From 0e2ddba30f8f52170a641eb1e0ea06180a7792ed Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Wed, 21 Jun 2023 02:58:24 +0800 Subject: [PATCH] Add the section of subset sum problem. (#558) --- .../cpp/chapter_backtracking/subset_sum_i.cpp | 57 +++ .../subset_sum_i_naive.cpp | 54 +++ .../chapter_backtracking/subset_sum_ii.cpp | 62 ++++ .../chapter_backtracking/subset_sum_i.java | 55 +++ .../subset_sum_i_naive.java | 53 +++ .../chapter_backtracking/subset_sum_ii.java | 60 ++++ .../chapter_backtracking/subset_sum_i.py | 48 +++ .../subset_sum_i_naive.py | 50 +++ .../chapter_backtracking/subset_sum_ii.py | 52 +++ .../subset_sum_i.png | Bin 0 -> 121652 bytes .../subset_sum_i_naive.png | Bin 0 -> 105568 bytes .../subset_sum_i_pruning.png | Bin 0 -> 95626 bytes .../subset_sum_ii.png | Bin 0 -> 124809 bytes .../subset_sum_ii_repeat.png | Bin 0 -> 65592 bytes .../subset_sum_problem.md | 324 ++++++++++++++++++ mkdocs.yml | 11 +- 16 files changed, 821 insertions(+), 5 deletions(-) create mode 100644 codes/cpp/chapter_backtracking/subset_sum_i.cpp create mode 100644 codes/cpp/chapter_backtracking/subset_sum_i_naive.cpp create mode 100644 codes/cpp/chapter_backtracking/subset_sum_ii.cpp create mode 100644 codes/java/chapter_backtracking/subset_sum_i.java create mode 100644 codes/java/chapter_backtracking/subset_sum_i_naive.java create mode 100644 codes/java/chapter_backtracking/subset_sum_ii.java create mode 100644 codes/python/chapter_backtracking/subset_sum_i.py create mode 100644 codes/python/chapter_backtracking/subset_sum_i_naive.py create mode 100644 codes/python/chapter_backtracking/subset_sum_ii.py create mode 100644 docs/chapter_backtracking/subset_sum_problem.assets/subset_sum_i.png create mode 100644 docs/chapter_backtracking/subset_sum_problem.assets/subset_sum_i_naive.png create mode 100644 docs/chapter_backtracking/subset_sum_problem.assets/subset_sum_i_pruning.png create mode 100644 docs/chapter_backtracking/subset_sum_problem.assets/subset_sum_ii.png create mode 100644 docs/chapter_backtracking/subset_sum_problem.assets/subset_sum_ii_repeat.png create mode 100644 docs/chapter_backtracking/subset_sum_problem.md diff --git a/codes/cpp/chapter_backtracking/subset_sum_i.cpp b/codes/cpp/chapter_backtracking/subset_sum_i.cpp new file mode 100644 index 000000000..d241deda9 --- /dev/null +++ b/codes/cpp/chapter_backtracking/subset_sum_i.cpp @@ -0,0 +1,57 @@ +/** + * 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 &state, int target, vector &choices, int start, vector> &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> subsetSumI(vector &nums, int target) { + vector 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> res; // 结果列表(子集列表) + backtrack(state, target, nums, start, res); + return res; +} + +/* Driver Code */ +int main() { + vector nums = {3, 4, 5}; + int target = 9; + + vector> 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; +} diff --git a/codes/cpp/chapter_backtracking/subset_sum_i_naive.cpp b/codes/cpp/chapter_backtracking/subset_sum_i_naive.cpp new file mode 100644 index 000000000..326dc6313 --- /dev/null +++ b/codes/cpp/chapter_backtracking/subset_sum_i_naive.cpp @@ -0,0 +1,54 @@ +/** + * 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 &state, int target, int total, vector &choices, vector> &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> subsetSumINaive(vector &nums, int target) { + vector state; // 状态(子集) + int total = 0; // 子集和 + vector> res; // 结果列表(子集列表) + backtrack(state, target, total, nums, res); + return res; +} + +/* Driver Code */ +int main() { + vector nums = {3, 4, 5}; + int target = 9; + + vector> 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; +} diff --git a/codes/cpp/chapter_backtracking/subset_sum_ii.cpp b/codes/cpp/chapter_backtracking/subset_sum_ii.cpp new file mode 100644 index 000000000..5c335cb4b --- /dev/null +++ b/codes/cpp/chapter_backtracking/subset_sum_ii.cpp @@ -0,0 +1,62 @@ +/** + * 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 &state, int target, vector &choices, int start, vector> &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> subsetSumII(vector &nums, int target) { + vector 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> res; // 结果列表(子集列表) + backtrack(state, target, nums, start, res); + return res; +} + +/* Driver Code */ +int main() { + vector nums = {4, 4, 5}; + int target = 9; + + vector> 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; +} diff --git a/codes/java/chapter_backtracking/subset_sum_i.java b/codes/java/chapter_backtracking/subset_sum_i.java new file mode 100644 index 000000000..47377bc0c --- /dev/null +++ b/codes/java/chapter_backtracking/subset_sum_i.java @@ -0,0 +1,55 @@ +/** + * File: subset_sum_i.java + * Created Time: 2023-06-21 + * Author: Krahets (krahets@163.com) + */ + +package chapter_backtracking; + +import java.util.*; + +public class subset_sum_i { + /* 回溯算法:子集和 I */ + static void backtrack(List state, int target, int[] choices, int start, List> res) { + // 子集和等于 target 时,记录解 + if (target == 0) { + res.add(new ArrayList<>(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.length; 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.add(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.remove(state.size() - 1); + } + } + + /* 求解子集和 I */ + static List> subsetSumI(int[] nums, int target) { + List state = new ArrayList<>(); // 状态(子集) + Arrays.sort(nums); // 对 nums 进行排序 + int start = 0; // 遍历起始点 + List> res = new ArrayList<>(); // 结果列表(子集列表) + backtrack(state, target, nums, start, res); + return res; + } + + public static void main(String[] args) { + int[] nums = { 3, 4, 5 }; + int target = 9; + + List> res = subsetSumI(nums, target); + + System.out.println("输入数组 nums = " + Arrays.toString(nums) + ", target = " + target); + System.out.println("所有和等于 " + target + " 的子集 res = " + res); + } +} diff --git a/codes/java/chapter_backtracking/subset_sum_i_naive.java b/codes/java/chapter_backtracking/subset_sum_i_naive.java new file mode 100644 index 000000000..15b7075e0 --- /dev/null +++ b/codes/java/chapter_backtracking/subset_sum_i_naive.java @@ -0,0 +1,53 @@ +/** + * File: subset_sum_i_naive.java + * Created Time: 2023-06-21 + * Author: Krahets (krahets@163.com) + */ + +package chapter_backtracking; + +import java.util.*; + +public class subset_sum_i_naive { + /* 回溯算法:子集和 I */ + static void backtrack(List state, int target, int total, int[] choices, List> res) { + // 子集和等于 target 时,记录解 + if (total == target) { + res.add(new ArrayList<>(state)); + return; + } + // 遍历所有选择 + for (int i = 0; i < choices.length; 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.add(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.remove(state.size() - 1); + } + } + + /* 求解子集和 I(包含重复子集) */ + static List> subsetSumINaive(int[] nums, int target) { + List state = new ArrayList<>(); // 状态(子集) + int total = 0; // 子集和 + List> res = new ArrayList<>(); // 结果列表(子集列表) + backtrack(state, target, total, nums, res); + return res; + } + + public static void main(String[] args) { + int[] nums = { 3, 4, 5 }; + int target = 9; + + List> res = subsetSumINaive(nums, target); + + System.out.println("输入数组 nums = " + Arrays.toString(nums) + ", target = " + target); + System.out.println("所有和等于 " + target + " 的子集 res = " + res); + System.out.println("请注意,该方法输出的结果包含重复集合"); + } +} diff --git a/codes/java/chapter_backtracking/subset_sum_ii.java b/codes/java/chapter_backtracking/subset_sum_ii.java new file mode 100644 index 000000000..5d72fb439 --- /dev/null +++ b/codes/java/chapter_backtracking/subset_sum_ii.java @@ -0,0 +1,60 @@ +/** + * File: subset_sum_ii.java + * Created Time: 2023-06-21 + * Author: Krahets (krahets@163.com) + */ + +package chapter_backtracking; + +import java.util.*; + +public class subset_sum_ii { + /* 回溯算法:子集和 II */ + static void backtrack(List state, int target, int[] choices, int start, List> res) { + // 子集和等于 target 时,记录解 + if (target == 0) { + res.add(new ArrayList<>(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.length; 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.add(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.remove(state.size() - 1); + } + } + + /* 求解子集和 II */ + static List> subsetSumII(int[] nums, int target) { + List state = new ArrayList<>(); // 状态(子集) + Arrays.sort(nums); // 对 nums 进行排序 + int start = 0; // 遍历起始点 + List> res = new ArrayList<>(); // 结果列表(子集列表) + backtrack(state, target, nums, start, res); + return res; + } + + public static void main(String[] args) { + int[] nums = { 4, 4, 5 }; + int target = 9; + + List> res = subsetSumII(nums, target); + + System.out.println("输入数组 nums = " + Arrays.toString(nums) + ", target = " + target); + System.out.println("所有和等于 " + target + " 的子集 res = " + res); + } +} diff --git a/codes/python/chapter_backtracking/subset_sum_i.py b/codes/python/chapter_backtracking/subset_sum_i.py new file mode 100644 index 000000000..cab0b395e --- /dev/null +++ b/codes/python/chapter_backtracking/subset_sum_i.py @@ -0,0 +1,48 @@ +""" +File: subset_sum_i.py +Created Time: 2023-06-17 +Author: Krahets (krahets@163.com) +""" + + +def backtrack( + state: list[int], target: int, choices: list[int], start: int, res: list[list[int]] +): + """回溯算法:子集和 I""" + # 子集和等于 target 时,记录解 + if target == 0: + res.append(list(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 i in range(start, len(choices)): + # 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.append(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() + + +def subset_sum_i(nums: list[int], target: int) -> list[list[int]]: + """求解子集和 I""" + state = [] # 状态(子集) + nums.sort() # 对 nums 进行排序 + start = 0 # 遍历起始点 + res = [] # 结果列表(子集列表) + backtrack(state, target, nums, start, res) + return res + + +"""Driver Code""" +if __name__ == "__main__": + nums = [3, 4, 5] + target = 9 + res = subset_sum_i(nums, target) + + print(f"输入数组 nums = {nums}, target = {target}") + print(f"所有和等于 {target} 的子集 res = {res}") diff --git a/codes/python/chapter_backtracking/subset_sum_i_naive.py b/codes/python/chapter_backtracking/subset_sum_i_naive.py new file mode 100644 index 000000000..4c19265de --- /dev/null +++ b/codes/python/chapter_backtracking/subset_sum_i_naive.py @@ -0,0 +1,50 @@ +""" +File: subset_sum_i_naive.py +Created Time: 2023-06-17 +Author: Krahets (krahets@163.com) +""" + + +def backtrack( + state: list[int], + target: int, + total: int, + choices: list[int], + res: list[list[int]], +): + """回溯算法:子集和 I""" + # 子集和等于 target 时,记录解 + if total == target: + res.append(list(state)) + return + # 遍历所有选择 + for i in range(len(choices)): + # 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.append(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() + + +def subset_sum_i_naive(nums: list[int], target: int) -> list[list[int]]: + """求解子集和 I(包含重复子集)""" + state = [] # 状态(子集) + total = 0 # 子集和 + res = [] # 结果列表(子集列表) + backtrack(state, target, total, nums, res) + return res + + +"""Driver Code""" +if __name__ == "__main__": + nums = [3, 4, 5] + target = 9 + res = subset_sum_i_naive(nums, target) + + print(f"输入数组 nums = {nums}, target = {target}") + print(f"所有和等于 {target} 的子集 res = {res}") + print(f"请注意,该方法输出的结果包含重复集合") diff --git a/codes/python/chapter_backtracking/subset_sum_ii.py b/codes/python/chapter_backtracking/subset_sum_ii.py new file mode 100644 index 000000000..130f0b321 --- /dev/null +++ b/codes/python/chapter_backtracking/subset_sum_ii.py @@ -0,0 +1,52 @@ +""" +File: subset_sum_ii.py +Created Time: 2023-06-17 +Author: Krahets (krahets@163.com) +""" + + +def backtrack( + state: list[int], target: int, choices: list[int], start: int, res: list[list[int]] +): + """回溯算法:子集和 II""" + # 子集和等于 target 时,记录解 + if target == 0: + res.append(list(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 i in range(start, len(choices)): + # 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 and choices[i] == choices[i - 1]: + continue + # 尝试:做出选择,更新 target, start + state.append(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() + + +def subset_sum_ii(nums: list[int], target: int) -> list[list[int]]: + """求解子集和 II""" + state = [] # 状态(子集) + nums.sort() # 对 nums 进行排序 + start = 0 # 遍历起始点 + res = [] # 结果列表(子集列表) + backtrack(state, target, nums, start, res) + return res + + +"""Driver Code""" +if __name__ == "__main__": + nums = [4, 4, 5] + target = 9 + res = subset_sum_ii(nums, target) + + print(f"输入数组 nums = {nums}, target = {target}") + print(f"所有和等于 {target} 的子集 res = {res}") diff --git 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