加所有顶点和边
for (const vector<Vertex *> &edge : edges) {
addVertex(edge[0]);
addVertex(edge[1]);
addEdge(edge[0], edge[1]);
}
}
/* 获取顶点数量 */
int size() {
return adjList.size();
}
/* 添加边 */
void addEdge(Vertex *vet1, Vertex *vet2) {
if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)
throw invalid_argument("不存在顶点");
// 添加边 vet1 - vet2
adjList[vet1].push_back(vet2);
adjList[vet2].push_back(vet1);
}
/* 删除边 */
void removeEdge(Vertex *vet1, Vertex *vet2) {
if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)
throw invalid_argument("不存在顶点");
// 删除边 vet1 - vet2
remove(adjList[vet1], vet2);
remove(adjList[vet2], vet1);
}
/* 添加顶点 */
void addVertex(Vertex *vet) {
if (adjList.count(vet))
return;
// 在邻接表中添加一个新链表
adjList[vet] = vector<Vertex *>();
}
/* 删除顶点 */
void removeVertex(Vertex *vet) {
if (!adjList.count(vet))
throw invalid_argument("不存在顶点");
// 在邻接表中删除顶点 vet 对应的链表
adjList.erase(vet);
// 遍历其他顶点的链表，删除所有包含 vet 的边
for (auto &adj : adjList) {
remove(adj.second, vet);
}
}
/* 打印邻接表 */
void print() {
cout << "邻接表 =" << endl;
for (auto &adj : adjList) {
const auto &key = adj.first;
const auto &vec = adj.second;
cout << key->val << ": ";
printVector(vetsToVals(vec));
}
}
};
// 测试样例请见 graph_adjacency_list_test.cpp

49
chapter_graph/graph_adjacency_list_test.cpp
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/**
* File: graph_adjacency_list_test.cpp
* Created Time: 2023-02-09
* Author: what-is-me (whatisme@outlook.jp), krahets (krahets@163.com)
*/
#include "./graph_adjacency_list.cpp"
/* Driver Code */
int main() {
/* 初始化无向图 */
vector<Vertex *> v = valsToVets(vector<int>{1, 3, 2, 5, 4});
vector<vector<Vertex *>> edges = {{v[0], v[1]}, {v[0], v[3]}, {v[1], v[2]},
{v[2], v[3]}, {v[2], v[4]}, {v[3], v[4]}};
GraphAdjList graph(edges);
cout << "\n初始化后，图为" << endl;
graph.print();
/* 添加边 */
// 顶点 1, 2 即 v[0], v[2]
graph.addEdge(v[0], v[2]);
cout << "\n添加边 1-2 后，图为" << endl;
graph.print();
/* 删除边 */
// 顶点 1, 3 即 v[0], v[1]
graph.removeEdge(v[0], v[1]);
cout << "\n删除边 1-3 后，图为" << endl;
graph.print();
/* 添加顶点 */
Vertex *v5 = new Vertex(6);
graph.addVertex(v5);
cout << "\n添加顶点 6 后，图为" << endl;
graph.print();
/* 删除顶点 */
// 顶点 3 即 v[1]
graph.removeVertex(v[1]);
cout << "\n删除顶点 3 后，图为" << endl;
graph.print();
// 释放内存
for (Vertex *vet : v) {
delete vet;
}
return 0;
}

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chapter_graph/graph_adjacency_matrix.cpp
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/**
* File: graph_adjacency_matrix.cpp
* Created Time: 2023-02-09
* Author: what-is-me (whatisme@outlook.jp)
*/
#include "../utils/common.hpp"
/* 基于邻接矩阵实现的无向图类 */
class GraphAdjMat {
vector<int> vertices; // 顶点列表，元素代表“顶点值”，索引代表“顶点索引”
vector<vector<int>> adjMat; // 邻接矩阵，行列索引对应“顶点索引”
public:
/* 构造方法 */
GraphAdjMat(const vector<int> &vertices, const vector<vector<int>> &edges) {
// 添加顶点
for (int val : vertices) {
addVertex(val);
}
// 添加边
// 请注意，edges 元素代表顶点索引，即对应 vertices 元素索引
for (const vector<int> &edge : edges) {
addEdge(edge[0], edge[1]);
}
}
/* 获取顶点数量 */
int size() const {
return vertices.size();
}
/* 添加顶点 */
void addVertex(int val) {
int n = size();
// 向顶点列表中添加新顶点的值
vertices.push_back(val);
// 在邻接矩阵中添加一行
adjMat.emplace_back(vector<int>(n, 0));
// 在邻接矩阵中添加一列
for (vector<int> &row : adjMat) {
row.push_back(0);
}
}
/* 删除顶点 */
void removeVertex(int index) {
if (index >= size()) {
throw out_of_range("顶点不存在");
}
// 在顶点列表中移除索引 index 的顶点
vertices.erase(vertices.begin() + index);
// 在邻接矩阵中删除索引 index 的行
adjMat.erase(adjMat.begin() + index);
// 在邻接矩阵中删除索引 index 的列
for (vector<int> &row : adjMat) {
row.erase(row.begin() + index);
}
}
/* 添加边 */
// 参数 i, j 对应 vertices 元素索引
void addEdge(int i, int j) {
// 索引越界与相等处理
if (i < 0 || j < 0 || i >= size() || j >= size() || i == j) {
throw out_of_range("顶点不存在");
}
// 在无向图中，邻接矩阵关于主对角线对称，即满足 (i, j) == (j, i)
adjMat[i][j] = 1;
adjMat[j][i] = 1;
}
/* 删除边 */
// 参数 i, j 对应 vertices 元素索引
void removeEdge(int i, int j) {
// 索引越界与相等处理
if (i < 0 || j < 0 || i >= size() || j >= size() || i == j) {
throw out_of_range("顶点不存在");
}
adjMat[i][j] = 0;
adjMat[j][i] = 0;
}
/* 打印邻接矩阵 */
void print() {
cout << "顶点列表 = ";
printVector(vertices);
cout << "邻接矩阵 =" << endl;
printVectorMatrix(adjMat);
}
};
/* Driver Code */
int main() {
/* 初始化无向图 */
// 请注意，edges 元素代表顶点索引，即对应 vertices 元素索引
vector<int> vertices = {1, 3, 2, 5, 4};
vector<vector<int>> edges = {{0, 1}, {0, 3}, {1, 2}, {2, 3}, {2, 4}, {3, 4}};
GraphAdjMat graph(vertices, edges);
cout << "\n初始化后，图为" << endl;
graph.print();
/* 添加边 */
// 顶点 1, 2 的索引分别为 0, 2
graph.addEdge(0, 2);
cout << "\n添加边 1-2 后，图为" << endl;
graph.print();
/* 删除边 */
// 顶点 1, 3 的索引分别为 0, 1
graph.removeEdge(0, 1);
cout << "\n删除边 1-3 后，图为" << endl;
graph.print();
/* 添加顶点 */
graph.addVertex(6);
cout << "\n添加顶点 6 后，图为" << endl;
graph.print();
/* 删除顶点 */
// 顶点 3 的索引为 1
graph.removeVertex(1);
cout << "\n删除顶点 3 后，图为" << endl;
graph.print();
return 0;
}

59
chapter_graph/graph_bfs.cpp
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/**
* File: graph_bfs.cpp
* Created Time: 2023-03-02
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
#include "./graph_adjacency_list.cpp"
/* 广度优先遍历 */
// 使用邻接表来表示图，以便获取指定顶点的所有邻接顶点
vector<Vertex *> graphBFS(GraphAdjList &graph, Vertex *startVet) {
// 顶点遍历序列
vector<Vertex *> res;
// 哈希表，用于记录已被访问过的顶点
unordered_set<Vertex *> visited = {startVet};
// 队列用于实现 BFS
queue<Vertex *> que;
que.push(startVet);
// 以顶点 vet 为起点，循环直至访问完所有顶点
while (!que.empty()) {
Vertex *vet = que.front();
que.pop(); // 队首顶点出队
res.push_back(vet); // 记录访问顶点
// 遍历该顶点的所有邻接顶点
for (auto adjVet : graph.adjList[vet]) {
if (visited.count(adjVet))
continue; // 跳过已被访问的顶点
que.push(adjVet); // 只入队未访问的顶点
visited.emplace(adjVet); // 标记该顶点已被访问
}
}
// 返回顶点遍历序列
return res;
}
/* Driver Code */
int main() {
/* 初始化无向图 */
vector<Vertex *> v = valsToVets({0, 1, 2, 3, 4, 5, 6, 7, 8, 9});
vector<vector<Vertex *>> edges = {{v[0], v[1]}, {v[0], v[3]}, {v[1], v[2]}, {v[1], v[4]},
{v[2], v[5]}, {v[3], v[4]}, {v[3], v[6]}, {v[4], v[5]},
{v[4], v[7]}, {v[5], v[8]}, {v[6], v[7]}, {v[7], v[8]}};
GraphAdjList graph(edges);
cout << "\n初始化后，图为\\n";
graph.print();
/* 广度优先遍历 */
vector<Vertex *> res = graphBFS(graph, v[0]);
cout << "\n广度优先遍历（BFS）顶点序列为" << endl;
printVector(vetsToVals(res));
// 释放内存
for (Vertex *vet : v) {
delete vet;
}
return 0;
}

55
chapter_graph/graph_dfs.cpp
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/**
* File: graph_dfs.cpp
* Created Time: 2023-03-02
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
#include "./graph_adjacency_list.cpp"
/* 深度优先遍历辅助函数 */
void dfs(GraphAdjList &graph, unordered_set<Vertex *> &visited, vector<Vertex *> &res, Vertex *vet) {
res.push_back(vet); // 记录访问顶点
visited.emplace(vet); // 标记该顶点已被访问
// 遍历该顶点的所有邻接顶点
for (Vertex *adjVet : graph.adjList[vet]) {
if (visited.count(adjVet))
continue; // 跳过已被访问的顶点
// 递归访问邻接顶点
dfs(graph, visited, res, adjVet);
}
}
/* 深度优先遍历 */
// 使用邻接表来表示图，以便获取指定顶点的所有邻接顶点
vector<Vertex *> graphDFS(GraphAdjList &graph, Vertex *startVet) {
// 顶点遍历序列
vector<Vertex *> res;
// 哈希表，用于记录已被访问过的顶点
unordered_set<Vertex *> visited;
dfs(graph, visited, res, startVet);
return res;
}
/* Driver Code */
int main() {
/* 初始化无向图 */
vector<Vertex *> v = valsToVets(vector<int>{0, 1, 2, 3, 4, 5, 6});
vector<vector<Vertex *>> edges = {{v[0], v[1]}, {v[0], v[3]}, {v[1], v[2]},
{v[2], v[5]}, {v[4], v[5]}, {v[5], v[6]}};
GraphAdjList graph(edges);
cout << "\n初始化后，图为" << endl;
graph.print();
/* 深度优先遍历 */
vector<Vertex *> res = graphDFS(graph, v[0]);
cout << "\n深度优先遍历（DFS）顶点序列为" << endl;
printVector(vetsToVals(res));
// 释放内存
for (Vertex *vet : v) {
delete vet;
}
return 0;
}

3
chapter_greedy/CMakeLists.txt
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add_executable(coin_change_greedy coin_change_greedy.cpp)
add_executable(fractional_knapsack fractional_knapsack.cpp)
add_executable(max_capacity max_capacity.cpp)

60
chapter_greedy/coin_change_greedy.cpp
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/**
* File: coin_change_greedy.cpp
* Created Time: 2023-07-20
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 零钱兑换：贪心 */
int coinChangeGreedy(vector<int> &coins, int amt) {
// 假设 coins 列表有序
int i = coins.size() - 1;
int count = 0;
// 循环进行贪心选择，直到无剩余金额
while (amt > 0) {
// 找到小于且最接近剩余金额的硬币
while (i > 0 && coins[i] > amt) {
i--;
}
// 选择 coins[i]
amt -= coins[i];
count++;
}
// 若未找到可行方案，则返回 -1
return amt == 0 ? count : -1;
}
/* Driver Code */
int main() {
// 贪心：能够保证找到全局最优解
vector<int> coins = {1, 5, 10, 20, 50, 100};
int amt = 186;
int res = coinChangeGreedy(coins, amt);
cout << "\ncoins = ";
printVector(coins);
cout << "amt = " << amt << endl;
cout << "凑到 " << amt << " 所需的最少硬币数量为 " << res << endl;
// 贪心：无法保证找到全局最优解
coins = {1, 20, 50};
amt = 60;
res = coinChangeGreedy(coins, amt);
cout << "\ncoins = ";
printVector(coins);
cout << "amt = " << amt << endl;
cout << "凑到 " << amt << " 所需的最少硬币数量为 " << res << endl;
cout << "实际上需要的最少数量为 3 ，即 20 + 20 + 20" << endl;
// 贪心：无法保证找到全局最优解
coins = {1, 49, 50};
amt = 98;
res = coinChangeGreedy(coins, amt);
cout << "\ncoins = ";
printVector(coins);
cout << "amt = " << amt << endl;
cout << "凑到 " << amt << " 所需的最少硬币数量为 " << res << endl;
cout << "实际上需要的最少数量为 2 ，即 49 + 49" << endl;
return 0;
}

56
chapter_greedy/fractional_knapsack.cpp
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/**
* File: fractional_knapsack.cpp
* Created Time: 2023-07-20
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 物品 */
class Item {
public:
int w; // 物品重量
int v; // 物品价值
Item(int w, int v) : w(w), v(v) {
}
};
/* 分数背包：贪心 */
double fractionalKnapsack(vector<int> &wgt, vector<int> &val, int cap) {
// 创建物品列表，包含两个属性：重量、价值
vector<Item> items;
for (int i = 0; i < wgt.size(); i++) {
items.push_back(Item(wgt[i], val[i]));
}
// 按照单位价值 item.v / item.w 从高到低进行排序
sort(items.begin(), items.end(), [](Item &a, Item &b) { return (double)a.v / a.w > (double)b.v / b.w; });
// 循环贪心选择
double res = 0;
for (auto &item : items) {
if (item.w <= cap) {
// 若剩余容量充足，则将当前物品整个装进背包
res += item.v;
cap -= item.w;
} else {
// 若剩余容量不足，则将当前物品的一部分装进背包
res += (double)item.v / item.w * cap;
// 已无剩余容量，因此跳出循环
break;
}
}
return res;
}
/* Driver Code */
int main() {
vector<int> wgt = {10, 20, 30, 40, 50};
vector<int> val = {50, 120, 150, 210, 240};
int cap = 50;
// 贪心算法
double res = fractionalKnapsack(wgt, val, cap);
cout << "不超过背包容量的最大物品价值为 " << res << endl;
return 0;
}

39
chapter_greedy/max_capacity.cpp
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/**
* File: max_capacity.cpp
* Created Time: 2023-07-21
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 最大容量：贪心 */
int maxCapacity(vector<int> &ht) {
// 初始化 i, j，使其分列数组两端
int i = 0, j = ht.size() - 1;
// 初始最大容量为 0
int res = 0;
// 循环贪心选择，直至两板相遇
while (i < j) {
// 更新最大容量
int cap = min(ht[i], ht[j]) * (j - i);
res = max(res, cap);
// 向内移动短板
if (ht[i] < ht[j]) {
i++;
} else {
j--;
}
}
return res;
}
/* Driver Code */
int main() {
vector<int> ht = {3, 8, 5, 2, 7, 7, 3, 4};
// 贪心算法
int res = maxCapacity(ht);
cout << "最大容量为 " << res << endl;
return 0;
}

39
chapter_greedy/max_product_cutting.cpp
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/**
* File: max_product_cutting.cpp
* Created Time: 2023-07-21
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 最大切分乘积：贪心 */
int maxProductCutting(int n) {
// 当 n <= 3 时，必须切分出一个 1
if (n <= 3) {
return 1 * (n - 1);
}
// 贪心地切分出 3 ，a 为 3 的个数，b 为余数
int a = n / 3;
int b = n % 3;
if (b == 1) {
// 当余数为 1 时，将一对 1 * 3 转化为 2 * 2
return (int)pow(3, a - 1) * 2 * 2;
}
if (b == 2) {
// 当余数为 2 时，不做处理
return (int)pow(3, a) * 2;
}
// 当余数为 0 时，不做处理
return (int)pow(3, a);
}
/* Driver Code */
int main() {
int n = 58;
// 贪心算法
int res = maxProductCutting(n);
cout << "最大切分乘积为" << res << endl;
return 0;
}

6
chapter_hashing/CMakeLists.txt
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add_executable(hash_map hash_map.cpp)
add_executable(array_hash_map_test array_hash_map_test.cpp)
add_executable(hash_map_chaining hash_map_chaining.cpp)
add_executable(hash_map_open_addressing hash_map_open_addressing.cpp)
add_executable(simple_hash simple_hash.cpp)
add_executable(built_in_hash built_in_hash.cpp)

110
chapter_hashing/array_hash_map.cpp
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/**
* File: array_hash_map.cpp
* Created Time: 2022-12-14
* Author: msk397 (machangxinq@gmail.com)
*/
#include "../utils/common.hpp"
/* 键值对 */
struct Pair {
public:
int key;
string val;
Pair(int key, string val) {
this->key = key;
this->val = val;
}
};
/* 基于数组实现的哈希表 */
class ArrayHashMap {
private:
vector<Pair *> buckets;
public:
ArrayHashMap() {
// 初始化数组，包含 100 个桶
buckets = vector<Pair *>(100);
}
~ArrayHashMap() {
// 释放内存
for (const auto &bucket : buckets) {
delete bucket;
}
buckets.clear();
}
/* 哈希函数 */
int hashFunc(int key) {
int index = key % 100;
return index;
}
/* 查询操作 */
string get(int key) {
int index = hashFunc(key);
Pair *pair = buckets[index];
if (pair == nullptr)
return "";
return pair->val;
}
/* 添加操作 */
void put(int key, string val) {
Pair *pair = new Pair(key, val);
int index = hashFunc(key);
buckets[index] = pair;
}
/* 删除操作 */
void remove(int key) {
int index = hashFunc(key);
// 释放内存并置为 nullptr
delete buckets[index];
buckets[index] = nullptr;
}
/* 获取所有键值对 */
vector<Pair *> pairSet() {
vector<Pair *> pairSet;
for (Pair *pair : buckets) {
if (pair != nullptr) {
pairSet.push_back(pair);
}
}
return pairSet;
}
/* 获取所有键 */
vector<int> keySet() {
vector<int> keySet;
for (Pair *pair : buckets) {
if (pair != nullptr) {
keySet.push_back(pair->key);
}
}
return keySet;
}
/* 获取所有值 */
vector<string> valueSet() {
vector<string> valueSet;
for (Pair *pair : buckets) {
if (pair != nullptr) {
valueSet.push_back(pair->val);
}
}
return valueSet;
}
/* 打印哈希表 */
void print() {
for (Pair *kv : pairSet()) {
cout << kv->key << " -> " << kv->val << endl;
}
}
};
// 测试样例请见 array_hash_map_test.cpp

52
chapter_hashing/array_hash_map_test.cpp
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/**
* File: array_hash_map_test.cpp
* Created Time: 2022-12-14
* Author: msk397 (machangxinq@gmail.com)
*/
#include "./array_hash_map.cpp"
/* Driver Code */
int main() {
/* 初始化哈希表 */
ArrayHashMap map = ArrayHashMap();
/* 添加操作 */
// 在哈希表中添加键值对 (key, value)
map.put(12836, "小哈");
map.put(15937, "小啰");
map.put(16750, "小算");
map.put(13276, "小法");
map.put(10583, "小鸭");
cout << "\n添加完成后，哈希表为\nKey -> Value" << endl;
map.print();
/* 查询操作 */
// 向哈希表中输入键 key ，得到值 value
string name = map.get(15937);
cout << "\n输入学号 15937 ，查询到姓名 " << name << endl;
/* 删除操作 */
// 在哈希表中删除键值对 (key, value)
map.remove(10583);
cout << "\n删除 10583 后，哈希表为\nKey -> Value" << endl;
map.print();
/* 遍历哈希表 */
cout << "\n遍历键值对 Key->Value" << endl;
for (auto kv : map.pairSet()) {
cout << kv->key << " -> " << kv->val << endl;
}
cout << "\n单独遍历键 Key" << endl;
for (auto key : map.keySet()) {
cout << key << endl;
}
cout << "\n单独遍历值 Value" << endl;
for (auto val : map.valueSet()) {
cout << val << endl;
}
return 0;
}

29
chapter_hashing/built_in_hash.cpp
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/**
* File: built_in_hash.cpp
* Created Time: 2023-06-21
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* Driver Code */
int main() {
int num = 3;
size_t hashNum = hash<int>()(num);
cout << "整数 " << num << " 的哈希值为 " << hashNum << "\n";
bool bol = true;
size_t hashBol = hash<bool>()(bol);
cout << "布尔量 " << bol << " 的哈希值为 " << hashBol << "\n";
double dec = 3.14159;
size_t hashDec = hash<double>()(dec);
cout << "小数 " << dec << " 的哈希值为 " << hashDec << "\n";
string str = "Hello 算法";
size_t hashStr = hash<string>()(str);
cout << "字符串 " << str << " 的哈希值为 " << hashStr << "\n";
// 在 C++ 中，内置 std:hash() 仅提供基本数据类型的哈希值计算
// 数组、对象的哈希值计算需要自行实现
}

46
chapter_hashing/hash_map.cpp
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/**
* File: hash_map.cpp
* Created Time: 2022-12-14
* Author: msk397 (machangxinq@gmail.com)
*/
#include "../utils/common.hpp"
/* Driver Code */
int main() {
/* 初始化哈希表 */
unordered_map<int, string> map;
/* 添加操作 */
// 在哈希表中添加键值对 (key, value)
map[12836] = "小哈";
map[15937] = "小啰";
map[16750] = "小算";
map[13276] = "小法";
map[10583] = "小鸭";
cout << "\n添加完成后，哈希表为\nKey -> Value" << endl;
printHashMap(map);
/* 查询操作 */
// 向哈希表中输入键 key ，得到值 value
string name = map[15937];
cout << "\n输入学号 15937 ，查询到姓名 " << name << endl;
/* 删除操作 */
// 在哈希表中删除键值对 (key, value)
map.erase(10583);
cout << "\n删除 10583 后，哈希表为\nKey -> Value" << endl;
printHashMap(map);
/* 遍历哈希表 */
cout << "\n遍历键值对 Key->Value" << endl;
for (auto kv : map) {
cout << kv.first << " -> " << kv.second << endl;
}
cout << "\n使用迭代器遍历 Key->Value" << endl;
for (auto iter = map.begin(); iter != map.end(); iter++) {
cout << iter->first << "->" << iter->second << endl;
}
return 0;
}

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chapter_hashing/hash_map_chaining.cpp
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/**
* File: hash_map_chaining.cpp
* Created Time: 2023-06-13
* Author: krahets (krahets@163.com)
*/
#include "./array_hash_map.cpp"
/* 链式地址哈希表 */
class HashMapChaining {
private:
int size; // 键值对数量
int capacity; // 哈希表容量
double loadThres; // 触发扩容的负载因子阈值
int extendRatio; // 扩容倍数
vector<vector<Pair *>> buckets; // 桶数组
public:
/* 构造方法 */
HashMapChaining() : size(0), capacity(4), loadThres(2.0 / 3.0), extendRatio(2) {
buckets.resize(capacity);
}
/* 析构方法 */
~HashMapChaining() {
for (auto &bucket : buckets) {
for (Pair *pair : bucket) {
// 释放内存
delete pair;
}
}
}
/* 哈希函数 */
int hashFunc(int key) {
return key % capacity;
}
/* 负载因子 */
double loadFactor() {
return (double)size / (double)capacity;
}
/* 查询操作 */
string get(int key) {
int index = hashFunc(key);
// 遍历桶，若找到 key ，则返回对应 val
for (Pair *pair : buckets[index]) {
if (pair->key == key) {
return pair->val;
}
}
// 若未找到 key ，则返回空字符串
return "";
}
/* 添加操作 */
void put(int key, string val) {
// 当负载因子超过阈值时，执行扩容
if (loadFactor() > loadThres) {
extend();
}
int index = hashFunc(key);
// 遍历桶，若遇到指定 key ，则更新对应 val 并返回
for (Pair *pair : buckets[index]) {
if (pair->key == key) {
pair->val = val;
return;
}
}
// 若无该 key ，则将键值对添加至尾部
buckets[index].push_back(new Pair(key, val));
size++;
}
/* 删除操作 */
void remove(int key) {
int index = hashFunc(key);
auto &bucket = buckets[index];
// 遍历桶，从中删除键值对
for (int i = 0; i < bucket.size(); i++) {
if (bucket[i]->key == key) {
Pair *tmp = bucket[i];
bucket.erase(bucket.begin() + i); // 从中删除键值对
delete tmp; // 释放内存
size--;
return;
}
}
}
/* 扩容哈希表 */
void extend() {
// 暂存原哈希表
vector<vector<Pair *>> bucketsTmp = buckets;
// 初始化扩容后的新哈希表
capacity *= extendRatio;
buckets.clear();
buckets.resize(capacity);
size = 0;
// 将键值对从原哈希表搬运至新哈希表
for (auto &bucket : bucketsTmp) {
for (Pair *pair : bucket) {
put(pair->key, pair->val);
// 释放内存
delete pair;
}
}
}
/* 打印哈希表 */
void print() {
for (auto &bucket : buckets) {
cout << "[";
for (Pair *pair : bucket) {
cout << pair->key << " -> " << pair->val << ", ";
}
cout << "]\n";
}
}
};
/* Driver Code */
int main() {
/* 初始化哈希表 */
HashMapChaining map = HashMapChaining();
/* 添加操作 */
// 在哈希表中添加键值对 (key, value)
map.put(12836, "小哈");
map.put(15937, "小啰");
map.put(16750, "小算");
map.put(13276, "小法");
map.put(10583, "小鸭");
cout << "\n添加完成后，哈希表为\nKey -> Value" << endl;
map.print();
/* 查询操作 */
// 向哈希表中输入键 key ，得到值 value
string name = map.get(13276);
cout << "\n输入学号 13276 ，查询到姓名 " << name << endl;
/* 删除操作 */
// 在哈希表中删除键值对 (key, value)
map.remove(12836);
cout << "\n删除 12836 后，哈希表为\nKey -> Value" << endl;
map.print();
return 0;
}

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chapter_hashing/hash_map_open_addressing.cpp
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/**
* File: hash_map_open_addressing.cpp
* Created Time: 2023-06-13
* Author: krahets (krahets@163.com)
*/
#include "./array_hash_map.cpp"
/* 开放寻址哈希表 */
class HashMapOpenAddressing {
private:
int size; // 键值对数量
int capacity = 4; // 哈希表容量
const double loadThres = 2.0 / 3.0; // 触发扩容的负载因子阈值
const int extendRatio = 2; // 扩容倍数
vector<Pair *> buckets; // 桶数组
Pair *TOMBSTONE = new Pair(-1, "-1"); // 删除标记
public:
/* 构造方法 */
HashMapOpenAddressing() : size(0), buckets(capacity, nullptr) {
}
/* 析构方法 */
~HashMapOpenAddressing() {
for (Pair *pair : buckets) {
if (pair != nullptr && pair != TOMBSTONE) {
delete pair;
}
}
delete TOMBSTONE;
}
/* 哈希函数 */
int hashFunc(int key) {
return key % capacity;
}
/* 负载因子 */
double loadFactor() {
return (double)size / capacity;
}
/* 搜索 key 对应的桶索引 */
int findBucket(int key) {
int index = hashFunc(key);
int firstTombstone = -1;
// 线性探测，当遇到空桶时跳出
while (buckets[index] != nullptr) {
// 若遇到 key ，返回对应的桶索引
if (buckets[index]->key == key) {
// 若之前遇到了删除标记，则将键值对移动至该索引处
if (firstTombstone != -1) {
buckets[firstTombstone] = buckets[index];
buckets[index] = TOMBSTONE;
return firstTombstone; // 返回移动后的桶索引
}
return index; // 返回桶索引
}
// 记录遇到的首个删除标记
if (firstTombstone == -1 && buckets[index] == TOMBSTONE) {
firstTombstone = index;
}
// 计算桶索引，越过尾部则返回头部
index = (index + 1) % capacity;
}
// 若 key 不存在，则返回添加点的索引
return firstTombstone == -1 ? index : firstTombstone;
}
/* 查询操作 */
string get(int key) {
// 搜索 key 对应的桶索引
int index = findBucket(key);
// 若找到键值对，则返回对应 val
if (buckets[index] != nullptr && buckets[index] != TOMBSTONE) {
return buckets[index]->val;
}
// 若键值对不存在，则返回空字符串
return "";
}
/* 添加操作 */
void put(int key, string val) {
// 当负载因子超过阈值时，执行扩容
if (loadFactor() > loadThres) {
extend();
}
// 搜索 key 对应的桶索引
int index = findBucket(key);
// 若找到键值对，则覆盖 val 并返回
if (buckets[index] != nullptr && buckets[index] != TOMBSTONE) {
buckets[index]->val = val;
return;
}
// 若键值对不存在，则添加该键值对
buckets[index] = new Pair(key, val);
size++;
}
/* 删除操作 */
void remove(int key) {
// 搜索 key 对应的桶索引
int index = findBucket(key);
// 若找到键值对，则用删除标记覆盖它
if (buckets[index] != nullptr && buckets[index] != TOMBSTONE) {
delete buckets[index];
buckets[index] = TOMBSTONE;
size--;
}
}
/* 扩容哈希表 */
void extend() {
// 暂存原哈希表
vector<Pair *> bucketsTmp = buckets;
// 初始化扩容后的新哈希表
capacity *= extendRatio;
buckets = vector<Pair *>(capacity, nullptr);
size = 0;
// 将键值对从原哈希表搬运至新哈希表
for (Pair *pair : bucketsTmp) {
if (pair != nullptr && pair != TOMBSTONE) {
put(pair->key, pair->val);
delete pair;
}
}
}
/* 打印哈希表 */
void print() {
for (Pair *pair : buckets) {
if (pair == nullptr) {
cout << "nullptr" << endl;
} else if (pair == TOMBSTONE) {
cout << "TOMBSTONE" << endl;
} else {
cout << pair->key << " -> " << pair->val << endl;
}
}
}
};
/* Driver Code */
int main() {
// 初始化哈希表
HashMapOpenAddressing hashmap;
// 添加操作
// 在哈希表中添加键值对 (key, val)
hashmap.put(12836, "小哈");
hashmap.put(15937, "小啰");
hashmap.put(16750, "小算");
hashmap.put(13276, "小法");
hashmap.put(10583, "小鸭");
cout << "\n添加完成后，哈希表为\nKey -> Value" << endl;
hashmap.print();
// 查询操作
// 向哈希表中输入键 key ，得到值 val
string name = hashmap.get(13276);
cout << "\n输入学号 13276 ，查询到姓名 " << name << endl;
// 删除操作
// 在哈希表中删除键值对 (key, val)
hashmap.remove(16750);
cout << "\n删除 16750 后，哈希表为\nKey -> Value" << endl;
hashmap.print();
return 0;
}

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chapter_hashing/simple_hash.cpp
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/**
* File: simple_hash.cpp
* Created Time: 2023-06-21
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 加法哈希 */
int addHash(string key) {
long long hash = 0;
const int MODULUS = 1000000007;
for (unsigned char c : key) {
hash = (hash + (int)c) % MODULUS;
}
return (int)hash;
}
/* 乘法哈希 */
int mulHash(string key) {
long long hash = 0;
const int MODULUS = 1000000007;
for (unsigned char c : key) {
hash = (31 * hash + (int)c) % MODULUS;
}
return (int)hash;
}
/* 异或哈希 */
int xorHash(string key) {
int hash = 0;
const int MODULUS = 1000000007;
for (unsigned char c : key) {
hash ^= (int)c;
}
return hash & MODULUS;
}
/* 旋转哈希 */
int rotHash(string key) {
long long hash = 0;
const int MODULUS = 1000000007;
for (unsigned char c : key) {
hash = ((hash << 4) ^ (hash >> 28) ^ (int)c) % MODULUS;
}
return (int)hash;
}
/* Driver Code */
int main() {
string key = "Hello dsad3241241dsa算123法";
int hash = addHash(key);
cout << "加法哈希值为 " << hash << endl;
hash = mulHash(key);
cout << "乘法哈希值为 " << hash << endl;
hash = xorHash(key);
cout << "异或哈希值为 " << hash << endl;
hash = rotHash(key);
cout << "旋转哈希值为 " << hash << endl;
return 0;
}

3
chapter_heap/CMakeLists.txt
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add_executable(heap heap.cpp)
add_executable(my_heap my_heap.cpp)
add_executable(top_k top_k.cpp)

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chapter_heap/heap.cpp
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/**
* File: heap.cpp
* Created Time: 2023-01-19
* Author: LoneRanger(836253168@qq.com)
*/
#include "../utils/common.hpp"
void testPush(priority_queue<int> &heap, int val) {
heap.push(val); // 元素入堆
cout << "\n元素 " << val << " 入堆后" << endl;
printHeap(heap);
}
void testPop(priority_queue<int> &heap) {
int val = heap.top();
heap.pop();
cout << "\n堆顶元素 " << val << " 出堆后" << endl;
printHeap(heap);
}
/* Driver Code */
int main() {
/* 初始化堆 */
// 初始化小顶堆
// priority_queue<int, vector<int>, greater<int>> minHeap;
// 初始化大顶堆
priority_queue<int, vector<int>, less<int>> maxHeap;
cout << "\n以下测试样例为大顶堆" << endl;
/* 元素入堆 */
testPush(maxHeap, 1);
testPush(maxHeap, 3);
testPush(maxHeap, 2);
testPush(maxHeap, 5);
testPush(maxHeap, 4);
/* 获取堆顶元素 */
int peek = maxHeap.top();
cout << "\n堆顶元素为 " << peek << endl;
/* 堆顶元素出堆 */
testPop(maxHeap);
testPop(maxHeap);
testPop(maxHeap);
testPop(maxHeap);
testPop(maxHeap);
/* 获取堆大小 */
int size = maxHeap.size();
cout << "\n堆元素数量为 " << size << endl;
/* 判断堆是否为空 */
bool isEmpty = maxHeap.empty();
cout << "\n堆是否为空 " << isEmpty << endl;
/* 输入列表并建堆 */
// 时间复杂度为 O(n) ，而非 O(nlogn)
vector<int> input{1, 3, 2, 5, 4};
priority_queue<int, vector<int>, greater<int>> minHeap(input.begin(), input.end());
cout << "输入列表并建立小顶堆后" << endl;
printHeap(minHeap);
return 0;
}

155
chapter_heap/my_heap.cpp
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/**
* File: my_heap.cpp
* Created Time: 2023-02-04
* Author: LoneRanger (836253168@qq.com), what-is-me (whatisme@outlook.jp)
*/
#include "../utils/common.hpp"
/* 大顶堆 */
class MaxHeap {
private:
// 使用动态数组，这样无须考虑扩容问题
vector<int> maxHeap;
/* 获取左子节点的索引 */
int left(int i) {
return 2 * i + 1;
}
/* 获取右子节点的索引 */
int right(int i) {
return 2 * i + 2;
}
/* 获取父节点的索引 */
int parent(int i) {
return (i - 1) / 2; // 向下整除
}
/* 从节点 i 开始，从底至顶堆化 */
void siftUp(int i) {
while (true) {
// 获取节点 i 的父节点
int p = parent(i);
// 当“越过根节点”或“节点无须修复”时，结束堆化
if (p < 0 || maxHeap[i] <= maxHeap[p])
break;
// 交换两节点
swap(maxHeap[i], maxHeap[p]);
// 循环向上堆化
i = p;
}
}
/* 从节点 i 开始，从顶至底堆化 */
void siftDown(int i) {
while (true) {
// 判断节点 i, l, r 中值最大的节点，记为 ma
int l = left(i), r = right(i), ma = i;
if (l < size() && maxHeap[l] > maxHeap[ma])
ma = l;
if (r < size() && maxHeap[r] > maxHeap[ma])
ma = r;
// 若节点 i 最大或索引 l, r 越界，则无须继续堆化，跳出
if (ma == i)
break;
swap(maxHeap[i], maxHeap[ma]);
// 循环向下堆化
i = ma;
}
}
public:
/* 构造方法，根据输入列表建堆 */
MaxHeap(vector<int> nums) {
// 将列表元素原封不动添加进堆
maxHeap = nums;
// 堆化除叶节点以外的其他所有节点
for (int i = parent(size() - 1); i >= 0; i--) {
siftDown(i);
}
}
/* 获取堆大小 */
int size() {
return maxHeap.size();
}
/* 判断堆是否为空 */
bool isEmpty() {
return size() == 0;
}
/* 访问堆顶元素 */
int peek() {
return maxHeap[0];
}
/* 元素入堆 */
void push(int val) {
// 添加节点
maxHeap.push_back(val);
// 从底至顶堆化
siftUp(size() - 1);
}
/* 元素出堆 */
void pop() {
// 判空处理
if (isEmpty()) {
throw out_of_range("堆为空");
}
// 交换根节点与最右叶节点（交换首元素与尾元素）
swap(maxHeap[0], maxHeap[size() - 1]);
// 删除节点
maxHeap.pop_back();
// 从顶至底堆化
siftDown(0);
}
/* 打印堆（二叉树）*/
void print() {
cout << "堆的数组表示：";
printVector(maxHeap);
cout << "堆的树状表示：" << endl;
TreeNode *root = vectorToTree(maxHeap);
printTree(root);
freeMemoryTree(root);
}
};
/* Driver Code */
int main() {
/* 初始化大顶堆 */
vector<int> vec{9, 8, 6, 6, 7, 5, 2, 1, 4, 3, 6, 2};
MaxHeap maxHeap(vec);
cout << "\n输入列表并建堆后" << endl;
maxHeap.print();
/* 获取堆顶元素 */
int peek = maxHeap.peek();
cout << "\n堆顶元素为 " << peek << endl;
/* 元素入堆 */
int val = 7;
maxHeap.push(val);
cout << "\n元素 " << val << " 入堆后" << endl;
maxHeap.print();
/* 堆顶元素出堆 */
peek = maxHeap.peek();
maxHeap.pop();
cout << "\n堆顶元素 " << peek << " 出堆后" << endl;
maxHeap.print();
/* 获取堆大小 */
int size = maxHeap.size();
cout << "\n堆元素数量为 " << size << endl;
/* 判断堆是否为空 */
bool isEmpty = maxHeap.isEmpty();
cout << "\n堆是否为空 " << isEmpty << endl;
return 0;
}

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chapter_heap/top_k.cpp
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/**
* File: top_k.cpp
* Created Time: 2023-06-12
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 基于堆查找数组中最大的 k 个元素 */
priority_queue<int, vector<int>, greater<int>> topKHeap(vector<int> &nums, int k) {
// 初始化小顶堆
priority_queue<int, vector<int>, greater<int>> heap;
// 将数组的前 k 个元素入堆
for (int i = 0; i < k; i++) {
heap.push(nums[i]);
}
// 从第 k+1 个元素开始，保持堆的长度为 k
for (int i = k; i < nums.size(); i++) {
// 若当前元素大于堆顶元素，则将堆顶元素出堆、当前元素入堆
if (nums[i] > heap.top()) {
heap.pop();
heap.push(nums[i]);
}
}
return heap;
}
// Driver Code
int main() {
vector<int> nums = {1, 7, 6, 3, 2};
int k = 3;
priority_queue<int, vector<int>, greater<int>> res = topKHeap(nums, k);
cout << "最大的 " << k << " 个元素为: ";
printHeap(res);
return 0;
}

4
chapter_searching/CMakeLists.txt
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add_executable(binary_search binary_search.cpp)
add_executable(binary_search_insertion binary_search_insertion.cpp)
add_executable(binary_search_edge binary_search_edge.cpp)
add_executable(two_sum two_sum.cpp)

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chapter_searching/binary_search.cpp
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/**
* File: binary_search.cpp
* Created Time: 2022-11-25
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 二分查找（双闭区间） */
int binarySearch(vector<int> &nums, int target) {
// 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
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 - 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
return m;
}
// 未找到目标元素，返回 -1
return -1;
}
/* 二分查找（左闭右开区间） */
int binarySearchLCRO(vector<int> &nums, int target) {
// 初始化左闭右开区间 [0, n) ，即 i, j 分别指向数组首元素、尾元素+1
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 - 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
return m;
}
// 未找到目标元素，返回 -1
return -1;
}
/* Driver Code */
int main() {
int target = 6;
vector<int> nums = {1, 3, 6, 8, 12, 15, 23, 26, 31, 35};
/* 二分查找（双闭区间） */
int index = binarySearch(nums, target);
cout << "目标元素 6 的索引 = " << index << endl;
/* 二分查找（左闭右开区间） */
index = binarySearchLCRO(nums, target);
cout << "目标元素 6 的索引 = " << index << endl;
return 0;
}

66
chapter_searching/binary_search_edge.cpp
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/**
* File: binary_search_edge.cpp
* Created Time: 2023-08-04
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 二分查找插入点（存在重复元素） */
int binarySearchInsertion(const vector<int> &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 {
j = m - 1; // 首个小于 target 的元素在区间 [i, m-1] 中
}
}
// 返回插入点 i
return i;
}
/* 二分查找最左一个 target */
int binarySearchLeftEdge(vector<int> &nums, int target) {
// 等价于查找 target 的插入点
int i = binarySearchInsertion(nums, target);
// 未找到 target ，返回 -1
if (i == nums.size() || nums[i] != target) {
return -1;
}
// 找到 target ，返回索引 i
return i;
}
/* 二分查找最右一个 target */
int binarySearchRightEdge(vector<int> &nums, int target) {
// 转化为查找最左一个 target + 1
int i = binarySearchInsertion(nums, target + 1);
// j 指向最右一个 target ，i 指向首个大于 target 的元素
int j = i - 1;
// 未找到 target ，返回 -1
if (j == -1 || nums[j] != target) {
return -1;
}
// 找到 target ，返回索引 j
return j;
}
/* Driver Code */
int main() {
// 包含重复元素的数组
vector<int> nums = {1, 3, 6, 6, 6, 6, 6, 10, 12, 15};
cout << "\n数组 nums = ";
printVector(nums);
// 二分查找左边界和右边界
for (int target : {6, 7}) {
int index = binarySearchLeftEdge(nums, target);
cout << "最左一个元素 " << target << " 的索引为 " << index << endl;
index = binarySearchRightEdge(nums, target);
cout << "最右一个元素 " << target << " 的索引为 " << index << endl;
}
return 0;
}

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chapter_searching/binary_search_insertion.cpp
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/**
* File: binary_search_insertion.cpp
* Created Time: 2023-08-04
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 二分查找插入点（无重复元素） */
int binarySearchInsertionSimple(vector<int> &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 {
return m; // 找到 target ，返回插入点 m
}
}
// 未找到 target ，返回插入点 i
return i;
}
/* 二分查找插入点（存在重复元素） */
int binarySearchInsertion(vector<int> &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
}
}
// 返回插入点 i
return i;
}
/* Driver Code */
int main() {
// 无重复元素的数组
vector<int> nums = {1, 3, 6, 8, 12, 15, 23, 26, 31, 35};
cout << "\n数组 nums = ";
printVector(nums);
// 二分查找插入点
for (int target : {6, 9}) {
int index = binarySearchInsertionSimple(nums, target);
cout << "元素 " << target << " 的插入点的索引为 " << index << endl;
}
// 包含重复元素的数组
nums = {1, 3, 6, 6, 6, 6, 6, 10, 12, 15};
cout << "\n数组 nums = ";
printVector(nums);
// 二分查找插入点
for (int target : {2, 6, 20}) {
int index = binarySearchInsertion(nums, target);
cout << "元素 " << target << " 的插入点的索引为 " << index << endl;
}
return 0;
}

53
chapter_searching/hashing_search.cpp
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/**
* File: hashing_search.cpp
* Created Time: 2022-11-25
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 哈希查找（数组） */
int hashingSearchArray(unordered_map<int, int> map, int target) {
// 哈希表的 key: 目标元素，value: 索引
// 若哈希表中无此 key ，返回 -1
if (map.find(target) == map.end())
return -1;
return map[target];
}
/* 哈希查找（链表） */
ListNode *hashingSearchLinkedList(unordered_map<int, ListNode *> map, int target) {
// 哈希表的 key: 目标节点值，value: 节点对象
// 若哈希表中无此 key ，返回 nullptr
if (map.find(target) == map.end())
return nullptr;
return map[target];
}
/* Driver Code */
int main() {
int target = 3;
/* 哈希查找（数组） */
vector<int> nums = {1, 5, 3, 2, 4, 7, 5, 9, 10, 8};
// 初始化哈希表
unordered_map<int, int> map;
for (int i = 0; i < nums.size(); i++) {
map[nums[i]] = i; // key: 元素，value: 索引
}
int index = hashingSearchArray(map, target);
cout << "目标元素 3 的索引 = " << index << endl;
/* 哈希查找（链表） */
ListNode *head = vecToLinkedList(nums);
// 初始化哈希表
unordered_map<int, ListNode *> map1;
while (head != nullptr) {
map1[head->val] = head; // key: 节点值，value: 节点
head = head->next;
}
ListNode *node = hashingSearchLinkedList(map1, target);
cout << "目标节点值 3 的对应节点对象为 " << node << endl;
return 0;
}

49
chapter_searching/linear_search.cpp
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/**
* File: linear_search.cpp
* Created Time: 2022-11-25
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 线性查找（数组） */
int linearSearchArray(vector<int> &nums, int target) {
// 遍历数组
for (int i = 0; i < nums.size(); i++) {
// 找到目标元素，返回其索引
if (nums[i] == target)
return i;
}
// 未找到目标元素，返回 -1
return -1;
}
/* 线性查找（链表） */
ListNode *linearSearchLinkedList(ListNode *head, int target) {
// 遍历链表
while (head != nullptr) {
// 找到目标节点，返回之
if (head->val == target)
return head;
head = head->next;
}
// 未找到目标节点，返回 nullptr
return nullptr;
}
/* Driver Code */
int main() {
int target = 3;
/* 在数组中执行线性查找 */
vector<int> nums = {1, 5, 3, 2, 4, 7, 5, 9, 10, 8};
int index = linearSearchArray(nums, target);
cout << "目标元素 3 的索引 = " << index << endl;
/* 在链表中执行线性查找 */
ListNode *head = vecToLinkedList(nums);
ListNode *node = linearSearchLinkedList(head, target);
cout << "目标节点值 3 的对应节点对象为 " << node << endl;
return 0;
}

54
chapter_searching/two_sum.cpp
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/**
* File: two_sum.cpp
* Created Time: 2022-11-25
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 方法一：暴力枚举 */
vector<int> twoSumBruteForce(vector<int> &nums, int target) {
int size = nums.size();
// 两层循环，时间复杂度为 O(n^2)
for (int i = 0; i < size - 1; i++) {
for (int j = i + 1; j < size; j++) {
if (nums[i] + nums[j] == target)
return {i, j};
}
}
return {};
}
/* 方法二：辅助哈希表 */
vector<int> twoSumHashTable(vector<int> &nums, int target) {
int size = nums.size();
// 辅助哈希表，空间复杂度为 O(n)
unordered_map<int, int> dic;
// 单层循环，时间复杂度为 O(n)
for (int i = 0; i < size; i++) {
if (dic.find(target - nums[i]) != dic.end()) {
return {dic[target - nums[i]], i};
}
dic.emplace(nums[i], i);
}
return {};
}
/* Driver Code */
int main() {
// ======= Test Case =======
vector<int> nums = {2, 7, 11, 15};
int target = 13;
// ====== Driver Code ======
// 方法一
vector<int> res = twoSumBruteForce(nums, target);
cout << "方法一 res = ";
printVector(res);
// 方法二
res = twoSumHashTable(nums, target);
cout << "方法二 res = ";
printVector(res);
return 0;
}

6
chapter_sorting/CMakeLists.txt
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add_executable(selection_sort selection_sort.cpp)
add_executable(bubble_sort bubble_sort.cpp)
add_executable(insertion_sort insertion_sort.cpp)
add_executable(merge_sort merge_sort.cpp)
add_executable(quick_sort quick_sort.cpp)
add_executable(heap_sort heap_sort.cpp)

56
chapter_sorting/bubble_sort.cpp
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/**
* File: bubble_sort.cpp
* Created Time: 2022-11-25
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 冒泡排序 */
void bubbleSort(vector<int> &nums) {
// 外循环：未排序区间为 [0, i]
for (int i = nums.size() - 1; i > 0; i--) {
// 内循环：将未排序区间 [0, i] 中的最大元素交换至该区间的最右端
for (int j = 0; j < i; j++) {
if (nums[j] > nums[j + 1]) {
// 交换 nums[j] 与 nums[j + 1]
// 这里使用了 std::swap() 函数
swap(nums[j], nums[j + 1]);
}
}
}
}
/* 冒泡排序（标志优化）*/
void bubbleSortWithFlag(vector<int> &nums) {
// 外循环：未排序区间为 [0, i]
for (int i = nums.size() - 1; i > 0; i--) {
bool flag = false; // 初始化标志位
// 内循环：将未排序区间 [0, i] 中的最大元素交换至该区间的最右端
for (int j = 0; j < i; j++) {
if (nums[j] > nums[j + 1]) {
// 交换 nums[j] 与 nums[j + 1]
// 这里使用了 std::swap() 函数
swap(nums[j], nums[j + 1]);
flag = true; // 记录交换元素
}
}
if (!flag)
break; // 此轮“冒泡”未交换任何元素，直接跳出
}
}
/* Driver Code */
int main() {
vector<int> nums = {4, 1, 3, 1, 5, 2};
bubbleSort(nums);
cout << "冒泡排序完成后 nums = ";
printVector(nums);
vector<int> nums1 = {4, 1, 3, 1, 5, 2};
bubbleSortWithFlag(nums1);
cout << "冒泡排序完成后 nums1 = ";
printVector(nums1);
return 0;
}

44
chapter_sorting/bucket_sort.cpp
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/**
* File: bucket_sort.cpp
* Created Time: 2023-03-30
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 桶排序 */
void bucketSort(vector<float> &nums) {
// 初始化 k = n/2 个桶，预期向每个桶分配 2 个元素
int k = nums.size() / 2;
vector<vector<float>> buckets(k);
// 1. 将数组元素分配到各个桶中
for (float num : nums) {
// 输入数据范围为 [0, 1)，使用 num * k 映射到索引范围 [0, k-1]
int i = num * k;
// 将 num 添加进桶 bucket_idx
buckets[i].push_back(num);
}
// 2. 对各个桶执行排序
for (vector<float> &bucket : buckets) {
// 使用内置排序函数，也可以替换成其他排序算法
sort(bucket.begin(), bucket.end());
}
// 3. 遍历桶合并结果
int i = 0;
for (vector<float> &bucket : buckets) {
for (float num : bucket) {
nums[i++] = num;
}
}
}
/* Driver Code */
int main() {
// 设输入数据为浮点数，范围为 [0, 1)
vector<float> nums = {0.49f, 0.96f, 0.82f, 0.09f, 0.57f, 0.43f, 0.91f, 0.75f, 0.15f, 0.37f};
bucketSort(nums);
cout << "桶排序完成后 nums = ";
printVector(nums);
return 0;
}

77
chapter_sorting/counting_sort.cpp
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/**
* File: counting_sort.cpp
* Created Time: 2023-03-17
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 计数排序 */
// 简单实现，无法用于排序对象
void countingSortNaive(vector<int> &nums) {
// 1. 统计数组最大元素 m
int m = 0;
for (int num : nums) {
m = max(m, num);
}
// 2. 统计各数字的出现次数
// counter[num] 代表 num 的出现次数
vector<int> counter(m + 1, 0);
for (int num : nums) {
counter[num]++;
}
// 3. 遍历 counter ，将各元素填入原数组 nums
int i = 0;
for (int num = 0; num < m + 1; num++) {
for (int j = 0; j < counter[num]; j++, i++) {
nums[i] = num;
}
}
}
/* 计数排序 */
// 完整实现，可排序对象，并且是稳定排序
void countingSort(vector<int> &nums) {
// 1. 统计数组最大元素 m
int m = 0;
for (int num : nums) {
m = max(m, num);
}
// 2. 统计各数字的出现次数
// counter[num] 代表 num 的出现次数
vector<int> counter(m + 1, 0);
for (int num : nums) {
counter[num]++;
}
// 3. 求 counter 的前缀和，将“出现次数”转换为“尾索引”
// 即 counter[num]-1 是 num 在 res 中最后一次出现的索引
for (int i = 0; i < m; i++) {
counter[i + 1] += counter[i];
}
// 4. 倒序遍历 nums ，将各元素填入结果数组 res
// 初始化数组 res 用于记录结果
int n = nums.size();
vector<int> res(n);
for (int i = n - 1; i >= 0; i--) {
int num = nums[i];
res[counter[num] - 1] = num; // 将 num 放置到对应索引处
counter[num]--; // 令前缀和自减 1 ，得到下次放置 num 的索引
}
// 使用结果数组 res 覆盖原数组 nums
nums = res;
}
/* Driver Code */
int main() {
vector<int> nums = {1, 0, 1, 2, 0, 4, 0, 2, 2, 4};
countingSortNaive(nums);
cout << "计数排序（无法排序对象）完成后 nums = ";
printVector(nums);
vector<int> nums1 = {1, 0, 1, 2, 0, 4, 0, 2, 2, 4};
countingSort(nums1);
cout << "计数排序完成后 nums1 = ";
printVector(nums1);
return 0;
}

54
chapter_sorting/heap_sort.cpp
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/**
* File: heap_sort.cpp
* Created Time: 2023-05-26
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 堆的长度为 n ，从节点 i 开始，从顶至底堆化 */
void siftDown(vector<int> &nums, int n, int i) {
while (true) {
// 判断节点 i, l, r 中值最大的节点，记为 ma
int l = 2 * i + 1;
int r = 2 * i + 2;
int ma = i;
if (l < n && nums[l] > nums[ma])
ma = l;
if (r < n && nums[r] > nums[ma])
ma = r;
// 若节点 i 最大或索引 l, r 越界，则无须继续堆化，跳出
if (ma == i) {
break;
}
// 交换两节点
swap(nums[i], nums[ma]);
// 循环向下堆化
i = ma;
}
}
/* 堆排序 */
void heapSort(vector<int> &nums) {
// 建堆操作：堆化除叶节点以外的其他所有节点
for (int i = nums.size() / 2 - 1; i >= 0; --i) {
siftDown(nums, nums.size(), i);
}
// 从堆中提取最大元素，循环 n-1 轮
for (int i = nums.size() - 1; i > 0; --i) {
// 交换根节点与最右叶节点（交换首元素与尾元素）
swap(nums[0], nums[i]);
// 以根节点为起点，从顶至底进行堆化
siftDown(nums, i, 0);
}
}
/* Driver Code */
int main() {
vector<int> nums = {4, 1, 3, 1, 5, 2};
heapSort(nums);
cout << "堆排序完成后 nums = ";
printVector(nums);
return 0;
}

31
chapter_sorting/insertion_sort.cpp
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/**
* File: insertion_sort.cpp
* Created Time: 2022-11-25
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 插入排序 */
void insertionSort(vector<int> &nums) {
// 外循环：已排序元素数量为 1, 2, ..., n
for (int i = 1; i < nums.size(); i++) {
int base = nums[i], j = i - 1;
// 内循环：将 base 插入到已排序部分的正确位置
while (j >= 0 && nums[j] > base) {
nums[j + 1] = nums[j]; // 将 nums[j] 向右移动一位
j--;
}
nums[j + 1] = base; // 将 base 赋值到正确位置
}
}
/* Driver Code */
int main() {
vector<int> nums = {4, 1, 3, 1, 5, 2};
insertionSort(nums);
cout << "插入排序完成后 nums = ";
printVector(nums);
return 0;
}

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chapter_sorting/merge_sort.cpp
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/**
* File: merge_sort.cpp
* Created Time: 2022-11-25
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 合并左子数组和右子数组 */
void merge(vector<int> &nums, int left, int mid, int right) {
// 左子数组区间为 [left, mid], 右子数组区间为 [mid+1, right]
// 创建一个临时数组 tmp ，用于存放合并后的结果
vector<int> tmp(right - left + 1);
// 初始化左子数组和右子数组的起始索引
int i = left, j = mid + 1, k = 0;
// 当左右子数组都还有元素时，进行比较并将较小的元素复制到临时数组中
while (i <= mid && j <= right) {
if (nums[i] <= nums[j])
tmp[k++] = nums[i++];
else
tmp[k++] = nums[j++];
}
// 将左子数组和右子数组的剩余元素复制到临时数组中
while (i <= mid) {
tmp[k++] = nums[i++];
}
while (j <= right) {
tmp[k++] = nums[j++];
}
// 将临时数组 tmp 中的元素复制回原数组 nums 的对应区间
for (k = 0; k < tmp.size(); k++) {
nums[left + k] = tmp[k];
}
}
/* 归并排序 */
void mergeSort(vector<int> &nums, int left, int right) {
// 终止条件
if (left >= right)
return; // 当子数组长度为 1 时终止递归
// 划分阶段
int mid = (left + right) / 2; // 计算中点
mergeSort(nums, left, mid); // 递归左子数组
mergeSort(nums, mid + 1, right); // 递归右子数组
// 合并阶段
merge(nums, left, mid, right);
}
/* Driver Code */
int main() {
/* 归并排序 */
vector<int> nums = {7, 3, 2, 6, 0, 1, 5, 4};
mergeSort(nums, 0, nums.size() - 1);
cout << "归并排序完成后 nums = ";
printVector(nums);
return 0;
}

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chapter_sorting/quick_sort.cpp
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/**
* File: quick_sort.cpp
* Created Time: 2022-11-25
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 快速排序类 */
class QuickSort {
private:
/* 元素交换 */
static void swap(vector<int> &nums, int i, int j) {
int tmp = nums[i];
nums[i] = nums[j];
nums[j] = tmp;
}
/* 哨兵划分 */
static int partition(vector<int> &nums, int left, int right) {
// 以 nums[left] 为基准数
int i = left, j = right;
while (i < j) {
while (i < j && nums[j] >= nums[left])
j--; // 从右向左找首个小于基准数的元素
while (i < j && nums[i] <= nums[left])
i++; // 从左向右找首个大于基准数的元素
swap(nums, i, j); // 交换这两个元素
}
swap(nums, i, left); // 将基准数交换至两子数组的分界线
return i; // 返回基准数的索引
}
public:
/* 快速排序 */
static void quickSort(vector<int> &nums, int left, int right) {
// 子数组长度为 1 时终止递归
if (left >= right)
return;
// 哨兵划分
int pivot = partition(nums, left, right);
// 递归左子数组、右子数组
quickSort(nums, left, pivot - 1);
quickSort(nums, pivot + 1, right);
}
};
/* 快速排序类（中位基准数优化） */
class QuickSortMedian {
private:
/* 元素交换 */
static void swap(vector<int> &nums, int i, int j) {
int tmp = nums[i];
nums[i] = nums[j];
nums[j] = tmp;
}
/* 选取三个候选元素的中位数 */
static int medianThree(vector<int> &nums, int left, int mid, int right) {
// 此处使用异或运算来简化代码
// 异或规则为 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1
if ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))
return left;
else if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))
return mid;
else
return right;
}
/* 哨兵划分（三数取中值） */
static int partition(vector<int> &nums, int left, int right) {
// 选取三个候选元素的中位数
int med = medianThree(nums, left, (left + right) / 2, right);
// 将中位数交换至数组最左端
swap(nums, left, med);
// 以 nums[left] 为基准数
int i = left, j = right;
while (i < j) {
while (i < j && nums[j] >= nums[left])
j--; // 从右向左找首个小于基准数的元素
while (i < j && nums[i] <= nums[left])
i++; // 从左向右找首个大于基准数的元素
swap(nums, i, j); // 交换这两个元素
}
swap(nums, i, left); // 将基准数交换至两子数组的分界线
return i; // 返回基准数的索引
}
public:
/* 快速排序 */
static void quickSort(vector<int> &nums, int left, int right) {
// 子数组长度为 1 时终止递归
if (left >= right)
return;
// 哨兵划分
int pivot = partition(nums, left, right);
// 递归左子数组、右子数组
quickSort(nums, left, pivot - 1);
quickSort(nums, pivot + 1, right);
}
};
/* 快速排序类（尾递归优化） */
class QuickSortTailCall {
private:
/* 元素交换 */
static void swap(vector<int> &nums, int i, int j) {
int tmp = nums[i];
nums[i] = nums[j];
nums[j] = tmp;
}
/* 哨兵划分 */
static int partition(vector<int> &nums, int left, int right) {
// 以 nums[left] 为基准数
int i = left, j = right;
while (i < j) {
while (i < j && nums[j] >= nums[left])
j--; // 从右向左找首个小于基准数的元素
while (i < j && nums[i] <= nums[left])
i++; // 从左向右找首个大于基准数的元素
swap(nums, i, j); // 交换这两个元素
}
swap(nums, i, left); // 将基准数交换至两子数组的分界线
return i; // 返回基准数的索引
}
public:
/* 快速排序（尾递归优化） */
static void quickSort(vector<int> &nums, int left, int right) {
// 子数组长度为 1 时终止
while (left < right) {
// 哨兵划分操作
int pivot = partition(nums, left, right);
// 对两个子数组中较短的那个执行快速排序
if (pivot - left < right - pivot) {
quickSort(nums, left, pivot - 1); // 递归排序左子数组
left = pivot + 1; // 剩余未排序区间为 [pivot + 1, right]
} else {
quickSort(nums, pivot + 1, right); // 递归排序右子数组
right = pivot - 1; // 剩余未排序区间为 [left, pivot - 1]
}
}
}
};
/* Driver Code */
int main() {
/* 快速排序 */
vector<int> nums{2, 4, 1, 0, 3, 5};
QuickSort::quickSort(nums, 0, nums.size() - 1);
cout << "快速排序完成后 nums = ";
printVector(nums);
/* 快速排序（中位基准数优化） */
vector<int> nums1 = {2, 4, 1, 0, 3, 5};
QuickSortMedian::quickSort(nums1, 0, nums1.size() - 1);
cout << "快速排序（中位基准数优化）完成后 nums = ";
printVector(nums);
/* 快速排序（尾递归优化） */
vector<int> nums2 = {2, 4, 1, 0, 3, 5};
QuickSortTailCall::quickSort(nums2, 0, nums2.size() - 1);
cout << "快速排序（尾递归优化）完成后 nums = ";
printVector(nums);
return 0;
}

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chapter_sorting/radix_sort.cpp
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/**
* File: radix_sort.cpp
* Created Time: 2023-03-26
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 获取元素 num 的第 k 位，其中 exp = 10^(k-1) */
int digit(int num, int exp) {
// 传入 exp 而非 k 可以避免在此重复执行昂贵的次方计算
return (num / exp) % 10;
}
/* 计数排序（根据 nums 第 k 位排序） */
void countingSortDigit(vector<int> &nums, int exp) {
// 十进制的位范围为 0~9 ，因此需要长度为 10 的桶数组
vector<int> counter(10, 0);
int n = nums.size();
// 统计 0~9 各数字的出现次数
for (int i = 0; i < n; i++) {
int d = digit(nums[i], exp); // 获取 nums[i] 第 k 位，记为 d
counter[d]++; // 统计数字 d 的出现次数
}
// 求前缀和，将“出现个数”转换为“数组索引”
for (int i = 1; i < 10; i++) {
counter[i] += counter[i - 1];
}
// 倒序遍历，根据桶内统计结果，将各元素填入 res
vector<int> res(n, 0);
for (int i = n - 1; i >= 0; i--) {
int d = digit(nums[i], exp);
int j = counter[d] - 1; // 获取 d 在数组中的索引 j
res[j] = nums[i]; // 将当前元素填入索引 j
counter[d]--; // 将 d 的数量减 1
}
// 使用结果覆盖原数组 nums
for (int i = 0; i < n; i++)
nums[i] = res[i];
}
/* 基数排序 */
void radixSort(vector<int> &nums) {
// 获取数组的最大元素，用于判断最大位数
int m = *max_element(nums.begin(), nums.end());
// 按照从低位到高位的顺序遍历
for (int exp = 1; exp <= m; exp *= 10)
// 对数组元素的第 k 位执行计数排序
// k = 1 -> exp = 1
// k = 2 -> exp = 10
// 即 exp = 10^(k-1)
countingSortDigit(nums, exp);
}
/* Driver Code */
int main() {
// 基数排序
vector<int> nums = {10546151, 35663510, 42865989, 34862445, 81883077,
88906420, 72429244, 30524779, 82060337, 63832996};
radixSort(nums);
cout << "基数排序完成后 nums = ";
printVector(nums);
return 0;
}

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chapter_sorting/selection_sort.cpp
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/**
* File: selection_sort.cpp
* Created Time: 2023-05-23
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 选择排序 */
void selectionSort(vector<int> &nums) {
int n = nums.size();
// 外循环：未排序区间为 [i, n-1]
for (int i = 0; i < n - 1; i++) {
// 内循环：找到未排序区间内的最小元素
int k = i;
for (int j = i + 1; j < n; j++) {
if (nums[j] < nums[k])
k = j; // 记录最小元素的索引
}
// 将该最小元素与未排序区间的首个元素交换
swap(nums[i], nums[k]);
}
}
/* Driver Code */
int main() {
vector<int> nums = {4, 1, 3, 1, 5, 2};
selectionSort(nums);
cout << "选择排序完成后 nums = ";
printVector(nums);
return 0;
}

9
chapter_stack_and_queue/CMakeLists.txt
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add_executable(array_deque array_deque.cpp)
add_executable(array_queue array_queue.cpp)
add_executable(array_stack array_stack.cpp)
add_executable(deque deque.cpp)
add_executable(linkedlist_deque linkedlist_deque.cpp)
add_executable(linkedlist_queue linkedlist_queue.cpp)
add_executable(linkedlist_stack linkedlist_stack.cpp)
add_executable(queue queue.cpp)
add_executable(stack stack.cpp)

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chapter_stack_and_queue/array_deque.cpp
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/**
* File: array_deque.cpp
* Created Time: 2023-03-02
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 基于环形数组实现的双向队列 */
class ArrayDeque {
private:
vector<int> nums; // 用于存储双向队列元素的数组
int front; // 队首指针，指向队首元素
int queSize; // 双向队列长度
public:
/* 构造方法 */
ArrayDeque(int capacity) {
nums.resize(capacity);
front = queSize = 0;
}
/* 获取双向队列的容量 */
int capacity() {
return nums.size();
}
/* 获取双向队列的长度 */
int size() {
return queSize;
}
/* 判断双向队列是否为空 */
bool isEmpty() {
return queSize == 0;
}
/* 计算环形数组索引 */
int index(int i) {
// 通过取余操作实现数组首尾相连
// 当 i 越过数组尾部后，回到头部
// 当 i 越过数组头部后，回到尾部
return (i + capacity()) % capacity();
}
/* 队首入队 */
void pushFirst(int num) {
if (queSize == capacity()) {
cout << "双向队列已满" << endl;
return;
}
// 队首指针向左移动一位
// 通过取余操作实现 front 越过数组头部后回到尾部
front = index(front - 1);
// 将 num 添加至队首
nums[front] = num;
queSize++;
}
/* 队尾入队 */
void pushLast(int num) {
if (queSize == capacity()) {
cout << "双向队列已满" << endl;
return;
}
// 计算队尾指针，指向队尾索引 + 1
int rear = index(front + queSize);
// 将 num 添加至队尾
nums[rear] = num;
queSize++;
}
/* 队首出队 */
int popFirst() {
int num = peekFirst();
// 队首指针向后移动一位
front = index(front + 1);
queSize--;
return num;
}
/* 队尾出队 */
int popLast() {
int num = peekLast();
queSize--;
return num;
}
/* 访问队首元素 */
int peekFirst() {
if (isEmpty())
throw out_of_range("双向队列为空");
return nums[front];
}
/* 访问队尾元素 */
int peekLast() {
if (isEmpty())
throw out_of_range("双向队列为空");
// 计算尾元素索引
int last = index(front + queSize - 1);
return nums[last];
}
/* 返回数组用于打印 */
vector<int> toVector() {
// 仅转换有效长度范围内的列表元素
vector<int> res(queSize);
for (int i = 0, j = front; i < queSize; i++, j++) {
res[i] = nums[index(j)];
}
return res;
}
};
/* Driver Code */
int main() {
/* 初始化双向队列 */
ArrayDeque *deque = new ArrayDeque(10);
deque->pushLast(3);
deque->pushLast(2);
deque->pushLast(5);
cout << "双向队列 deque = ";
printVector(deque->toVector());
/* 访问元素 */
int peekFirst = deque->peekFirst();
cout << "队首元素 peekFirst = " << peekFirst << endl;
int peekLast = deque->peekLast();
cout << "队尾元素 peekLast = " << peekLast << endl;
/* 元素入队 */
deque->pushLast(4);
cout << "元素 4 队尾入队后 deque = ";
printVector(deque->toVector());
deque->pushFirst(1);
cout << "元素 1 队首入队后 deque = ";
printVector(deque->toVector());
/* 元素出队 */
int popLast = deque->popLast();
cout << "队尾出队元素 = " << popLast << "，队尾出队后 deque = ";
printVector(deque->toVector());
int popFirst = deque->popFirst();
cout << "队首出队元素 = " << popFirst << "，队首出队后 deque = ";
printVector(deque->toVector());
/* 获取双向队列的长度 */
int size = deque->size();
cout << "双向队列长度 size = " << size << endl;
/* 判断双向队列是否为空 */
bool isEmpty = deque->isEmpty();
cout << "双向队列是否为空 = " << boolalpha << isEmpty << endl;
return 0;
}

129
chapter_stack_and_queue/array_queue.cpp
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/**
* File: array_queue.cpp
* Created Time: 2022-11-25
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 基于环形数组实现的队列 */
class ArrayQueue {
private:
int *nums; // 用于存储队列元素的数组
int front; // 队首指针，指向队首元素
int queSize; // 队列长度
int queCapacity; // 队列容量
public:
ArrayQueue(int capacity) {
// 初始化数组
nums = new int[capacity];
queCapacity = capacity;
front = queSize = 0;
}
~ArrayQueue() {
delete[] nums;
}
/* 获取队列的容量 */
int capacity() {
return queCapacity;
}
/* 获取队列的长度 */
int size() {
return queSize;
}
/* 判断队列是否为空 */
bool isEmpty() {
return size() == 0;
}
/* 入队 */
void push(int num) {
if (queSize == queCapacity) {
cout << "队列已满" << endl;
return;
}
// 计算队尾指针，指向队尾索引 + 1
// 通过取余操作实现 rear 越过数组尾部后回到头部
int rear = (front + queSize) % queCapacity;
// 将 num 添加至队尾
nums[rear] = num;
queSize++;
}
/* 出队 */
int pop() {
int num = peek();
// 队首指针向后移动一位，若越过尾部，则返回到数组头部
front = (front + 1) % queCapacity;
queSize--;
return num;
}
/* 访问队首元素 */
int peek() {
if (isEmpty())
throw out_of_range("队列为空");
return nums[front];
}
/* 将数组转化为 Vector 并返回 */
vector<int> toVector() {
// 仅转换有效长度范围内的列表元素
vector<int> arr(queSize);
for (int i = 0, j = front; i < queSize; i++, j++) {
arr[i] = nums[j % queCapacity];
}
return arr;
}
};
/* Driver Code */
int main() {
/* 初始化队列 */
int capacity = 10;
ArrayQueue *queue = new ArrayQueue(capacity);
/* 元素入队 */
queue->push(1);
queue->push(3);
queue->push(2);
queue->push(5);
queue->push(4);
cout << "队列 queue = ";
printVector(queue->toVector());
/* 访问队首元素 */
int peek = queue->peek();
cout << "队首元素 peek = " << peek << endl;
/* 元素出队 */
peek = queue->pop();
cout << "出队元素 pop = " << peek << "，出队后 queue = ";
printVector(queue->toVector());
/* 获取队列的长度 */
int size = queue->size();
cout << "队列长度 size = " << size << endl;
/* 判断队列是否为空 */
bool empty = queue->isEmpty();
cout << "队列是否为空 = " << empty << endl;
/* 测试环形数组 */
for (int i = 0; i < 10; i++) {
queue->push(i);
queue->pop();
cout << "第 " << i << " 轮入队 + 出队后 queue = ";
printVector(queue->toVector());
}
// 释放内存
delete queue;
return 0;
}

85
chapter_stack_and_queue/array_stack.cpp
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/**
* File: array_stack.cpp
* Created Time: 2022-11-28
* Author: qualifier1024 (2539244001@qq.com)
*/
#include "../utils/common.hpp"
/* 基于数组实现的栈 */
class ArrayStack {
private:
vector<int> stack;
public:
/* 获取栈的长度 */
int size() {
return stack.size();
}
/* 判断栈是否为空 */
bool isEmpty() {
return stack.size() == 0;
}
/* 入栈 */
void push(int num) {
stack.push_back(num);
}
/* 出栈 */
int pop() {
int num = top();
stack.pop_back();
return num;
}
/* 访问栈顶元素 */
int top() {
if (isEmpty())
throw out_of_range("栈为空");
return stack.back();
}
/* 返回 Vector */
vector<int> toVector() {
return stack;
}
};
/* Driver Code */
int main() {
/* 初始化栈 */
ArrayStack *stack = new ArrayStack();
/* 元素入栈 */
stack->push(1);
stack->push(3);
stack->push(2);
stack->push(5);
stack->push(4);
cout << "栈 stack = ";
printVector(stack->toVector());
/* 访问栈顶元素 */
int top = stack->top();
cout << "栈顶元素 top = " << top << endl;
/* 元素出栈 */
top = stack->pop();
cout << "出栈元素 pop = " << top << "，出栈后 stack = ";
printVector(stack->toVector());
/* 获取栈的长度 */
int size = stack->size();
cout << "栈的长度 size = " << size << endl;
/* 判断是否为空 */
bool empty = stack->isEmpty();
cout << "栈是否为空 = " << empty << endl;
// 释放内存
delete stack;
return 0;
}

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chapter_stack_and_queue/deque.cpp
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/**
* File: deque.cpp
* Created Time: 2022-11-25
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* Driver Code */
int main() {
/* 初始化双向队列 */
deque<int> deque;
/* 元素入队 */
deque.push_back(2);
deque.push_back(5);
deque.push_back(4);
deque.push_front(3);
deque.push_front(1);
cout << "双向队列 deque = ";
printDeque(deque);
/* 访问元素 */
int front = deque.front();
cout << "队首元素 front = " << front << endl;
int back = deque.back();
cout << "队尾元素 back = " << back << endl;
/* 元素出队 */
deque.pop_front();
cout << "队首出队元素 popFront = " << front << "，队首出队后 deque = ";
printDeque(deque);
deque.pop_back();
cout << "队尾出队元素 popLast = " << back << "，队尾出队后 deque = ";
printDeque(deque);
/* 获取双向队列的长度 */
int size = deque.size();
cout << "双向队列长度 size = " << size << endl;
/* 判断双向队列是否为空 */
bool empty = deque.empty();
cout << "双向队列是否为空 = " << empty << endl;
return 0;
}

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chapter_stack_and_queue/linkedlist_deque.cpp
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/**
* File: linkedlist_deque.cpp
* Created Time: 2023-03-02
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 双向链表节点 */
struct DoublyListNode {
int val; // 节点值
DoublyListNode *next; // 后继节点指针
DoublyListNode *prev; // 前驱节点指针
DoublyListNode(int val) : val(val), prev(nullptr), next(nullptr) {
}
};
/* 基于双向链表实现的双向队列 */
class LinkedListDeque {
private:
DoublyListNode *front, *rear; // 头节点 front ，尾节点 rear
int queSize = 0; // 双向队列的长度
public:
/* 构造方法 */
LinkedListDeque() : front(nullptr), rear(nullptr) {
}
/* 析构方法 */
~LinkedListDeque() {
// 遍历链表删除节点，释放内存
DoublyListNode *pre, *cur = front;
while (cur != nullptr) {
pre = cur;
cur = cur->next;
delete pre;
}
}
/* 获取双向队列的长度 */
int size() {
return queSize;
}
/* 判断双向队列是否为空 */
bool isEmpty() {
return size() == 0;
}
/* 入队操作 */
void push(int num, bool isFront) {
DoublyListNode *node = new DoublyListNode(num);
// 若链表为空，则令 front 和 rear 都指向 node
if (isEmpty())
front = rear = node;
// 队首入队操作
else if (isFront) {
// 将 node 添加至链表头部
front->prev = node;
node->next = front;
front = node; // 更新头节点
// 队尾入队操作
} else {
// 将 node 添加至链表尾部
rear->next = node;
node->prev = rear;
rear = node; // 更新尾节点
}
queSize++; // 更新队列长度
}
/* 队首入队 */
void pushFirst(int num) {
push(num, true);
}
/* 队尾入队 */
void pushLast(int num) {
push(num, false);
}
/* 出队操作 */
int pop(bool isFront) {
if (isEmpty())
throw out_of_range("队列为空");
int val;
// 队首出队操作
if (isFront) {
val = front->val; // 暂存头节点值
// 删除头节点
DoublyListNode *fNext = front->next;
if (fNext != nullptr) {
fNext->prev = nullptr;
front->next = nullptr;
}
delete front;
front = fNext; // 更新头节点
// 队尾出队操作
} else {
val = rear->val; // 暂存尾节点值
// 删除尾节点
DoublyListNode *rPrev = rear->prev;
if (rPrev != nullptr) {
rPrev->next = nullptr;
rear->prev = nullptr;
}
delete rear;
rear = rPrev; // 更新尾节点
}
queSize--; // 更新队列长度
return val;
}
/* 队首出队 */
int popFirst() {
return pop(true);
}
/* 队尾出队 */
int popLast() {
return pop(false);
}
/* 访问队首元素 */
int peekFirst() {
if (isEmpty())
throw out_of_range("双向队列为空");
return front->val;
}
/* 访问队尾元素 */
int peekLast() {
if (isEmpty())
throw out_of_range("双向队列为空");
return rear->val;
}
/* 返回数组用于打印 */
vector<int> toVector() {
DoublyListNode *node = front;
vector<int> res(size());
for (int i = 0; i < res.size(); i++) {
res[i] = node->val;
node = node->next;
}
return res;
}
};
/* Driver Code */
int main() {
/* 初始化双向队列 */
LinkedListDeque *deque = new LinkedListDeque();
deque->pushLast(3);
deque->pushLast(2);
deque->pushLast(5);
cout << "双向队列 deque = ";
printVector(deque->toVector());
/* 访问元素 */
int peekFirst = deque->peekFirst();
cout << "队首元素 peekFirst = " << peekFirst << endl;
int peekLast = deque->peekLast();
cout << "队尾元素 peekLast = " << peekLast << endl;
/* 元素入队 */
deque->pushLast(4);
cout << "元素 4 队尾入队后 deque =";
printVector(deque->toVector());
deque->pushFirst(1);
cout << "元素 1 队首入队后 deque = ";
printVector(deque->toVector());
/* 元素出队 */
int popLast = deque->popLast();
cout << "队尾出队元素 = " << popLast << "，队尾出队后 deque = ";
printVector(deque->toVector());
int popFirst = deque->popFirst();
cout << "队首出队元素 = " << popFirst << "，队首出队后 deque = ";
printVector(deque->toVector());
/* 获取双向队列的长度 */
int size = deque->size();
cout << "双向队列长度 size = " << size << endl;
/* 判断双向队列是否为空 */
bool isEmpty = deque->isEmpty();
cout << "双向队列是否为空 = " << boolalpha << isEmpty << endl;
// 释放内存
delete deque;
return 0;
}

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/**
* File: linkedlist_queue.cpp
* Created Time: 2022-11-25
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 基于链表实现的队列 */
class LinkedListQueue {
private:
ListNode *front, *rear; // 头节点 front ，尾节点 rear
int queSize;
public:
LinkedListQueue() {
front = nullptr;
rear = nullptr;
queSize = 0;
}
~LinkedListQueue() {
// 遍历链表删除节点，释放内存
freeMemoryLinkedList(front);
}
/* 获取队列的长度 */
int size() {
return queSize;
}
/* 判断队列是否为空 */
bool isEmpty() {
return queSize == 0;
}
/* 入队 */
void push(int num) {
// 在尾节点后添加 num
ListNode *node = new ListNode(num);
// 如果队列为空，则令头、尾节点都指向该节点
if (front == nullptr) {
front = node;
rear = node;
}
// 如果队列不为空，则将该节点添加到尾节点后
else {
rear->next = node;
rear = node;
}
queSize++;
}
/* 出队 */
int pop() {
int num = peek();
// 删除头节点
ListNode *tmp = front;
front = front->next;
// 释放内存
delete tmp;
queSize--;
return num;
}
/* 访问队首元素 */
int peek() {
if (size() == 0)
throw out_of_range("队列为空");
return front->val;
}
/* 将链表转化为 Vector 并返回 */
vector<int> toVector() {
ListNode *node = front;
vector<int> res(size());
for (int i = 0; i < res.size(); i++) {
res[i] = node->val;
node = node->next;
}
return res;
}
};
/* Driver Code */
int main() {
/* 初始化队列 */
LinkedListQueue *queue = new LinkedListQueue();
/* 元素入队 */
queue->push(1);
queue->push(3);
queue->push(2);
queue->push(5);
queue->push(4);
cout << "队列 queue = ";
printVector(queue->toVector());
/* 访问队首元素 */
int peek = queue->peek();
cout << "队首元素 peek = " << peek << endl;
/* 元素出队 */
peek = queue->pop();
cout << "出队元素 pop = " << peek << "，出队后 queue = ";
printVector(queue->toVector());
/* 获取队列的长度 */
int size = queue->size();
cout << "队列长度 size = " << size << endl;
/* 判断队列是否为空 */
bool empty = queue->isEmpty();
cout << "队列是否为空 = " << empty << endl;
// 释放内存
delete queue;
return 0;
}

109
chapter_stack_and_queue/linkedlist_stack.cpp
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/**
* File: linkedlist_stack.cpp
* Created Time: 2022-11-28
* Author: qualifier1024 (2539244001@qq.com)
*/
#include "../utils/common.hpp"
/* 基于链表实现的栈 */
class LinkedListStack {
private:
ListNode *stackTop; // 将头节点作为栈顶
int stkSize; // 栈的长度
public:
LinkedListStack() {
stackTop = nullptr;
stkSize = 0;
}
~LinkedListStack() {
// 遍历链表删除节点，释放内存
freeMemoryLinkedList(stackTop);
}
/* 获取栈的长度 */
int size() {
return stkSize;
}
/* 判断栈是否为空 */
bool isEmpty() {
return size() == 0;
}
/* 入栈 */
void push(int num) {
ListNode *node = new ListNode(num);
node->next = stackTop;
stackTop = node;
stkSize++;
}
/* 出栈 */
int pop() {
int num = top();
ListNode *tmp = stackTop;
stackTop = stackTop->next;
// 释放内存
delete tmp;
stkSize--;
return num;
}
/* 访问栈顶元素 */
int top() {
if (isEmpty())
throw out_of_range("栈为空");
return stackTop->val;
}
/* 将 List 转化为 Array 并返回 */
vector<int> toVector() {
ListNode *node = stackTop;
vector<int> res(size());
for (int i = res.size() - 1; i >= 0; i--) {
res[i] = node->val;
node = node->next;
}
return res;
}
};
/* Driver Code */
int main() {
/* 初始化栈 */
LinkedListStack *stack = new LinkedListStack();
/* 元素入栈 */
stack->push(1);
stack->push(3);
stack->push(2);
stack->push(5);
stack->push(4);
cout << "栈 stack = ";
printVector(stack->toVector());
/* 访问栈顶元素 */
int top = stack->top();
cout << "栈顶元素 top = " << top << endl;
/* 元素出栈 */
top = stack->pop();
cout << "出栈元素 pop = " << top << "，出栈后 stack = ";
printVector(stack->toVector());
/* 获取栈的长度 */
int size = stack->size();
cout << "栈的长度 size = " << size << endl;
/* 判断是否为空 */
bool empty = stack->isEmpty();
cout << "栈是否为空 = " << empty << endl;
// 释放内存
delete stack;
return 0;
}

41
chapter_stack_and_queue/queue.cpp
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/**
* File: queue.cpp
* Created Time: 2022-11-28
* Author: qualifier1024 (2539244001@qq.com)
*/
#include "../utils/common.hpp"
/* Driver Code */
int main() {
/* 初始化队列 */
queue<int> queue;
/* 元素入队 */
queue.push(1);
queue.push(3);
queue.push(2);
queue.push(5);
queue.push(4);
cout << "队列 queue = ";
printQueue(queue);
/* 访问队首元素 */
int front = queue.front();
cout << "队首元素 front = " << front << endl;
/* 元素出队 */
queue.pop();
cout << "出队元素 front = " << front << "，出队后 queue = ";
printQueue(queue);
/* 获取队列的长度 */
int size = queue.size();
cout << "队列长度 size = " << size << endl;
/* 判断队列是否为空 */
bool empty = queue.empty();
cout << "队列是否为空 = " << empty << endl;
return 0;
}

41
chapter_stack_and_queue/stack.cpp
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/**
* File: stack.cpp
* Created Time: 2022-11-28
* Author: qualifier1024 (2539244001@qq.com)
*/
#include "../utils/common.hpp"
/* Driver Code */
int main() {
/* 初始化栈 */
stack<int> stack;
/* 元素入栈 */
stack.push(1);
stack.push(3);
stack.push(2);
stack.push(5);
stack.push(4);
cout << "栈 stack = ";
printStack(stack);
/* 访问栈顶元素 */
int top = stack.top();
cout << "栈顶元素 top = " << top << endl;
/* 元素出栈 */
stack.pop(); // 无返回值
cout << "出栈元素 pop = " << top << "，出栈后 stack = ";
printStack(stack);
/* 获取栈的长度 */
int size = stack.size();
cout << "栈的长度 size = " << size << endl;
/* 判断是否为空 */
bool empty = stack.empty();
cout << "栈是否为空 = " << empty << endl;
return 0;
}

6
chapter_tree/CMakeLists.txt
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add_executable(avl_tree avl_tree.cpp)
add_executable(binary_search_tree binary_search_tree.cpp)
add_executable(binary_tree binary_tree.cpp)
add_executable(binary_tree_bfs binary_tree_bfs.cpp)
add_executable(binary_tree_dfs binary_tree_dfs.cpp)
add_executable(array_binary_tree array_binary_tree.cpp)

137
chapter_tree/array_binary_tree.cpp
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/**
* File: array_binary_tree.cpp
* Created Time: 2023-07-19
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 数组表示下的二叉树类 */
class ArrayBinaryTree {
public:
/* 构造方法 */
ArrayBinaryTree(vector<int> arr) {
tree = arr;
}
/* 列表容量 */
int size() {
return tree.size();
}
/* 获取索引为 i 节点的值 */
int val(int i) {
// 若索引越界，则返回 INT_MAX ，代表空位
if (i < 0 || i >= size())
return INT_MAX;
return tree[i];
}
/* 获取索引为 i 节点的左子节点的索引 */
int left(int i) {
return 2 * i + 1;
}
/* 获取索引为 i 节点的右子节点的索引 */
int right(int i) {
return 2 * i + 2;
}
/* 获取索引为 i 节点的父节点的索引 */
int parent(int i) {
return (i - 1) / 2;
}
/* 层序遍历 */
vector<int> levelOrder() {
vector<int> res;
// 直接遍历数组
for (int i = 0; i < size(); i++) {
if (val(i) != INT_MAX)
res.push_back(val(i));
}
return res;
}
/* 前序遍历 */
vector<int> preOrder() {
vector<int> res;
dfs(0, "pre", res);
return res;
}
/* 中序遍历 */
vector<int> inOrder() {
vector<int> res;
dfs(0, "in", res);
return res;
}
/* 后序遍历 */
vector<int> postOrder() {
vector<int> res;
dfs(0, "post", res);
return res;
}
private:
vector<int> tree;
/* 深度优先遍历 */
void dfs(int i, string order, vector<int> &res) {
// 若为空位，则返回
if (val(i) == INT_MAX)
return;
// 前序遍历
if (order == "pre")
res.push_back(val(i));
dfs(left(i), order, res);
// 中序遍历
if (order == "in")
res.push_back(val(i));
dfs(right(i), order, res);
// 后序遍历
if (order == "post")
res.push_back(val(i));
}
};
/* Driver Code */
int main() {
// 初始化二叉树
// 使用 INT_MAX 代表空位 nullptr
vector<int> arr = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};
TreeNode *root = vectorToTree(arr);
cout << "\n初始化二叉树\n";
cout << "二叉树的数组表示：\n";
printVector(arr);
cout << "二叉树的链表表示：\n";
printTree(root);
// 数组表示下的二叉树类
ArrayBinaryTree abt(arr);
// 访问节点
int i = 1;
int l = abt.left(i), r = abt.right(i), p = abt.parent(i);
cout << "\n当前节点的索引为 " << i << "，值为 " << abt.val(i) << "\n";
cout << "其左子节点的索引为 " << l << "，值为 " << (l != INT_MAX ? to_string(abt.val(l)) : "nullptr") << "\n";
cout << "其右子节点的索引为 " << r << "，值为 " << (r != INT_MAX ? to_string(abt.val(r)) : "nullptr") << "\n";
cout << "其父节点的索引为 " << p << "，值为 " << (p != INT_MAX ? to_string(abt.val(p)) : "nullptr") << "\n";
// 遍历树
vector<int> res = abt.levelOrder();
cout << "\n层序遍历为： ";
printVector(res);
res = abt.preOrder();
cout << "前序遍历为： ";
printVector(res);
res = abt.inOrder();
cout << "中序遍历为： ";
printVector(res);
res = abt.postOrder();
cout << "后序遍历为： ";
printVector(res);
return 0;
}

233
chapter_tree/avl_tree.cpp
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/**
* File: avl_tree.cpp
* Created Time: 2023-02-03
* Author: what-is-me (whatisme@outlook.jp)
*/
#include "../utils/common.hpp"
/* AVL 树 */
class AVLTree {
public:
TreeNode *root; // 根节点
private:
/* 更新节点高度 */
void updateHeight(TreeNode *node) {
// 节点高度等于最高子树高度 + 1
node->height = max(height(node->left), height(node->right)) + 1;
}
/* 右旋操作 */
TreeNode *rightRotate(TreeNode *node) {
TreeNode *child = node->left;
TreeNode *grandChild = child->right;
// 以 child 为原点，将 node 向右旋转
child->right = node;
node->left = grandChild;
// 更新节点高度
updateHeight(node);
updateHeight(child);
// 返回旋转后子树的根节点
return child;
}
/* 左旋操作 */
TreeNode *leftRotate(TreeNode *node) {
TreeNode *child = node->right;
TreeNode *grandChild = child->left;
// 以 child 为原点，将 node 向左旋转
child->left = node;
node->right = grandChild;
// 更新节点高度
updateHeight(node);
updateHeight(child);
// 返回旋转后子树的根节点
return child;
}
/* 执行旋转操作，使该子树重新恢复平衡 */
TreeNode *rotate(TreeNode *node) {
// 获取节点 node 的平衡因子
int _balanceFactor = balanceFactor(node);
// 左偏树
if (_balanceFactor > 1) {
if (balanceFactor(node->left) >= 0) {
// 右旋
return rightRotate(node);
} else {
// 先左旋后右旋
node->left = leftRotate(node->left);
return rightRotate(node);
}
}
// 右偏树
if (_balanceFactor < -1) {
if (balanceFactor(node->right) <= 0) {
// 左旋
return leftRotate(node);
} else {
// 先右旋后左旋
node->right = rightRotate(node->right);
return leftRotate(node);
}
}
// 平衡树，无须旋转，直接返回
return node;
}
/* 递归插入节点（辅助方法） */
TreeNode *insertHelper(TreeNode *node, int val) {
if (node == nullptr)
return new TreeNode(val);
/* 1. 查找插入位置并插入节点 */
if (val < node->val)
node->left = insertHelper(node->left, val);
else if (val > node->val)
node->right = insertHelper(node->right, val);
else
return node; // 重复节点不插入，直接返回
updateHeight(node); // 更新节点高度
/* 2. 执行旋转操作，使该子树重新恢复平衡 */
node = rotate(node);
// 返回子树的根节点
return node;
}
/* 递归删除节点（辅助方法） */
TreeNode *removeHelper(TreeNode *node, int val) {
if (node == nullptr)
return nullptr;
/* 1. 查找节点并删除 */
if (val < node->val)
node->left = removeHelper(node->left, val);
else if (val > node->val)
node->right = removeHelper(node->right, val);
else {
if (node->left == nullptr || node->right == nullptr) {
TreeNode *child = node->left != nullptr ? node->left : node->right;
// 子节点数量 = 0 ，直接删除 node 并返回
if (child == nullptr) {
delete node;
return nullptr;
}
// 子节点数量 = 1 ，直接删除 node
else {
delete node;
node = child;
}
} else {
// 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
TreeNode *temp = node->right;
while (temp->left != nullptr) {
temp = temp->left;
}
int tempVal = temp->val;
node->right = removeHelper(node->right, temp->val);
node->val = tempVal;
}
}
updateHeight(node); // 更新节点高度
/* 2. 执行旋转操作，使该子树重新恢复平衡 */
node = rotate(node);
// 返回子树的根节点
return node;
}
public:
/* 获取节点高度 */
int height(TreeNode *node) {
// 空节点高度为 -1 ，叶节点高度为 0
return node == nullptr ? -1 : node->height;
}
/* 获取平衡因子 */
int balanceFactor(TreeNode *node) {
// 空节点平衡因子为 0
if (node == nullptr)
return 0;
// 节点平衡因子 = 左子树高度 - 右子树高度
return height(node->left) - height(node->right);
}
/* 插入节点 */
void insert(int val) {
root = insertHelper(root, val);
}
/* 删除节点 */
void remove(int val) {
root = removeHelper(root, val);
}
/* 查找节点 */
TreeNode *search(int val) {
TreeNode *cur = root;
// 循环查找，越过叶节点后跳出
while (cur != nullptr) {
// 目标节点在 cur 的右子树中
if (cur->val < val)
cur = cur->right;
// 目标节点在 cur 的左子树中
else if (cur->val > val)
cur = cur->left;
// 找到目标节点，跳出循环
else
break;
}
// 返回目标节点
return cur;
}
/*构造方法*/
AVLTree() : root(nullptr) {
}
/*析构方法*/
~AVLTree() {
freeMemoryTree(root);
}
};
void testInsert(AVLTree &tree, int val) {
tree.insert(val);
cout << "\n插入节点 " << val << " 后，AVL 树为" << endl;
printTree(tree.root);
}
void testRemove(AVLTree &tree, int val) {
tree.remove(val);
cout << "\n删除节点 " << val << " 后，AVL 树为" << endl;
printTree(tree.root);
}
/* Driver Code */
int main() {
/* 初始化空 AVL 树 */
AVLTree avlTree;
/* 插入节点 */
// 请关注插入节点后，AVL 树是如何保持平衡的
testInsert(avlTree, 1);
testInsert(avlTree, 2);
testInsert(avlTree, 3);
testInsert(avlTree, 4);
testInsert(avlTree, 5);
testInsert(avlTree, 8);
testInsert(avlTree, 7);
testInsert(avlTree, 9);
testInsert(avlTree, 10);
testInsert(avlTree, 6);
/* 插入重复节点 */
testInsert(avlTree, 7);
/* 删除节点 */
// 请关注删除节点后，AVL 树是如何保持平衡的
testRemove(avlTree, 8); // 删除度为 0 的节点
testRemove(avlTree, 5); // 删除度为 1 的节点
testRemove(avlTree, 4); // 删除度为 2 的节点
/* 查询节点 */
TreeNode *node = avlTree.search(7);
cout << "\n查找到的节点对象为 " << node << "，节点值 = " << node->val << endl;
}

170
chapter_tree/binary_search_tree.cpp
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/**
* File: binary_search_tree.cpp
* Created Time: 2022-11-25
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 二叉搜索树 */
class BinarySearchTree {
private:
TreeNode *root;
public:
/* 构造方法 */
BinarySearchTree() {
// 初始化空树
root = nullptr;
}
/* 析构方法 */
~BinarySearchTree() {
freeMemoryTree(root);
}
/* 获取二叉树根节点 */
TreeNode *getRoot() {
return root;
}
/* 查找节点 */
TreeNode *search(int num) {
TreeNode *cur = root;
// 循环查找，越过叶节点后跳出
while (cur != nullptr) {
// 目标节点在 cur 的右子树中
if (cur->val < num)
cur = cur->right;
// 目标节点在 cur 的左子树中
else if (cur->val > num)
cur = cur->left;
// 找到目标节点，跳出循环
else
break;
}
// 返回目标节点
return cur;
}
/* 插入节点 */
void insert(int num) {
// 若树为空，则初始化根节点
if (root == nullptr) {
root = new TreeNode(num);
return;
}
TreeNode *cur = root, *pre = nullptr;
// 循环查找，越过叶节点后跳出
while (cur != nullptr) {
// 找到重复节点，直接返回
if (cur->val == num)
return;
pre = cur;
// 插入位置在 cur 的右子树中
if (cur->val < num)
cur = cur->right;
// 插入位置在 cur 的左子树中
else
cur = cur->left;
}
// 插入节点
TreeNode *node = new TreeNode(num);
if (pre->val < num)
pre->right = node;
else
pre->left = node;
}
/* 删除节点 */
void remove(int num) {
// 若树为空，直接提前返回
if (root == nullptr)
return;
TreeNode *cur = root, *pre = nullptr;
// 循环查找，越过叶节点后跳出
while (cur != nullptr) {
// 找到待删除节点，跳出循环
if (cur->val == num)
break;
pre = cur;
// 待删除节点在 cur 的右子树中
if (cur->val < num)
cur = cur->right;
// 待删除节点在 cur 的左子树中
else
cur = cur->left;
}
// 若无待删除节点，则直接返回
if (cur == nullptr)
return;
// 子节点数量 = 0 or 1
if (cur->left == nullptr || cur->right == nullptr) {
// 当子节点数量 = 0 / 1 时， child = nullptr / 该子节点
TreeNode *child = cur->left != nullptr ? cur->left : cur->right;
// 删除节点 cur
if (cur != root) {
if (pre->left == cur)
pre->left = child;
else
pre->right = child;
} else {
// 若删除节点为根节点，则重新指定根节点
root = child;
}
// 释放内存
delete cur;
}
// 子节点数量 = 2
else {
// 获取中序遍历中 cur 的下一个节点
TreeNode *tmp = cur->right;
while (tmp->left != nullptr) {
tmp = tmp->left;
}
int tmpVal = tmp->val;
// 递归删除节点 tmp
remove(tmp->val);
// 用 tmp 覆盖 cur
cur->val = tmpVal;
}
}
};
/* Driver Code */
int main() {
/* 初始化二叉搜索树 */
BinarySearchTree *bst = new BinarySearchTree();
// 请注意，不同的插入顺序会生成不同的二叉树，该序列可以生成一个完美二叉树
vector<int> nums = {8, 4, 12, 2, 6, 10, 14, 1, 3, 5, 7, 9, 11, 13, 15};
for (int num : nums) {
bst->insert(num);
}
cout << endl << "初始化的二叉树为\n" << endl;
printTree(bst->getRoot());
/* 查找节点 */
TreeNode *node = bst->search(7);
cout << endl << "查找到的节点对象为 " << node << "，节点值 = " << node->val << endl;
/* 插入节点 */
bst->insert(16);
cout << endl << "插入节点 16 后，二叉树为\n" << endl;
printTree(bst->getRoot());
/* 删除节点 */
bst->remove(1);
cout << endl << "删除节点 1 后，二叉树为\n" << endl;
printTree(bst->getRoot());
bst->remove(2);
cout << endl << "删除节点 2 后，二叉树为\n" << endl;
printTree(bst->getRoot());
bst->remove(4);
cout << endl << "删除节点 4 后，二叉树为\n" << endl;
printTree(bst->getRoot());
// 释放内存
delete bst;
return 0;
}

43
chapter_tree/binary_tree.cpp
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/**
* File: binary_tree.cpp
* Created Time: 2022-11-25
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* Driver Code */
int main() {
/* 初始化二叉树 */
// 初始化节点
TreeNode *n1 = new TreeNode(1);
TreeNode *n2 = new TreeNode(2);
TreeNode *n3 = new TreeNode(3);
TreeNode *n4 = new TreeNode(4);
TreeNode *n5 = new TreeNode(5);
// 构建节点之间的引用（指针）
n1->left = n2;
n1->right = n3;
n2->left = n4;
n2->right = n5;
cout << endl << "初始化二叉树\n" << endl;
printTree(n1);
/* 插入与删除节点 */
TreeNode *P = new TreeNode(0);
// 在 n1 -> n2 中间插入节点 P
n1->left = P;
P->left = n2;
cout << endl << "插入节点 P 后\n" << endl;
printTree(n1);
// 删除节点 P
n1->left = n2;
delete P; // 释放内存
cout << endl << "删除节点 P 后\n" << endl;
printTree(n1);
// 释放内存
freeMemoryTree(n1);
return 0;
}

42
chapter_tree/binary_tree_bfs.cpp
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/**
* File: binary_tree_bfs.cpp
* Created Time: 2022-11-25
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
/* 层序遍历 */
vector<int> levelOrder(TreeNode *root) {
// 初始化队列，加入根节点
queue<TreeNode *> queue;
queue.push(root);
// 初始化一个列表，用于保存遍历序列
vector<int> vec;
while (!queue.empty()) {
TreeNode *node = queue.front();
queue.pop(); // 队列出队
vec.push_back(node->val); // 保存节点值
if (node->left != nullptr)
queue.push(node->left); // 左子节点入队
if (node->right != nullptr)
queue.push(node->right); // 右子节点入队
}
return vec;
}
/* Driver Code */
int main() {
/* 初始化二叉树 */
// 这里借助了一个从数组直接生成二叉树的函数
TreeNode *root = vectorToTree(vector<int>{1, 2, 3, 4, 5, 6, 7});
cout << endl << "初始化二叉树\n" << endl;
printTree(root);
/* 层序遍历 */
vector<int> vec = levelOrder(root);
cout << endl << "层序遍历的节点打印序列 = ";
printVector(vec);
return 0;
}

69
chapter_tree/binary_tree_dfs.cpp
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/**
* File: binary_tree_dfs.cpp
* Created Time: 2022-11-25
* Author: krahets (krahets@163.com)
*/
#include "../utils/common.hpp"
// 初始化列表，用于存储遍历序列
vector<int> vec;
/* 前序遍历 */
void preOrder(TreeNode *root) {
if (root == nullptr)
return;
// 访问优先级：根节点 -> 左子树 -> 右子树
vec.push_back(root->val);
preOrder(root->left);
preOrder(root->right);
}
/* 中序遍历 */
void inOrder(TreeNode *root) {
if (root == nullptr)
return;
// 访问优先级：左子树 -> 根节点 -> 右子树
inOrder(root->left);
vec.push_back(root->val);
inOrder(root->right);
}
/* 后序遍历 */
void postOrder(TreeNode *root) {
if (root == nullptr)
return;
// 访问优先级：左子树 -> 右子树 -> 根节点
postOrder(root->left);
postOrder(root->right);
vec.push_back(root->val);
}
/* Driver Code */
int main() {
/* 初始化二叉树 */
// 这里借助了一个从数组直接生成二叉树的函数
TreeNode *root = vectorToTree(vector<int>{1, 2, 3, 4, 5, 6, 7});
cout << endl << "初始化二叉树\n" << endl;
printTree(root);
/* 前序遍历 */
vec.clear();
preOrder(root);
cout << endl << "前序遍历的节点打印序列 = ";
printVector(vec);
/* 中序遍历 */
vec.clear();
inOrder(root);
cout << endl << "中序遍历的节点打印序列 = ";
printVector(vec);
/* 后序遍历 */
vec.clear();
postOrder(root);
cout << endl << "后序遍历的节点打印序列 = ";
printVector(vec);
return 0;
}

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test/output/priority_queue_demo.exe
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28
test/priority_queue_demo.cpp
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#include <queue>
#include <iostream>
struct Person {
std::string name;
int age;
};
struct CompareByAge {
bool operator()(const Person& p1, const Person& p2) const {
return p1.age < p2.age;
}
};
int main() {
std::priority_queue<Person, std::vector<Person>, CompareByAge> pq;
pq.push({"Alice", 25});
pq.push({"Bob", 30});
pq.push({"Charlie", 20});
while (!pq.empty()) {
std::cout << pq.top().name << " ";
pq.pop();
}
return 0;
}

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