From 76f11ae168747bc27de10e1d66437a532b28ea6e Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Fri, 21 Jul 2023 15:16:51 +0800 Subject: [PATCH] Add the section of max capacity problem. (#639) --- codes/cpp/chapter_greedy/CMakeLists.txt | 3 +- codes/cpp/chapter_greedy/max_capacity.cpp | 39 +++++ codes/java/chapter_greedy/max_capacity.java | 38 +++++ codes/python/chapter_greedy/max_capacity.py | 33 ++++ .../max_capacity_example.png | Bin 0 -> 42003 bytes .../max_capacity_greedy_step1.png | Bin 0 -> 48327 bytes .../max_capacity_greedy_step2.png | Bin 0 -> 54060 bytes .../max_capacity_greedy_step3.png | Bin 0 -> 54593 bytes .../max_capacity_greedy_step4.png | Bin 0 -> 55331 bytes .../max_capacity_greedy_step5.png | Bin 0 -> 55619 bytes .../max_capacity_greedy_step6.png | Bin 0 -> 56198 bytes .../max_capacity_greedy_step7.png | Bin 0 -> 56233 bytes .../max_capacity_greedy_step8.png | Bin 0 -> 56692 bytes .../max_capacity_greedy_step9.png | Bin 0 -> 47079 bytes .../max_capacity_initial_state.png | Bin 0 -> 48705 bytes .../max_capacity_moving_long_board.png | Bin 0 -> 52028 bytes .../max_capacity_moving_short_board.png | Bin 0 -> 51707 bytes .../max_capacity_skipped_states.png | Bin 0 -> 74512 bytes docs/chapter_greedy/max_capacity_problem.md | 158 ++++++++++++++++++ mkdocs.yml | 2 + 20 files changed, 272 insertions(+), 1 deletion(-) create mode 100644 codes/cpp/chapter_greedy/max_capacity.cpp create mode 100644 codes/java/chapter_greedy/max_capacity.java create mode 100644 codes/python/chapter_greedy/max_capacity.py create mode 100644 docs/chapter_greedy/max_capacity_problem.assets/max_capacity_example.png create mode 100644 docs/chapter_greedy/max_capacity_problem.assets/max_capacity_greedy_step1.png create mode 100644 docs/chapter_greedy/max_capacity_problem.assets/max_capacity_greedy_step2.png create mode 100644 docs/chapter_greedy/max_capacity_problem.assets/max_capacity_greedy_step3.png create mode 100644 docs/chapter_greedy/max_capacity_problem.assets/max_capacity_greedy_step4.png create mode 100644 docs/chapter_greedy/max_capacity_problem.assets/max_capacity_greedy_step5.png create mode 100644 docs/chapter_greedy/max_capacity_problem.assets/max_capacity_greedy_step6.png create mode 100644 docs/chapter_greedy/max_capacity_problem.assets/max_capacity_greedy_step7.png create mode 100644 docs/chapter_greedy/max_capacity_problem.assets/max_capacity_greedy_step8.png create mode 100644 docs/chapter_greedy/max_capacity_problem.assets/max_capacity_greedy_step9.png create mode 100644 docs/chapter_greedy/max_capacity_problem.assets/max_capacity_initial_state.png create mode 100644 docs/chapter_greedy/max_capacity_problem.assets/max_capacity_moving_long_board.png create mode 100644 docs/chapter_greedy/max_capacity_problem.assets/max_capacity_moving_short_board.png create mode 100644 docs/chapter_greedy/max_capacity_problem.assets/max_capacity_skipped_states.png create mode 100644 docs/chapter_greedy/max_capacity_problem.md diff --git a/codes/cpp/chapter_greedy/CMakeLists.txt b/codes/cpp/chapter_greedy/CMakeLists.txt index 96ee4d218..91788668d 100644 --- a/codes/cpp/chapter_greedy/CMakeLists.txt +++ b/codes/cpp/chapter_greedy/CMakeLists.txt @@ -1,2 +1,3 @@ add_executable(coin_change_greedy coin_change_greedy.cpp) -add_executable(fractional_knapsack fractional_knapsack.cpp) \ No newline at end of file +add_executable(fractional_knapsack fractional_knapsack.cpp) +add_executable(max_capacity max_capacity.cpp) \ No newline at end of file diff --git a/codes/cpp/chapter_greedy/max_capacity.cpp b/codes/cpp/chapter_greedy/max_capacity.cpp new file mode 100644 index 000000000..99926c42a --- /dev/null +++ b/codes/cpp/chapter_greedy/max_capacity.cpp @@ -0,0 +1,39 @@ +/** + * 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 &ht) { + // 初始化 i, j 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 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; +} diff --git a/codes/java/chapter_greedy/max_capacity.java b/codes/java/chapter_greedy/max_capacity.java new file mode 100644 index 000000000..85e743399 --- /dev/null +++ b/codes/java/chapter_greedy/max_capacity.java @@ -0,0 +1,38 @@ +/** + * File: max_capacity.java + * Created Time: 2023-07-21 + * Author: Krahets (krahets@163.com) + */ + +package chapter_greedy; + +public class max_capacity { + /* 最大容量:贪心 */ + static int maxCapacity(int[] ht) { + // 初始化 i, j 分列数组两端 + int i = 0, j = ht.length - 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 = Math.min(ht[i], ht[j]) * (j - i); + res = Math.max(res, cap); + // 向内移动短板 + if (ht[i] < ht[j]) { + i++; + } else { + j--; + } + } + return res; + } + + public static void main(String[] args) { + int[] ht = { 3, 8, 5, 2, 7, 7, 3, 4 }; + + // 贪心算法 + int res = maxCapacity(ht); + System.out.println("最大容量为 " + res); + } +} diff --git a/codes/python/chapter_greedy/max_capacity.py b/codes/python/chapter_greedy/max_capacity.py new file mode 100644 index 000000000..12377b408 --- /dev/null +++ b/codes/python/chapter_greedy/max_capacity.py @@ -0,0 +1,33 @@ +""" +File: max_capacity.py +Created Time: 2023-07-21 +Author: Krahets (krahets@163.com) +""" + + +def max_capacity(ht: list[int]) -> int: + """最大容量:贪心""" + # 初始化 i, j 分列数组两端 + i, j = 0, len(ht) - 1 + # 初始最大容量为 0 + res = 0 + # 循环贪心选择,直至两板相遇 + while i < j: + # 更新最大容量 + cap = min(ht[i], ht[j]) * (j - i) + res = max(res, cap) + # 向内移动短板 + if ht[i] < ht[j]: + i += 1 + else: + j -= 1 + return res + + +"""Driver Code""" +if __name__ == "__main__": + ht = [3, 8, 5, 2, 7, 7, 3, 4] + + # 贪心算法 + res = max_capacity(ht) + print(f"最大容量为 {res}") diff --git 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,数组中的每个元素代表一个垂直隔板的高度。数组中的任意两个隔板,以及它们之间的空间可以组成一个容器。容器的容量等于高度和宽度的乘积(即面积),其中高度由较短的隔板决定,宽度是两个隔板的数组索引之差。 + + 请在数组中选择两个隔板,使得组成的容器的容量最大,返回最大容量。 + +![最大容量问题的示例数据](max_capacity_problem.assets/max_capacity_example.png) + +**第一步:问题分析** + +容器由任意两个隔板围成,**因此本题的状态为两个隔板的索引,记为 $[i, j]$** 。 + +根据定义,容量等于高度乘以宽度,其中高度由短板决定,宽度是两隔板的索引之差。设容量为 $cap[i, j]$ ,可得计算公式: + +$$ +cap[i, j] = \min(ht[i], ht[j]) \times (j - i) +$$ + +设数组长度为 $n$ ,两个隔板的组合数量(即状态总数)为 $C_n^2 = \frac{n(n - 1)}{2}$ 个。最直接地,**我们可以穷举所有状态**,从而求得最大容量,时间复杂度为 $O(n^2)$ 。 + +**第二步:贪心策略确定** + +当然,这道题还有更高效率的解法。如下图所示,现选取一个状态 $[i, j]$ ,其满足索引 $i < j$ 且高度 $ht[i] < ht[j]$ ,即 $i$ 为短板、 $j$ 为长板。 + +![初始状态](max_capacity_problem.assets/max_capacity_initial_state.png) + +我们发现,**如果将长板 $j$ 向短板 $i$ 靠近,则容量一定变小**。这是因为在移动长板 $j$ 后: + +- 宽度 $j-i$ 肯定变小; +- 高度由短板决定,因此高度只可能不变( $i$ 仍为短板)或变小(移动后的 $j$ 成为短板); + +![向内移动长板后的状态](max_capacity_problem.assets/max_capacity_moving_long_board.png) + +反向思考,**我们只有向内收缩短板 $i$ ,才有可能使容量变大**。因为虽然宽度一定变小,**但高度可能会变大**(移动后的短板 $i$ 变长了)。 + +![向内移动长板后的状态](max_capacity_problem.assets/max_capacity_moving_short_board.png) + +由此便可推出本题的贪心策略: + +1. 初始状态下,指针 $i$ , $j$ 分列与数组两端。 +2. 计算当前状态的容量 $cap[i, j]$ ,并更新最大容量。 +3. 比较板 $i$ 和 板 $j$ 的高度,并将短板向内移动一格。 +4. 循环执行第 `2.` , `3.` 步,直至 $i$ 和 $j$ 相遇时结束。 + +=== "<1>" + ![最大容量问题的贪心过程](max_capacity_problem.assets/max_capacity_greedy_step1.png) + +=== "<2>" + ![max_capacity_greedy_step2](max_capacity_problem.assets/max_capacity_greedy_step2.png) + +=== "<3>" + ![max_capacity_greedy_step3](max_capacity_problem.assets/max_capacity_greedy_step3.png) + +=== "<4>" + ![max_capacity_greedy_step4](max_capacity_problem.assets/max_capacity_greedy_step4.png) + +=== "<5>" + ![max_capacity_greedy_step5](max_capacity_problem.assets/max_capacity_greedy_step5.png) + +=== "<6>" + ![max_capacity_greedy_step6](max_capacity_problem.assets/max_capacity_greedy_step6.png) + +=== "<7>" + ![max_capacity_greedy_step7](max_capacity_problem.assets/max_capacity_greedy_step7.png) + +=== "<8>" + ![max_capacity_greedy_step8](max_capacity_problem.assets/max_capacity_greedy_step8.png) + +=== "<9>" + ![max_capacity_greedy_step9](max_capacity_problem.assets/max_capacity_greedy_step9.png) + +代码实现如下所示。最多循环 $n$ 轮,**因此时间复杂度为 $O(n)$** 。变量 $i$ , $j$ , $res$ 使用常数大小额外空间,**因此空间复杂度为 $O(1)$** 。 + +=== "Java" + + ```java title="max_capacity.java" + [class]{max_capacity}-[func]{maxCapacity} + ``` + +=== "C++" + + ```cpp title="max_capacity.cpp" + [class]{}-[func]{maxCapacity} + ``` + +=== "Python" + + ```python title="max_capacity.py" + [class]{}-[func]{max_capacity} + ``` + +=== "Go" + + ```go title="max_capacity.go" + [class]{}-[func]{maxCapacity} + ``` + +=== "JavaScript" + + ```javascript title="max_capacity.js" + [class]{}-[func]{maxCapacity} + ``` + +=== "TypeScript" + + ```typescript title="max_capacity.ts" + [class]{}-[func]{maxCapacity} + ``` + +=== "C" + + ```c title="max_capacity.c" + [class]{}-[func]{maxCapacity} + ``` + +=== "C#" + + ```csharp title="max_capacity.cs" + [class]{max_capacity}-[func]{maxCapacity} + ``` + +=== "Swift" + + ```swift title="max_capacity.swift" + [class]{}-[func]{maxCapacity} + ``` + +=== "Zig" + + ```zig title="max_capacity.zig" + [class]{}-[func]{maxCapacity} + ``` + +=== "Dart" + + ```dart title="max_capacity.dart" + [class]{}-[func]{maxCapacity} + ``` + +**第三步:正确性证明** + +之所以贪心比穷举更快,是因为每轮的贪心选择都会“跳过”一些状态。 + +比如在状态 $cap[i, j]$ 下,$i$ 为短板、$j$ 为长板。若贪心地将短板 $i$ 向内移动一格,会导致以下状态被“跳过”,**意味着之后无法验证这些状态的容量大小**。 + +$$ +cap[i, i+1], cap[i, i+2], \cdots, cap[i, j-2], cap[i, j-1] +$$ + +![移动短板导致被跳过的状态](max_capacity_problem.assets/max_capacity_skipped_states.png) + +观察发现,**这些被跳过的状态实际上就是将长板 $j$ 向内移动的所有状态**。而在第二步中,我们已经证明内移长板一定会导致容量变小,也就是说这些被跳过的状态的容量一定更小。 + +也就是说,被跳过的状态都不可能是最优解,**跳过它们不会导致错过最优解**。 + +以上的分析说明,**移动短板的操作是“安全”的**,贪心策略是有效的。 diff --git a/mkdocs.yml b/mkdocs.yml index c6224906f..b79ebcd4d 100644 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -269,6 +269,8 @@ nav: - 15.1.   贪心算法: chapter_greedy/greedy_algorithm.md # [status: new] - 15.2.   分数背包问题: chapter_greedy/fractional_knapsack_problem.md + # [status: new] + - 15.3.   最大容量问题: chapter_greedy/max_capacity_problem.md - 16.   附录: # [icon: material/help-circle-outline] - chapter_appendix/index.md