From 785f30672673c44c4806da821c9733fd85a4d78e Mon Sep 17 00:00:00 2001 From: CGH0S7 <776459475@qq.com> Date: Thu, 18 Dec 2025 16:00:22 +0800 Subject: [PATCH] first commit --- dynamic/knapsack | Bin 0 -> 22504 bytes dynamic/knapsack.cpp | 159 + dynamic/labtemplate.typ | 166 + dynamic/main.pdf | 9892 +++++++++++++++++ dynamic/main.typ | 281 + dynamic/ops_vs_n.png | Bin 0 -> 224147 bytes dynamic/plotter.py | 56 + dynamic/results.csv | 16 + dynamic/task.txt | 12 + dynamic/time_vs_n.png | Bin 0 -> 186996 bytes dynamic/算法分析实验二-动态规划-c++.pdf | 9892 +++++++++++++++++ dynamic/算法分析实验二.zip | Bin 0 -> 958984 bytes greed/labtemplate.typ | 166 + greed/main.pdf | 9442 ++++++++++++++++ greed/main.typ | 202 + greed/multimachine | Bin 0 -> 46640 bytes greed/results/algo_comparison.csv | 331 + greed/results/gpu_sim_plots.png | Bin 0 -> 35535 bytes greed/results/gpu_sim_results.csv | 10 + greed/results/ratio_boxplot.png | Bin 0 -> 27675 bytes greed/results/ratio_vs_n.png | Bin 0 -> 62094 bytes greed/results/time_comparison.png | Bin 0 -> 38618 bytes greed/results/worst_case_verification.csv | 5 + greed/src/gpu_sim.py | 224 + greed/src/multimachine.cpp | 273 + greed/src/plot_algo.py | 58 + greed/task.txt | 27 + horse_travel/README.md | 27 + horse_travel/main | Bin 0 -> 2426622 bytes horse_travel/main.go | 149 + horse_travel/实验4-go.docx | Bin 0 -> 38399 bytes horse_travel/实验报告.md | 148 + horse_travel/屏幕截图_20251113_154902.png | Bin 0 -> 91892 bytes sort_closet/code-sorting/BubbleSort.cpp | 35 + .../code-sorting/DualPivotQuickSort.cpp | 67 + sort_closet/code-sorting/InsertSort.cpp | 33 + sort_closet/code-sorting/Makefile | 36 + sort_closet/code-sorting/MergeSort.cpp | 86 + sort_closet/code-sorting/QuickSort.cpp | 48 + sort_closet/code-sorting/QuickSort3Way.cpp | 70 + .../code-sorting/QuickSortOptimized.cpp | 81 + sort_closet/code-sorting/README-zh.md | 82 + sort_closet/code-sorting/README.md | 77 + sort_closet/code-sorting/REPORT.md | 98 + sort_closet/code-sorting/ShellSort.cpp | 32 + sort_closet/code-sorting/algorithm.h | 20 + .../average_comparisons_vs_size.png | Bin 0 -> 157436 bytes .../code-sorting/average_moves_vs_size.png | Bin 0 -> 155625 bytes .../code-sorting/average_time_vs_size.png | Bin 0 -> 150401 bytes sort_closet/code-sorting/main.cpp | 110 + sort_closet/code-sorting/out.cpp | 16 + sort_closet/code-sorting/out.h | 6 + sort_closet/code-sorting/plot_results.py | 138 + sort_closet/code-sorting/实验一-C.zip | Bin 0 -> 1687569 bytes sort_closet/code-sorting/实验一-c.docx | Bin 0 -> 1259826 bytes sort_closet/code_template-DP.tar | Bin 0 -> 30208 bytes sort_closet/code_template-DP/Makefile | 16 + sort_closet/code_template-DP/README.md | 29 + sort_closet/code_template-DP/algorithm.cpp | 144 + sort_closet/code_template-DP/algorithm.h | 17 + sort_closet/code_template-DP/analyser.cpp | 98 + sort_closet/code_template-DP/data_gen.cpp | 81 + sort_closet/code_template-DP/main.cpp | 57 + sort_closet/code_template-DP/test.cpp | 122 + sort_closet/code_template-DP/test.h | 14 + sort_closet/code_template-DP/test.sh | 22 + sort_closet/code_template-sorting.zip | Bin 0 -> 2688 bytes ...报告模板-提交时重命名为“实验x-编程语言”.docx | Bin 0 -> 22906 bytes ...报告模板-提交时重命名为“实验x-编程语言”.docx | Bin 0 -> 22906 bytes 69 files changed, 33171 insertions(+) create mode 100755 dynamic/knapsack create mode 100644 dynamic/knapsack.cpp create mode 100644 dynamic/labtemplate.typ create mode 100644 dynamic/main.pdf create mode 100644 dynamic/main.typ create mode 100644 dynamic/ops_vs_n.png create mode 100644 dynamic/plotter.py create mode 100644 dynamic/results.csv create mode 100644 dynamic/task.txt create mode 100644 dynamic/time_vs_n.png create mode 100644 dynamic/算法分析实验二-动态规划-c++.pdf create mode 100644 dynamic/算法分析实验二.zip create mode 100644 greed/labtemplate.typ create mode 100644 greed/main.pdf create mode 100644 greed/main.typ create mode 100755 greed/multimachine create mode 100644 greed/results/algo_comparison.csv create mode 100644 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+1,159 @@ +#include +#include +#include +#include +#include +#include + +struct Item { + int weight; // 物品重量 + int value; // 物品价值 + int count; // 物品数量（用于多重背包问题） +}; + +// 全局操作计数器，用于性能分析 +long long ops_count = 0; + +// ================================================================= +// 完全背包问题算法实现 +// ================================================================= + +// 算法1：朴素动态规划（三层循环） +int complete_knapsack_v1(const std::vector& items, int capacity) { + ops_count = 0; // 重置操作计数器 + int n = items.size(); + if (n == 0) return 0; + std::vector> dp(n + 1, std::vector(capacity + 1, 0)); + + // 遍历每个物品 + for (int i = 1; i <= n; ++i) { + int w = items[i - 1].weight; + int v = items[i - 1].value; + // 遍历每个容量 + for (int j = 0; j <= capacity; ++j) { + dp[i][j] = dp[i-1][j]; // 不选择第i个物品 + ops_count++; + // 尝试选择k个第i个物品 + for (int k = 1; k * w <= j; ++k) { + ops_count++; + if (dp[i-1][j - k * w] + k * v > dp[i][j]) { + dp[i][j] = dp[i-1][j - k * w] + k * v; + } + } + } + } + return dp[n][capacity]; +} + +// 算法2：优化的二维动态规划（两层循环） +int complete_knapsack_v2(const std::vector& items, int capacity) { + ops_count = 0; // 重置操作计数器 + int n = items.size(); + if (n == 0) return 0; + std::vector> dp(n + 1, std::vector(capacity + 1, 0)); + + // 遍历每个物品 + for (int i = 1; i <= n; ++i) { + int w = items[i - 1].weight; + int v = items[i - 1].value; + // 遍历每个容量 + for (int j = 0; j <= capacity; ++j) { + ops_count++; + if (j < w) { + dp[i][j] = dp[i - 1][j]; // 容量不足，不能选择第i个物品 + } else { + // 选择不选第i个物品或选择第i个物品的最大值 + dp[i][j] = std::max(dp[i - 1][j], dp[i][j - w] + v); + } + } + } + return dp[n][capacity]; +} + +// 算法3：空间优化的动态规划（一维数组） +int complete_knapsack_v3(const std::vector& items, int capacity) { + ops_count = 0; // 重置操作计数器 + std::vector dp(capacity + 1, 0); // 一维DP数组 + + // 遍历每个物品 + for (const auto& item : items) { + // 从物品重量开始遍历容量（完全背包） + for (int j = item.weight; j <= capacity; ++j) { + ops_count++; + dp[j] = std::max(dp[j], dp[j - item.weight] + item.value); + } + } + return dp[capacity]; +} + + +// ================================================================= +// 基准测试运行器 +// ================================================================= +void run_experiments(int min_n, int max_n, int step_n, int trials, int capacity) { + // 初始化随机数生成器 + std::random_device rd; + std::mt19937 gen(rd()); + std::uniform_int_distribution<> weight_dist(1, 40); // 重量范围：1-40 + std::uniform_int_distribution<> value_dist(1, 100); // 价值范围：1-100 + + // 输出CSV格式的表头 + std::cout << "n,algo,time_us,ops\n"; + + // 对不同物品数量进行测试 + for (int n = min_n; n <= max_n; n += step_n) { + if (n==0) continue; + long long total_time_v1 = 0, total_ops_v1 = 0; + long long total_time_v2 = 0, total_ops_v2 = 0; + long long total_time_v3 = 0, total_ops_v3 = 0; + + // 进行多次试验取平均值 + for (int t = 0; t < trials; ++t) { + // 生成随机物品 + std::vector items(n); + for (int i = 0; i < n; ++i) { + items[i] = {weight_dist(gen), value_dist(gen), 0}; + } + + // 测试算法1的运行时间 + auto start_v1 = std::chrono::high_resolution_clock::now(); + complete_knapsack_v1(items, capacity); + auto end_v1 = std::chrono::high_resolution_clock::now(); + total_time_v1 += std::chrono::duration_cast(end_v1 - start_v1).count(); + total_ops_v1 += ops_count; + + // 测试算法2的运行时间 + auto start_v2 = std::chrono::high_resolution_clock::now(); + complete_knapsack_v2(items, capacity); + auto end_v2 = std::chrono::high_resolution_clock::now(); + total_time_v2 += std::chrono::duration_cast(end_v2 - start_v2).count(); + total_ops_v2 += ops_count; + + // 测试算法3的运行时间 + auto start_v3 = std::chrono::high_resolution_clock::now(); + complete_knapsack_v3(items, capacity); + auto end_v3 = std::chrono::high_resolution_clock::now(); + total_time_v3 += std::chrono::duration_cast(end_v3 - start_v3).count(); + total_ops_v3 += ops_count; + } + + // 输出每个算法的平均结果 + std::cout << n << ",v1," << total_time_v1 / trials << "," << total_ops_v1 / trials << "\n"; + std::cout << n << ",v2," << total_time_v2 / trials << "," << total_ops_v2 / trials << "\n"; + std::cout << n << ",v3," << total_time_v3 / trials << "," << total_ops_v3 / trials << "\n"; + } +} + +int main(int argc, char* argv[]) { + // 实验参数：最小物品数、最大物品数、步长、每个物品数的试验次数、背包容量 + // 为保持算法1的合理运行时间，使用较小的容量和物品数量 + int min_n = 5; // 最小物品数 + int max_n = 25; // 最大物品数 + int step_n = 5; // 物品数步长 + int trials = 10; // 每个物品数的试验次数 + int capacity = 100; // 背包容量 + + run_experiments(min_n, max_n, step_n, trials, capacity); + + return 0; +} diff --git a/dynamic/labtemplate.typ b/dynamic/labtemplate.typ new file mode 100644 index 0000000..2af79c0 --- /dev/null +++ b/dynamic/labtemplate.typ @@ -0,0 +1,166 @@ +#let times = "Times LT Pro" +#let times = "Times New Roman" +#let song = (times, "Noto Serif CJK SC") +#let hei = (times, "Noto Sans CJK SC") +#let kai = (times, "Noto Serif CJK SC") +#let xbsong = (times, "Noto Serif CJK SC") +#let fsong = (times, "Noto Serif CJK SC") +#let code = (times, "JetBrains Mono") +#let nudtlabpaper(title: "", + author1: "", + id1: "", + advisor: "", + jobtitle: "", + lab: "", + date: "", + header_str: "", + minimal_cover: false, + body) = { + // Set the document's basic properties. + set document(author: author1, title: title) + set page( + + margin: (left: 30mm, right: 30mm, top: 30mm, bottom: 30mm), + ) + + // If minimal_cover is requested, render an otherwise-empty first page + // that only displays the "实验时间" near the bottom center. + if minimal_cover { + v(158pt) + align(center)[ + #block(text(weight: 700, size: 30pt, font: hei, tracking: 1pt, "2025秋 -《算法设计与分析》")) + ] + align(center)[ + #block(text(weight: 700, size: 24pt, font: song, tracking: 1pt, "动态规划算法分析实验报告")) + ] + + + // Keep standard margins but push content down toward the bottom. + v(220pt) + align(center)[ + #block(text(size: 14pt, font: song, tracking: 9pt, "实验时间")) + ] + v(2pt) + align(center)[ + #block(text(size: 16pt, font: song, date)) + ] + pagebreak() + } else { + // Title row. + v(158pt) + align(center)[ + #block(text(weight: 700, size: 30pt, font: hei, tracking: 1pt, "2025秋 -《算法设计与分析》")) + ] + align(center)[ + #block(text(weight: 700, size: 24pt, font: song, tracking: 1pt, "动态规划算法分析实验报告")) + ] + + v(103pt) + pad( + left: 1em, + right: 1em, + grid( + // columns: (80pt, 1fr), + // rows: (17pt, auto), + // text(weight: 700, size: 16pt, font: song, "实验名称："), + // align(center, text(weight: "regular", size: 16pt, font: song, title)), + // text(""), + // line(length: 100%) + ) + // #block(text(weight: 700, 1.75em, title)) + // underline(text(weight: 700, size: 16pt, font: song, title)) + ) + + // Author information. + + v(62.5pt) + + grid( + columns: (0.25fr, 0.25fr, 0.25fr, 0.25fr), + rows: (20pt, 8pt, 20pt, 8pt, 20pt, 8pt, 20pt, 12pt), + text(size: 14pt, font: song, tracking: 9pt, "学员姓名"), + align(center, text(size: 14pt, font: song, author1)), + text(size: 14pt, font: song, tracking: 54pt, "学号"), + align(center, text(size: 14pt, font: times, id1)), + text(""), + line(length: 100%), + text(""), + line(length: 100%), + text(size: 14pt, font: song, tracking: 9pt, "指导教员"), + align(center, text(size: 14pt, font: song, advisor)), + text(size: 14pt, font: song, tracking: 54pt, "职称"), + align(center, text(size: 14pt, font: song, jobtitle)), + text(""), + line(length: 100%), + text(""), + line(length: 100%), + text(size: 14pt, font: song, tracking: 9pt, "实验室"), + align(center, text(size: 14pt, font: song, lab)), + text(size: 14pt, font: song, tracking: 9pt, "实验时间"), + align(center, text(size: 14pt, font: song, date)), + text(""), + line(length: 100%), + text(""), + line(length: 100%), + ) + + v(50.5pt) + align(center, text(font: hei, size: 15pt, "国防科技大学教育训练部制")) + + pagebreak() + } + + set page( + margin: (left: 30mm, right: 30mm, top: 30mm, bottom: 30mm), + numbering: "i", + number-align: center, + ) + + v(14pt) + align(center)[ + #block(text(font: hei, size: 14pt, "《本科实验报告》填写说明")) + ] + + v(14pt) + text("") + par(first-line-indent: 2em, text(font: song, size: 12pt, "实验报告内容编排应符合以下要求：")) + + par(first-line-indent: 2em, text(font: fsong, size: 12pt, "（1）采用A4（21cm×29.7cm）白色复印纸，单面黑字。上下左右各侧的页边距均为3cm；缺省文档网格：字号为小4号，中文为宋体，英文和阿拉伯数字为Times New Roman，每页30行，每行36字；页脚距边界为2.5cm，页码置于页脚、居中，采用小5号阿拉伯数字从1开始连续编排，封面不编页码。")) + + par(first-line-indent: 2em, text(font: fsong, size: 12pt, "（2）报告正文最多可设四级标题，字体均为黑体，第一级标题字号为4号，其余各级标题为小4号；标题序号第一级用“一、”、“二、”……，第二级用“（一）”、“（二）” ……，第三级用“1.”、“2.” ……，第四级用“（1）”、“（2）” ……，分别按序连续编排。")) + + par(first-line-indent: 2em, text(font: fsong, size: 12pt, "（3）正文插图、表格中的文字字号均为5号。")) + + pagebreak() + + set page( + margin: (left: 30mm, right: 30mm, top: 30mm, bottom: 30mm), + numbering: "1", + number-align: center, + ) + + set heading(numbering: "1.1") + // set text(font: hei, lang: "zh") + + show heading: it => box(width: 100%)[ + #v(0.50em) + #set text(font: hei) + #counter(heading).display() + // #h(0.5em) + #it.body + ] + // Main body. + set par(justify: true) + + body +} + +#let para(t) = par(first-line-indent: 2em, text(font: song, size: 10.5pt, t)) +#let subpara(t) = 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0 R 97 0 R 97 0 R 97 0 R 94 0 R 94 0 R 96 0 R 94 0 R 94 0 R 94 0 R 94 0 R 95 0 R 95 0 R 95 0 R 95 0 R 94 0 R 94 0 R 93 0 R 93 0 R 93 0 R] +endobj + +28 0 obj +[92 0 R 92 0 R 89 0 R 89 0 R 69 0 R 69 0 R 86 0 R 69 0 R 69 0 R 69 0 R 69 0 R 85 0 R 85 0 R 69 0 R 69 0 R 69 0 R 69 0 R 69 0 R 84 0 R 84 0 R 84 0 R 84 0 R 84 0 R 84 0 R 84 0 R 69 0 R 69 0 R 69 0 R 83 0 R 69 0 R 69 0 R 69 0 R 82 0 R 69 0 R 69 0 R 69 0 R 69 0 R 69 0 R 69 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 81 0 R 69 0 R 69 0 R 69 0 R 80 0 R 80 0 R 69 0 R 79 0 R 79 0 R 69 0 R 78 0 R 78 0 R 69 0 R 69 0 R 69 0 R 77 0 R 69 0 R 69 0 R 69 0 R 76 0 R 69 0 R 69 0 R 69 0 R 75 0 R 69 0 R 69 0 R 69 0 R 69 0 R 69 0 R 74 0 R 74 0 R 74 0 R 74 0 R 74 0 R 74 0 R 74 0 R 74 0 R 69 0 R 69 0 R 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62 0 R 62 0 R 61 0 R 61 0 R 61 0 R 61 0 R 61 0 R 61 0 R 61 0 R 61 0 R 61 0 R 61 0 R 61 0 R 61 0 R 61 0 R 61 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 57 0 R 57 0 R 57 0 R 57 0 R 57 0 R 57 0 R 57 0 R 57 0 R 57 0 R 57 0 R 57 0 R 57 0 R 57 0 R 57 0 R 57 0 R 57 0 R 57 0 R 57 0 R 57 0 R 57 0 R 57 0 R 57 0 R 57 0 R 57 0 R 57 0 R 57 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 55 0 R 55 0 R 54 0 R 54 0 R 53 0 R 53 0 R 52 0 R 52 0 R 52 0 R 52 0 R 52 0 R 52 0 R 52 0 R 52 0 R 52 0 R 52 0 R 51 0 R 49 0 R 49 0 R 49 0 R 49 0 R 49 0 R 49 0 R 47 0 R 47 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 44 0 R 34 0 R 34 0 R 34 0 R 43 0 R 43 0 R 34 0 R 34 0 R 42 0 R 42 0 R 42 0 R 42 0 R 34 0 R 34 0 R 34 0 R 41 0 R 41 0 R 41 0 R 41 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 40 0 R 34 0 R 34 0 R 34 0 R 39 0 R 39 0 R 39 0 R 39 0 R 39 0 R 39 0 R 39 0 R 39 0 R 39 0 R 39 0 R 34 0 R 34 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 34 0 R 34 0 R 37 0 R 34 0 R 34 0 R 34 0 R 34 0 R 36 0 R 34 0 R 34 0 R 34 0 R 35 0 R 35 0 R 34 0 R 34 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 33 0 R 33 0 R 33 0 R 30 0 R 30 0 R 30 0 R 32 0 R 32 0 R 32 0 R 32 0 R 30 0 R 30 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 30 0 R] +endobj + +29 0 obj +<< + /Type /StructElem + /S /Document + /P 19 0 R + /K [430 0 R 429 0 R 428 0 R 427 0 R 426 0 R 423 0 R 359 0 R 356 0 R 347 0 R 344 0 R 343 0 R 340 0 R 339 0 R 318 0 R 315 0 R 314 0 R 298 0 R 297 0 R 290 0 R 283 0 R 268 0 R 246 0 R 244 0 R 241 0 R 229 0 R 209 0 R 207 0 R 204 0 R 194 0 R 182 0 R 180 0 R 177 0 R 169 0 R 159 0 R 156 0 R 153 0 R 149 0 R 144 0 R 135 0 R 134 0 R 131 0 R 129 0 R 125 0 R 121 0 R 120 0 R 103 0 R 102 0 R 99 0 R 98 0 R 94 0 R 93 0 R 90 0 R 87 0 R 69 0 R 50 0 R 48 0 R 45 0 R 34 0 R 30 0 R] +>> +endobj + +30 0 obj +<< + /Type /StructElem + /S /P + /P 29 0 R + /K [505 506 507 508 509 510 511 512 513 514 515 516 33 0 R 520 521 522 32 0 R 527 528 31 0 R 538] + /Pg 571 0 R +>> +endobj + +31 0 obj +<< + /Type /StructElem + /S /Formula + /P 30 0 R + /K [529 530 531 532 533 534 535 536 537] + /Pg 571 0 R +>> +endobj + +32 0 obj +<< + /Type /StructElem + /S /Formula + /P 30 0 R + /K [523 524 525 526] + /Pg 571 0 R +>> +endobj + +33 0 obj +<< + /Type /StructElem + /S /Formula + /P 30 0 R + /K [517 518 519] + /Pg 571 0 R +>> +endobj + +34 0 obj +<< + /Type /StructElem + /S /P + /P 29 0 R + /K [429 430 431 432 433 434 435 436 44 0 R 438 439 440 43 0 R 443 444 42 0 R 449 450 451 41 0 R 456 457 458 459 460 461 462 40 0 R 464 465 466 39 0 R 477 478 38 0 R 490 491 37 0 R 493 494 495 496 36 0 R 498 499 500 35 0 R 503 504] + /Pg 571 0 R +>> +endobj + +35 0 obj +<< + /Type /StructElem + /S /Formula + /P 34 0 R + /K [501 502] + /Pg 571 0 R +>> +endobj + +36 0 obj +<< + /Type /StructElem + /S /Formula + /P 34 0 R + /K [497] + /Pg 571 0 R +>> +endobj + +37 0 obj +<< + /Type /StructElem + /S /Formula + /P 34 0 R + /K [492] + /Pg 571 0 R +>> +endobj + +38 0 obj +<< + /Type /StructElem + /S /Formula + /P 34 0 R + /K [479 480 481 482 483 484 485 486 487 488 489] + /Pg 571 0 R +>> +endobj + +39 0 obj +<< + /Type /StructElem + /S /Formula + /P 34 0 R + /K [467 468 469 470 471 472 473 474 475 476] + /Pg 571 0 R +>> +endobj + +40 0 obj +<< + /Type /StructElem + /S /Formula + /P 34 0 R + /K [463] + /Pg 571 0 R +>> +endobj + +41 0 obj +<< + /Type /StructElem + /S /Formula + /P 34 0 R + /K [452 453 454 455] + /Pg 571 0 R +>> +endobj + +42 0 obj +<< + /Type /StructElem + /S /Formula + /P 34 0 R + /K [445 446 447 448] + /Pg 571 0 R +>> +endobj + +43 0 obj +<< + /Type /StructElem + /S /Formula + /P 34 0 R + /K [441 442] + /Pg 571 0 R +>> +endobj + +44 0 obj +<< + /Type /StructElem + /S /Formula + /P 34 0 R + /K [437] + /Pg 571 0 R +>> +endobj + +45 0 obj +<< + /Type /StructElem + /S /P + /P 29 0 R + /K [46 0 R] +>> +endobj + +46 0 obj +<< + /Type /StructElem + /S /H2 + /P 45 0 R + /T + /K [47 0 R] +>> +endobj + +47 0 obj +<< + /Type /StructElem + /S /P + /P 46 0 R + /K [427 428] + /Pg 571 0 R +>> +endobj + +48 0 obj +<< + /Type /StructElem + /S /P + /P 29 0 R + /K [49 0 R] +>> +endobj + +49 0 obj +<< + /Type /StructElem + /S /Em + /P 48 0 R + /K [421 422 423 424 425 426] + /Pg 571 0 R +>> +endobj + +50 0 obj +<< + /Type /StructElem + /S /Code + /P 29 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K 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+<0085> <9488> +<0086> <5728> +<0087> <4EE3> +<0088> <7801> +<0089> <4E2D> +<008A> <5982> +<008B> <67E5> +<008C> <8868> +<008D> <7B49> +<008E> <5C0F> +<008F> <56FA> +<0090> <5B9A> +<0091> <4EA7> +<0092> <751F> +<0093> <548C> +<0094> <753B> +<0095> <5230> +<0096> <7406> +<0097> <8BBA> +<0098> <503C> +<0099> <7167> +<009A> <4F7F> +<009B> <65B9> +<009C> <89E3> +<009D> <51B3> +<009E> <6574> +<009F> <5176> +<00A0> <5373> +<00A1> <6BCF> +<00A2> <6709> +<00A3> <9650> +<00A4> <7B2C> +<00A5> <4E2A> +<00A6> +<00A7> <66F2> +<00A8> <7EBF> +<00A9> <91C7> +<00AA> <5750> +<00AB> <6807> +<00AC> <663E> +<00AD> <793A> +<00AE> <66F4> +<00AF> <597D> +<00B0> <5730> +<00B1> <5C55> +<00B2> <7EA7> +<00B3> <5DEE> +<00B4> <5F02> +<00B5> <65E0> +<00B6> <662F> +<00B7> <8FD8> +<00B8> <90FD> +<00B9> <8457> +<00BA> <9AD8> +<00BB> <589E> +<00BC> <957F> +<00BD> <8D8B> +<00BE> <52BF> +<00BF> <590D> +<00C0> <6742> +<00C1> <5EA6> +<00C2> <543B> +<00C3> <5408> +<00C4> <660E> +<00C5> <8BE5> +<00C6> <9645> +<00C7> <5E94> +<00C8> <6548> +endbfchar +41 beginbfchar +<00C9> <7387> +<00CA> <6781> +<00CB> <4F4E> +<00CC> <5C24> +<00CD> <7A0D> +<00CE> <5448> +<00CF> <51FA> +<00D0> <76F8> +<00D1> <8FD9> +<00D2> <81F4> +<00D3> <4E86> +<00D4> <5E45> +<00D5> <63D0> +<00D6> <5347> +<00D7> <4F3C> +<00D8> <5B83> +<00D9> <4E5F> +<00DA> <4FDD> +<00DB> <6301> +<00DC> <8BC1> +<00DD> <524D> +<00DE> <964D> +<00DF> <6D88> +<00E0> <8017> +<00E1> <867D> +<00E2> <7136> +<00E3> <4F46> +<00E4> <7531> +<00E5> <5B58> +<00E6> <8BBF> +<00E7> <914D> +<00E8> <7F13> +<00E9> <5C40> +<00EA> <90E8> +<00EB> <67D0> +<00EC> <4E9B> +<00ED> <60C5> +<00EE> <51B5> +<00EF> <4F1A> +<00F0> <7EC6> +<00F1> <5FAE> +endbfchar +endcmap +CMapName currentdict /CMap defineresource pop +end +end +%%EndResource +%%EOF +endstream +endobj + +493 0 obj +<< + /Length 43285 + /Filter /FlateDecode +>> +stream +xyt#y'GfFDF'n$@$A.YUUduT,VjutlGz~k,ݒגFݒyvtZeɲط D[~C ddd~ gI_!೏ѧp?>Gz#`>?/=vg?T~ 0cwvF{їHkO=_ M7;ϽQ ް?P{3oyw|獛$;%Ϸsw +PldX`l@@ 0Xr\%- +@K]@ 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DHYK.叅9CFͨ>;9K{ 7d=dkwO:oB / +s}@Vo>tЎ2nll<裇zYg :裏3rO>%/z C߯zE%xB T32~;Kwʯ<֯_QlMy!n{=_z> [---^{N30;$n' =dy[8$hi _ܷo444K.y^rm@嵴d#۶;@f\P=V^=cƌ뮻nȐ!Gyd~ۯ_#y3c64WWD7w=+qHݿ~֬Hzs+H"] ۰a[eeOoT$DbxߟO|~굛^ in"]@ t3*'QGrꙍ> Lt. ]ŻGfHzg{H"] kjjj\.7rȬSli8dB tQ8$}՚WGUi"]@ t3*o„ !\.裏fzfO!"]@ zaz;qH䳕Yj Ӏ t. ES?OfܹY/pȄHŬ?Hx}"- ]@ t.p}@͟?y2ӦMz9FC&DH˻xŇN?tw>zuT^t. DHi>֭[Wr7|s˩g642!DſY8$sؐQ/8QuzOyH"] Lr)<3}"]@xʘğKf:Q/I. DHi> f*rYfeeO!"]@ ],tQߙ8$d:ToH. DHi> +gqFiT?r꓍> Lt.v5ǜ5.qH^ut. DHi> +k׮=#K2r^Qѧ . .^=qHf܃ŬH4kt. DHZvmSergϮY&u}"]@bM!;ϐ 6 2]@ t.p}@&aÆ7˕FeJ9?O?s:묬*eO!"]@ =]۰MJn|!R5 HC"]@4\S1e WsJ R642!D^Ŧ-Ͼ!+ny=u t. EDy 0dFF{{۳^dOzCZ{!{ſ"]@ t32Qni&dvF)_rC2ij6l%İTlF{{O=8$s֥7mޖ= ꘍P(0!S.[^ \@@;&8$so'nc 1=tAXCd߿Z.] ꘍'}ݟ-_s}uFF>_A>zE=]@ .x!NOWd:jX=A"]@4\@ѧ񏮭-At.κGC2O·_e:j[M=D"]@4\@ѧ . SX!L!ֲWGͫ4"]@ t3r7__+WܣXQFk> Lt.nx/?yt9t^$Qj4 QH"] L 8?\.xXc=ѧ . G/S&ĥf:D-=M"]@4\k?rӻӧp-끍> Lt.X٪qW#3zuԏK*@"]@4\rƍ7v 6mjll,=裏5}"]@wWky=C2MW֫VPH"] [re.vi]~ΰa)w넍> Lt.vՆso̝\,^ uҀDH" gT޿Lt뭷dxn]cѧ . nk<䂦![[( t. 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00000 n +0000359035 00000 n +0000360832 00000 n +0000361233 00000 n +0000363931 00000 n +0000364368 00000 n +0000372679 00000 n +0000378140 00000 n +0000548302 00000 n +0000553763 00000 n +0000758202 00000 n +0000758358 00000 n +0000759515 00000 n +trailer +<< + /Size 580 + /Root 579 0 R + /Info 577 0 R + /ID [(Q+qmWLoJItDFiJaAQa8ykw==) (Q+qmWLoJItDFiJaAQa8ykw==)] +>> +startxref +759774 +%%EOF \ No newline at end of file diff --git a/dynamic/main.typ b/dynamic/main.typ new file mode 100644 index 0000000..0dcd3c9 --- /dev/null +++ b/dynamic/main.typ @@ -0,0 +1,281 @@ +#import "labtemplate.typ": * +#show: nudtlabpaper.with( + author1: "程景愉", + id1: "202302723005", + advisor: " 胡罡", + jobtitle: "教授", + lab: "306-707", + date: "2025.11.28", + header_str: "动态规划算法分析实验报告", + minimal_cover: true, +) + +#set page(header: [ + #set par(spacing: 6pt) + #align(center)[#text(size: 11pt)[《算法设计与分析》实验报告]] + #v(-0.3em) + #line(length: 100%, stroke: (thickness: 1pt)) +],) + +#show heading: it => box(width: 100%)[ + #v(0.50em) + #set text(font: hei) + #it.body +] + +#outline(title: "目录",depth: 3, indent: 1em) +// #pagebreak() +#outline( + title: [图目录], + target: figure.where(kind: image), +) + +#show heading: it => box(width: 100%)[ + #v(0.50em) + #set text(font: hei) + #counter(heading).display() + #it.body +] +#set enum(indent: 0.5em,body-indent: 0.5em,) +#pagebreak() + += 实验介绍 +#para[ +动态规划（Dynamic Programming, DP）是一种通过把原问题分解为相对简单的子问题的方式来求解复杂问题的方法。它常用于优化问题，其中，问题的最优解可以通过子问题的最优解来构造。本实验旨在深入理解动态规划算法在解决背包问题中的应用，特别是完全背包问题及其优化，并通过实验数据分析不同实现方式的性能差异。 +] + += 实验内容 +#para[ +本实验主要围绕动态规划算法解决完全背包问题展开，并涉及多重背包问题的初步分析。具体内容包括： +] + + 实现两种基于不同递推公式的完全背包动态规划算法。, + + 对所实现的算法进行插桩，记录关键操作次数。, + + 以物品种类数量 $n$ 为输入规模，通过大量随机测试样本，统计不同算法的平均运行时间与关键操作次数。, + + 改变物品种类规模 $n$，对比分析不同规模下各算法的性能，并利用 Python 绘制数据图。, + + 实现完全背包问题的一维数组空间优化版本，并与上述算法进行对比。, + + （附加）对多重背包问题实现至少两种动态规划算法，并进行性能分析。, += 实验要求 +#para[ +运用动态规划算法求解完全背包问题并进行分析，具体要求如下： +] + + 针对完全背包问题，实现基于两种递推公式的动态规划算法。 + + 在代码中插桩，记录关键操作次数（如查表次数等）。 + + 以物品种类的大小n为输入规模，固定n，随机产生大量测试样本，统计两种算法的平均运行时间和关键操作次数，并进行记录。 + + 改变物品种类规模，对不同规模问题各算法的结果对比分析，通过统计python画图插入到报告中记录，与理论值进行对照分析。 + + 使用一维数组的方式解决整数背包问题，并记录其平均运行时间和关键操作次数，与上述两种算法进行对比。 +#para[ +附加：运用动态规划算法求解多重背包问题并进行分析，具体要求如下： +] + + 多重背包即每种物品的数量有限，第i种物品的数量上限为ki个； + + 对多重背包问题实现两种以上动态规划算法，并对其性能进行分析。 += 实验步骤 +== 算法设计 +=== 完全背包算法一：朴素三重循环动态规划 +#para[ +该算法是完全背包问题的一种直观解法，其递推关系考虑了对每个物品 $i$，我们可以选择不取，或者取 $k$ 件，其中 $k$ 可以是 1 到容量允许的最大值。 +设 $"dp"[i][j]$ 表示在前 $i$ 种物品中选择，背包容量为 $j$ 时的最大价值。 +递推公式为： +#box(fill: luma(240), radius: 3pt, inset: 8pt)[ + #set text(size: 0.9em) + #align(center)[ + $ "dp"[i][j] = "max"("dp"[i-1][j], "max"_(k=1)^(j/w_i)("dp"[i-1][j - k dot w_i] + k dot v_i)) $ + ] +] +其中 $w_i$ 和 $v_i$ 分别表示第 $i$ 种物品的重量和价值。 +该算法的时间复杂度为 $O(n dot W dot (W/w_"min"))$，其中 $n$ 为物品种类数，$W$ 为背包容量，$w_"min"$ 为物品的最小重量。 +] +```cpp +int complete_knapsack_v1(const std::vector& items, int capacity) { + ops_count = 0; + int n = items.size(); + if (n == 0) return 0; + std::vector> dp(n + 1, std::vector(capacity + 1, 0)); + for (int i = 1; i <= n; ++i) { + int w = items[i - 1].weight; + int v = items[i - 1].value; + for (int j = 0; j <= capacity; ++j) { + dp[i][j] = dp[i-1][j]; // Option to not take item i + ops_count++; + for (int k = 1; k * w <= j; ++k) { + ops_count++; + if (dp[i-1][j - k * w] + k * v > dp[i][j]) { + dp[i][j] = dp[i-1][j - k * w] + k * v; + } + } + } + } + return dp[n][capacity]; +} +``` +#align(center)[_代码 1: 完全背包算法一 C++ 实现_] + +=== 完全背包算法二：优化二维动态规划 +#para[ +该算法是完全背包问题更常用且更高效的二维动态规划解法。它利用了完全背包的特性：在考虑第 $i$ 种物品时，如果选择放入该物品，那么接下来的决策仍然可以在包含第 $i$ 种物品的集合中进行。 +递推公式为： +#box(fill: luma(240), radius: 3pt, inset: 8pt)[ + #set text(size: 0.9em) + #align(center)[ + $ "dp"[i][j] = "max"("dp"[i-1][j], "dp"[i][j - w_i] + v_i) $ + ] +] +其中 $"dp"[i-1][j]$ 表示不选择第 $i$ 种物品的最大价值，而 $"dp"[i][j - w_i] + v_i$ 表示选择至少一件第 $i$ 种物品，并在剩余容量 $j - w_i$ 中继续考虑第 $i$ 种物品（以及之前的物品）。 +该算法的时间复杂度为 $O(n dot W)$，空间复杂度为 $O(n dot W)$。 +] +```cpp +int complete_knapsack_v2(const std::vector& items, int capacity) { + ops_count = 0; + int n = items.size(); + if (n == 0) return 0; + std::vector> dp(n + 1, std::vector(capacity + 1, 0)); + for (int i = 1; i <= n; ++i) { + int w = items[i - 1].weight; + int v = items[i - 1].value; + for (int j = 0; j <= capacity; ++j) { + ops_count++; + if (j < w) { + dp[i][j] = dp[i - 1][j]; + } else { + dp[i][j] = std::max(dp[i - 1][j], dp[i][j - w] + v); + } + } + } + return dp[n][capacity]; +} +``` +#align(center)[_代码 2: 完全背包算法二 C++ 实现_] + +=== 完全背包算法三：空间优化一维动态规划 +#para[ +该算法是对算法二的空间优化版本，它将二维 $"dp"$ 数组优化为一维 $"dp"$ 数组。由于计算 $"dp"[i][j]$ 时只依赖于 $"dp"[i-1]$ 和 $"dp"[i]$ 自身（通过 $"dp"[j-w_i]$），因此可以通过在一维数组上正序遍历容量来实现。 +递推公式为： +#box(fill: luma(240), radius: 3pt, inset: 8pt)[ + #set text(size: 0.9em) + #align(center)[ + $ "dp"[j] = "max"("dp"[j], "dp"[j - w_i] + v_i) $ + ] +] +该算法的时间复杂度仍为 $O(n dot W)$，但空间复杂度优化为 $O(W)$，极大地节省了内存。 +] +```cpp +int complete_knapsack_v3(const std::vector& items, int capacity) { + ops_count = 0; + std::vector dp(capacity + 1, 0); + for (const auto& item : items) { + for (int j = item.weight; j <= capacity; ++j) { + ops_count++; + dp[j] = std::max(dp[j], dp[j - item.weight] + item.value); + } + } + return dp[capacity]; +} +``` +#align(center)[_代码 3: 完全背包算法三 C++ 实现_] + +== 实验环境与参数设置 +#para[ +本实验在 Linux 操作系统环境下进行，C++ 代码使用 #link("https://gcc.gnu.org/")[GCC] 编译器 (`g++`) 进行编译，并以 (`-O2`) 级别进行优化。数据分析与绘图使用 #link("https://www.python.org/")[Python] 编程语言，依赖 #link("https://pandas.pydata.org/")[pandas]、#link("https://matplotlib.org/")[matplotlib] 和 #link("https://seaborn.pydata.org/")[seaborn] 等库。 +] + +#para[ +实验中，我们固定背包容量 $W=100$，并随机生成物品。物品的重量在 $[1, 40]$ 范围内均匀分布，价值在 $[1, 100]$ 范围内均匀分布。为了消除随机性带来的误差，每个 $n$ 值（物品种类数）进行 $10$ 次独立实验，并取其平均运行时间及关键操作次数。物品种类数 $n$ 从 $5$ 递增到 $25$，步长为 $5$。 +] +#para[ +我们定义“关键操作次数”为动态规划表中状态值的更新或访问次数。具体在 C++ 代码中，通过全局变量 (`ops_count`) 在每次 (`dp`) 数组赋值或比较时进行累加。 +] + +== 数据收集与可视化 +#para[ +实验数据由 C++ 程序 (`knapsack`) 收集。该程序在每次运行完一个算法后，将物品种类数 $n$、算法名称（v1、v2、v3）、平均运行时间（微秒）和平均关键操作次数输出到标准输出，并重定向保存至 (`results.csv`) 文件。 +] +#para[ +Python 脚本 (`plotter.py`) 负责读取 (`results.csv`) 文件，使用 (`matplotlib`) 和 (`seaborn`) 库生成两幅图表： +] + + 平均运行时间与物品种类数 $n$ 的关系图。 + + 平均关键操作次数与物品种类数 $n$ 的关系图，其中关键操作次数曲线采用对数坐标显示以更好地展现数量级差异。 +#para[ +这些图表将直观地展示不同算法的性能随问题规模变化的趋势。 +] += 实验结果 +#para[ +本节展示了不同动态规划算法在解决完全背包问题时，其平均运行时间与关键操作次数随物品种类数 $n$ 变化的实验结果。 +] + +#figure( + image("time_vs_n.png", width: 80%), + caption: [平均运行时间与物品种类数的关系], +) + +#figure( + image("ops_vs_n.png", width: 80%), + caption: [平均关键操作次数与物品种类数的关系], +) + +#para[ +从上述图表中，我们可以观察到以下趋势： +] + - *算法一 (Naive DP)*：无论是在运行时间还是关键操作次数上，算法一都显著高于算法二和算法三。其增长趋势与其理论分析的 $O(n dot W dot (W/w_min))$ 复杂度吻合，表明该方法在实际应用中效率极低，尤其是在问题规模稍大时。 + - *算法二 (Optimized 2D DP)*：算法二的运行时间和关键操作次数都呈现出与 $n$ 线性相关的增长趋势，这与其理论时间复杂度 $O(n dot W)$ 一致。与算法一相比，其性能有了大幅提升。 + - *算法三 (Space-Optimized 1D DP)*：算法三在运行时间上与算法二表现相似，同样呈现出与 $n$ 线性相关的增长。在关键操作次数上，它也与算法二保持一致的增长模式。这验证了空间优化版本在不改变时间复杂度的前提下，能有效降低空间消耗。虽然理论上时间复杂度相同，但由于内存访问模式的改变（更少的内存分配，更好的缓存局部性），在某些情况下可能会有细微的性能提升，但在本实验的数据规模下，这种差异不明显。 + +#para[ +总体而言，算法二和算法三在处理完全背包问题上表现出良好的可伸缩性，而算法三更是在空间效率上具有优势。算法一作为一种直观但效率低下的实现，仅适合理解概念，不适用于实际大规模问题。 +] += 实验总结 +#para[ +本实验通过实现和比较三种基于动态规划的完全背包算法，深入分析了不同递推关系和优化策略对算法性能的影响。实验结果清晰地表明，算法一（朴素三重循环）由于其较高的复杂性，在运行时间与关键操作次数上均表现出最差的性能，验证了其不适用于实际应用。 +] +#para[ +相比之下，算法二（优化二维动态规划）和算法三（空间优化一维动态规划）均展示出优越的性能，其时间复杂度为 $O(n dot W)$，运行时间随问题规模 $n$ 呈线性增长。特别是算法三，在保持与算法二相同时间复杂度的同时，将空间复杂度优化至 $O(W)$，这在处理大容量背包问题时具有显著优势。 +] +#para[ +本次实验不仅加深了对动态规划解决完全背包问题的理解，也强调了算法设计中选择合适的递推关系和进行空间优化的重要性。未来工作可以扩展到更复杂的背包问题，例如多重背包的更高效实现（如二进制优化）及其在更大规模数据下的性能分析。 +] + +#pagebreak() += 附加：多重背包问题分析 + +== 多重背包算法一：朴素动态规划 +#para[ +多重背包问题与完全背包问题类似，但每种物品的数量是有限的。对于第 $i$ 种物品，其数量上限为 $k_i$ 个。 +设 $"dp"[i][j]$ 表示在前 $i$ 种物品中选择，背包容量为 $j$ 时的最大价值。 +递推公式为： +#box(fill: luma(240), radius: 3pt, inset: 8pt)[ + #set text(size: 0.9em) + #align(center)[ + $ "dp"[i][j] = "max"_(0 <= c <= "min"(k_i, j/w_i))("dp"[i-1][j - c dot w_i] + c dot v_i) $ + ] +] +其中 $w_i$、$v_i$、$k_i$ 分别表示第 $i$ 种物品的重量、价值和数量上限，$c$ 表示选择第 $i$ 种物品的件数。 +该算法的时间复杂度为 $O(W dot "sum" k_i)$，在最坏情况下，如果 $k_i$ 很大，其性能会接近完全背包的朴素解法。若 $"sum" k_i$ 可以简化为 $K_"max"$，则复杂度为 $O(n dot W dot K_"max")$。 +] +```cpp +// Algorithm for Multiple Knapsack (Direct DP) +int multiple_knapsack_v1(const std::vector& items, int capacity) { + int n = items.size(); + if (n == 0) return 0; + std::vector> dp(n + 1, std::vector(capacity + 1, 0)); + for (int i = 1; i <= n; ++i) { + int w = items[i - 1].weight; + int v = items[i - 1].value; + int k = items[i - 1].count; // Max count for this item + for (int j = 0; j <= capacity; ++j) { + dp[i][j] = dp[i-1][j]; + for (int c = 1; c <= k && c * w <= j; ++c) { + dp[i][j] = std::max(dp[i][j], dp[i - 1][j - c * w] + c * v); + } + } + } + return dp[n][capacity]; +} +``` +#align(center)[_代码 4: 多重背包算法一 C++ 实现_] + +== 多重背包算法二：二进制优化 +#para[ +二进制优化是解决多重背包问题的一种高效方法。其核心思想是将每种数量有限的物品拆分成若干件特殊的“物品”，使得这些特殊物品的组合可以表示原物品的任意数量。具体来说，对于第 $i$ 种物品，如果其数量上限为 $k_i$，我们可以将其拆分为重量和价值分别为 $c dot w_i$ 和 $c dot v_i$ 的“物品”，其中 $c$ 取 $1, 2, 4, "dots", 2^p$，以及剩余的 $k_i - (2^(p+1)-1)$。这些 $c$ 的和可以表示从 $1$ 到 $k_i$ 之间的任何一个整数。 +] +#para[ +拆分后，多重背包问题就转化为了一个 0/1 背包问题。我们可以使用 0/1 背包问题的标准动态规划方法（如与完全背包算法三类似的一维 DP 优化）来解决。 +转化后的物品总数将从 $"sum" k_i$ 减少到 $"sum" "log" k_i$，从而将时间复杂度优化为 $O(W dot "sum" "log" k_i)$。 +] diff --git a/dynamic/ops_vs_n.png b/dynamic/ops_vs_n.png new file mode 100644 index 0000000000000000000000000000000000000000..a74516cb69b58750323edd804888f4da7be2a22f GIT binary patch literal 224147 zcmeEuXHZnz)@`GR96*FalqhgiqJWY!N>C61$vKJSoI?W&C_%u30!pSyO_Os5Z3zmJ zqtGPDIXBo%cx&rD_kQ*My{h-)RlU2bENJYodxbg29AnHCsqt9(3JC)V3u+I2D?m5Oa%T?6#Uv3{E+Ze)c4eKdFJV3;cg96v+#6vbn$ewv%KzY?e1ac;(YhE zpulZ0-s`rWo~|Ae{QOS;e#UJVcN_jsMqa((UH))=VBi6R-R6M)5%}iHdcz1{u!jor z+PxN%a)5JZN-W@gBEFa0_e?IxSzkCaP z{y&$kHM3Cw_~d_I*NN*c{`&z;gNhLIKmHf_$Azu`{3mRfkih4EJhXoiBKbca3d?_F z{?EgZi~o$p zf5XLp#^S%>;@@NO|9QAr!Ay|;8jF)Y28W`*Yb9bOx9U7kb~}v}Ke8{rxljJEPip_$ zi%v%8I*SI}TA`8G2V3}Eo8Cu0Q(k(w?cnnirC6p9GN%VaLSAdbLOxrVOMR;&g?j#5 z^~;7{;snXvDK0G^uo#a%p#bd0#;(MA(Oc)bxtGZ%a~Xkq4=0=(HwO<^hblXHipuUa 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print(f"错误：找不到文件 {csv_file_path}") + sys.exit(1) + + # 算法名称映射 + algo_names = { + 'v1': '朴素动态规划（三层循环）', + 'v2': '优化二维动态规划', + 'v3': '空间优化一维动态规划' + } + df['algo_name'] = df['algo'].map(algo_names) + + # 设置绘图风格 + sns.set_theme(style="whitegrid") + + # 图表1：运行时间与物品数量的关系 + plt.figure(figsize=(10, 6)) + time_plot = sns.lineplot(data=df, x='n', y='time_us', hue='algo_name', marker='o', palette='viridis') + plt.title('平均运行时间与物品种类数量(n)的关系') + plt.xlabel('物品种类数量(n)') + plt.ylabel('平均运行时间（微秒）') + time_plot.legend(title='算法') + plt.grid(True, which='both', linestyle='--') + plt.savefig('time_vs_n.png', dpi=300) + plt.close() + + # 图表2：操作次数与物品数量的关系 + plt.figure(figsize=(10, 6)) + ops_plot = sns.lineplot(data=df, x='n', y='ops', hue='algo_name', marker='o', palette='plasma') + plt.title('关键操作次数与物品种类数量(n)的关系') + plt.xlabel('物品种类数量(n)') + plt.ylabel('平均关键操作次数（对数尺度）') + plt.yscale('log') + ops_plot.legend(title='算法') + plt.grid(True, which='both', linestyle='--') + plt.savefig('ops_vs_n.png', dpi=300) + plt.close() + +if __name__ == '__main__': + if len(sys.argv) > 1: + # 使用命令行参数指定的CSV文件 + create_plots(sys.argv[1]) + else: + # 如果没有提供参数，使用默认文件名 + create_plots('results.csv') diff --git a/dynamic/results.csv b/dynamic/results.csv new file mode 100644 index 0000000..a691316 --- /dev/null +++ b/dynamic/results.csv @@ -0,0 +1,16 @@ +n,algo,time_us,ops +5,v1,2,3011 +5,v2,0,505 +5,v3,0,399 +10,v1,3,5641 +10,v2,1,1010 +10,v3,0,795 +15,v1,7,8723 +15,v2,1,1515 +15,v3,0,1231 +20,v1,9,14664 +20,v2,1,2020 +20,v3,0,1641 +25,v1,9,14808 +25,v2,2,2525 +25,v3,0,2007 diff --git a/dynamic/task.txt b/dynamic/task.txt new file mode 100644 index 0000000..5a70a9c --- /dev/null +++ b/dynamic/task.txt @@ -0,0 +1,12 @@ +运用动态规划算法求解完全背包问题并进行分析，具体要求如下： + + 针对完全背包问题，实现基于两种递推公式的动态规划算法； + 在代码中插桩，记录关键操作次数（如查表次数等）； + 以物品种类的大小n为输入规模，固定n，随机产生大量测试样本，统计两种算法的平均运行时间和关键操作次数，并进行记录； + 改变物品种类规模，对不同规模问题各算法的结果对比分析，通过统计python画图插入到报告中记录，与理论值进行对照分析； + 使用一维数组的方式解决整数背包问题，并记录其平均运行时间和关键操作次数，与上述两种算法进行对比。 + +附加：运用动态规划算法求解多重背包问题并进行分析，具体要求如下： 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+<0085> <9488> +<0086> <5728> +<0087> <4EE3> +<0088> <7801> +<0089> <4E2D> +<008A> <5982> +<008B> <67E5> +<008C> <8868> +<008D> <7B49> +<008E> <5C0F> +<008F> <56FA> +<0090> <5B9A> +<0091> <4EA7> +<0092> <751F> +<0093> <548C> +<0094> <753B> +<0095> <5230> +<0096> <7406> +<0097> <8BBA> +<0098> <503C> +<0099> <7167> +<009A> <4F7F> +<009B> <65B9> +<009C> <89E3> +<009D> <51B3> +<009E> <6574> +<009F> <5176> +<00A0> <5373> +<00A1> <6BCF> +<00A2> <6709> +<00A3> <9650> +<00A4> <7B2C> +<00A5> <4E2A> +<00A6> +<00A7> <66F2> +<00A8> <7EBF> +<00A9> <91C7> +<00AA> <5750> +<00AB> <6807> +<00AC> <663E> +<00AD> <793A> +<00AE> <66F4> +<00AF> <597D> +<00B0> <5730> +<00B1> <5C55> +<00B2> <7EA7> +<00B3> <5DEE> +<00B4> <5F02> +<00B5> <65E0> +<00B6> <662F> +<00B7> <8FD8> +<00B8> <90FD> +<00B9> <8457> +<00BA> <9AD8> +<00BB> <589E> +<00BC> <957F> +<00BD> <8D8B> +<00BE> <52BF> +<00BF> <590D> +<00C0> <6742> +<00C1> <5EA6> +<00C2> <543B> +<00C3> <5408> +<00C4> <660E> +<00C5> <8BE5> +<00C6> <9645> +<00C7> <5E94> +<00C8> <6548> +endbfchar +41 beginbfchar +<00C9> <7387> +<00CA> <6781> +<00CB> <4F4E> +<00CC> <5C24> +<00CD> <7A0D> +<00CE> <5448> +<00CF> <51FA> +<00D0> <76F8> +<00D1> <8FD9> +<00D2> <81F4> +<00D3> <4E86> +<00D4> <5E45> +<00D5> <63D0> +<00D6> <5347> +<00D7> <4F3C> +<00D8> <5B83> +<00D9> <4E5F> +<00DA> <4FDD> +<00DB> <6301> +<00DC> <8BC1> +<00DD> <524D> +<00DE> <964D> +<00DF> <6D88> +<00E0> <8017> +<00E1> <867D> +<00E2> <7136> +<00E3> <4F46> +<00E4> <7531> +<00E5> <5B58> +<00E6> <8BBF> +<00E7> <914D> +<00E8> <7F13> +<00E9> <5C40> +<00EA> <90E8> +<00EB> <67D0> +<00EC> <4E9B> +<00ED> <60C5> +<00EE> <51B5> +<00EF> <4F1A> +<00F0> <7EC6> +<00F1> <5FAE> +endbfchar +endcmap +CMapName currentdict /CMap defineresource pop +end +end +%%EndResource +%%EOF +endstream +endobj + +493 0 obj +<< + /Length 43285 + /Filter /FlateDecode +>> +stream +xyt#y'GfFDF'n$@$A.YUUduT,VjutlGz~k,ݒגFݒyvtZeɲط D[~C ddd~ gI_!೏ѧp?>Gz#`>?/=vg?T~ 0cwvF{їHkO=_ M7;ϽQ ް?P{3oyw|獛$;%Ϸsw +PldX`l@@ 0Xr\%- +@K]@ 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DHYK.叅9CFͨ>;9K{ 7d=dkwO:oB / +s}@Vo>tЎ2nll<裇zYg :裏3rO>%/z C߯zE%xB T32~;Kwʯ<֯_QlMy!n{=_z> [---^{N30;$n' =dy[8$hi _ܷo444K.y^rm@嵴d#۶;@f\P=V^=cƌ뮻nȐ!Gyd~ۯ_#y3c64WWD7w=+qHݿ~֬Hzs+H"] ۰a[eeOoT$DbxߟO|~굛^ in"]@ t3*'QGrꙍ> Lt. ]ŻGfHzg{H"] kjjj\.7rȬSli8dB tQ8$}՚WGUi"]@ t3*o„ !\.裏fzfO!"]@ zaz;qH䳕Yj Ӏ t. ES?OfܹY/pȄHŬ?Hx}"- ]@ t.p}@͟?y2ӦMz9FC&DH˻xŇN?tw>zuT^t. DHi>֭[Wr7|s˩g642!DſY8$sؐQ/8QuzOyH"] Lr)<3}"]@xʘğKf:Q/I. DHi> f*rYfeeO!"]@ ],tQߙ8$d:ToH. DHi> +gqFiT?r꓍> Lt.v5ǜ5.qH^ut. DHi> +k׮=#K2r^Qѧ . .^=qHf܃ŬH4kt. DHZvmSergϮY&u}"]@bM!;ϐ 6 2]@ t.p}@&aÆ7˕FeJ9?O?s:묬*eO!"]@ =]۰MJn|!R5 HC"]@4\S1e WsJ R642!D^Ŧ-Ͼ!+ny=u t. EDy 0dFF{{۳^dOzCZ{!{ſ"]@ t32Qni&dvF)_rC2ij6l%İTlF{{O=8$s֥7mޖ= ꘍P(0!S.[^ \@@;&8$so'nc 1=tAXCd߿Z.] ꘍'}ݟ-_s}uFF>_A>zE=]@ .x!NOWd:jX=A"]@4\@ѧ񏮭-At.κGC2O·_e:j[M=D"]@4\@ѧ . SX!L!ֲWGͫ4"]@ t3r7__+WܣXQFk> Lt.nx/?yt9t^$Qj4 QH"] L 8?\.xXc=ѧ . G/S&ĥf:D-=M"]@4\k?rӻӧp-끍> Lt.X٪qW#3zuԏK*@"]@4\rƍ7v 6mjll,=裏5}"]@wWky=C2MW֫VPH"] [re.vi]~ΰa)w넍> Lt.vՆso̝\,^ uҀDH" gT޿Lt뭷dxn]cѧ . nk<䂦![[( t. 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"JetBrains Mono") +#let nudtlabpaper(title: "", + author1: "", + id1: "", + advisor: "", + jobtitle: "", + lab: "", + date: "", + header_str: "", + minimal_cover: false, + body) = { + // Set the document's basic properties. + set document(author: author1, title: title) + set page( + + margin: (left: 30mm, right: 30mm, top: 30mm, bottom: 30mm), + ) + + // If minimal_cover is requested, render an otherwise-empty first page + // that only displays the "实验时间" near the bottom center. + if minimal_cover { + v(158pt) + align(center)[ + #block(text(weight: 700, size: 30pt, font: hei, tracking: 1pt, "2025秋 -《算法设计与分析》")) + ] + align(center)[ + #block(text(weight: 700, size: 24pt, font: song, tracking: 1pt, "贪心算法分析实验报告")) + ] + + + // Keep standard margins but push content down toward the bottom. + v(220pt) + align(center)[ + #block(text(size: 14pt, font: song, tracking: 9pt, "实验时间")) + ] + v(2pt) + align(center)[ + #block(text(size: 16pt, font: song, date)) + ] + pagebreak() + } else { + // Title row. + v(158pt) + align(center)[ + #block(text(weight: 700, size: 30pt, font: hei, tracking: 1pt, "2025秋 -《算法设计与分析》")) + ] + align(center)[ + #block(text(weight: 700, size: 24pt, font: song, tracking: 1pt, "动态规划算法分析实验报告")) + ] + + v(103pt) + pad( + left: 1em, + right: 1em, + grid( + // columns: (80pt, 1fr), + // rows: (17pt, auto), + // text(weight: 700, size: 16pt, font: song, "实验名称："), + // align(center, text(weight: "regular", size: 16pt, font: song, title)), + // text(""), + // line(length: 100%) + ) + // #block(text(weight: 700, 1.75em, title)) + // underline(text(weight: 700, size: 16pt, font: song, title)) + ) + + // Author information. + + v(62.5pt) + + grid( + columns: (0.25fr, 0.25fr, 0.25fr, 0.25fr), + rows: (20pt, 8pt, 20pt, 8pt, 20pt, 8pt, 20pt, 12pt), + text(size: 14pt, font: song, tracking: 9pt, "学员姓名"), + align(center, text(size: 14pt, font: song, author1)), + text(size: 14pt, font: song, tracking: 54pt, "学号"), + align(center, text(size: 14pt, font: times, id1)), + text(""), + line(length: 100%), + text(""), + line(length: 100%), + text(size: 14pt, font: song, tracking: 9pt, "指导教员"), + align(center, text(size: 14pt, font: song, advisor)), + text(size: 14pt, font: song, tracking: 54pt, "职称"), + align(center, text(size: 14pt, font: song, jobtitle)), + text(""), + line(length: 100%), + text(""), + line(length: 100%), + text(size: 14pt, font: song, tracking: 9pt, "实验室"), + align(center, text(size: 14pt, font: song, lab)), + text(size: 14pt, font: song, tracking: 9pt, "实验时间"), + align(center, text(size: 14pt, font: song, date)), + text(""), + line(length: 100%), + text(""), + line(length: 100%), + ) + + v(50.5pt) + align(center, text(font: hei, size: 15pt, "国防科技大学教育训练部制")) + + pagebreak() + } + + set page( + margin: (left: 30mm, right: 30mm, top: 30mm, bottom: 30mm), + numbering: "i", + number-align: center, + ) + + v(14pt) + align(center)[ + #block(text(font: hei, size: 14pt, "《本科实验报告》填写说明")) + ] + + v(14pt) + text("") + par(first-line-indent: 2em, text(font: song, size: 12pt, "实验报告内容编排应符合以下要求：")) + + par(first-line-indent: 2em, text(font: fsong, size: 12pt, "（1）采用A4（21cm×29.7cm）白色复印纸，单面黑字。上下左右各侧的页边距均为3cm；缺省文档网格：字号为小4号，中文为宋体，英文和阿拉伯数字为Times New Roman，每页30行，每行36字；页脚距边界为2.5cm，页码置于页脚、居中，采用小5号阿拉伯数字从1开始连续编排，封面不编页码。")) + + par(first-line-indent: 2em, text(font: fsong, size: 12pt, "（2）报告正文最多可设四级标题，字体均为黑体，第一级标题字号为4号，其余各级标题为小4号；标题序号第一级用“一、”、“二、”……，第二级用“（一）”、“（二）” ……，第三级用“1.”、“2.” ……，第四级用“（1）”、“（2）” ……，分别按序连续编排。")) + + par(first-line-indent: 2em, text(font: fsong, size: 12pt, "（3）正文插图、表格中的文字字号均为5号。")) + + pagebreak() + + set page( + margin: (left: 30mm, right: 30mm, top: 30mm, bottom: 30mm), + numbering: "1", + number-align: center, + ) + + set heading(numbering: "1.1") + // set text(font: hei, lang: "zh") + + show heading: it => box(width: 100%)[ + #v(0.50em) + #set text(font: hei) + #counter(heading).display() + // #h(0.5em) + #it.body + ] + // Main body. + set par(justify: true) + + body +} + +#let para(t) = par(first-line-indent: 2em, text(font: song, size: 10.5pt, t)) +#let subpara(t) = par(first-line-indent: 2em, text(font: song, size: 10pt, t)) +#let cb(t) = block( + text(font: ("Consolas","FangSong_GB2312"), t), + fill: luma(240), + inset: 1pt, + radius: 4pt, + // width: 100%, +) diff --git a/greed/main.pdf b/greed/main.pdf new file mode 100644 index 0000000..0897072 --- /dev/null +++ b/greed/main.pdf @@ -0,0 +1,9442 @@ +%PDF-1.7 +% + +1 0 obj +<< + /Type /Pages + /Count 8 + /Kids [500 0 R 502 0 R 527 0 R 529 0 R 531 0 R 533 0 R 535 0 R 537 0 R] +>> +endobj + +2 0 obj +<< + /Type /Outlines + /First 3 0 R + /Last 17 0 R + /Count 7 +>> +endobj + +3 0 obj +<< + /Parent 2 0 R + /Next 4 0 R + /Title + /Dest 477 0 R +>> +endobj + +4 0 obj +<< + /Parent 2 0 R + /Next 5 0 R + /Prev 3 0 R + /Title + /Dest 478 0 R +>> +endobj + +5 0 obj +<< + /Parent 2 0 R + /Next 6 0 R + /Prev 4 0 R + /Title + /Dest 479 0 R +>> +endobj + +6 0 obj +<< + /Parent 2 0 R + /Next 15 0 R + /Prev 5 0 R + /First 7 0 R + /Last 14 0 R + /Count -3 + 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<7840> +<00D3> <589E> +<00D4> <4E86> +<00D5> <9884> +<00D6> <5904> +<00D7> <6B65> +<00D8> <9AA4> +<00D9> <6240> +<00DA> <6709> +<00DB> <9012> +<00DC> <51CF> +<00DD> <7136> +<00DE> <540E> +<00DF> <64CD> +<00E0> <8F83> +<00E1> <5927> +<00E2> <88AB> +<00E3> <800C> +<00E4> <907F> +<00E5> <514D> +<00E6> <5269> +<00E7> <4E2A> +<00E8> <5BFC> +<00E9> <81F4> +<00EA> <6781> +<00EB> <8861> +<00EC> <60C5> +<00ED> <51B5> +<00EE> <7531> +<00EF> <51B3> +<00F0> <5B9A> +<00F1> <8BC4> +<00F2> <4F30> +<00F3> <80FD> +<00F4> <6211> +<00F5> <4EEC> +<00F6> <9700> +<00F7> <641C> +<00F8> <7D22> +<00F9> <652F> +<00FA> <9650> +<00FB> <526A> +<00FC> <679D> +<00FD> <53F0> +<00FE> <957F> +<00FF> <7D27> +<0100> <63A5> +<0101> <7740> +<0102> <4F1A> +<0103> <5300> +<0104> <7EC8> +<0105> <53D8> +<0106> <72EC> +<0107> <6B64> +<0108> <4EE3> +<0109> <8BC1> +<010A> <7ED3> +<010B> <679C> +<010C> <4E0E> +<010D> <5B8C> +<010E> <5168> +<010F> <574F> +<0110> <4F8B> +<0111> <543B> +<0112> <8FDB> +<0113> <91CF> +<0114> <968F> +<0115> <6D4B> +<0116> <8BD5> +<0117> <7BB1> +<0118> <5E38> +<0119> <4E4B> +<011A> <8D8A> +<011B> <4E14> +<011C> <7A33> +<011D> <76F8> +<011E> <5E03> +<011F> <5BBD> +<0120> <5E73> +<0121> <5DEE> +<0122> <5347> +<0123> <9AD8> +<0124> <8D8B> +<0125> <52BF> +<0126> <5F80> +<0127> <964D> +<0128> <8FD9> +<0129> <56E0> +<012A> <586B> +<012B> <5F02> +<012C> <5219> +endbfchar +62 beginbfchar +<012D> <4FDD> +<012E> <6301> +<012F> <6548> +<0130> <8FD0> +<0131> <77ED> +<0132> <5FAE> +<0133> <79D2> +<0134> <5DF2> +<0135> <96BE> +<0136> <8BA1> +<0137> <6790> +<0138> <660E> +<0139> <867D> +<013A> <7B80> +<013B> <4F46> +<013C> <5E26> +<013D> <5DE8> +<013E> <63D0> +<013F> <7EDD> +<0140> <53CA> +<0141> <6784> +<0142> <9020> +<0143> <90FD> +<0144> <83B7> +<0145> <542F> +<0146> <793A> +<0147> <56F0> +<0148> <52A1> +<0149> <5173> +<014A> <91CD> +<014B> <62E5> +<014C> <5757> +<014D> <8FBE> +<014E> <670D> +<014F> <6CCA> +<0150> <677E> +<0151> <6267> +<0152> <5B58> +<0153> <7387> +<0154> <635F> +<0155> <8017> +<0156> <5B50> +<0157> <7CFB> +<0158> <7EDF> +<0159> <76EE> +<015A> <8F7B> +<015B> <663E> +<015C> <9500> +<015D> <8D44> +<015E> <6E90> +<015F> <76F2> +<0160> <8FFD> +<0161> <6FC0> +<0162> <53D6> +<0163> <611F> +<0164> <77E5> +<0165> <52A8> +<0166> <6001> +<0167> <7C92> +<0168> <66F4> +<0169> <65B9> +<016A> <6848> +endbfchar +endcmap +CMapName currentdict /CMap defineresource pop +end +end +%%EndResource +%%EOF +endstream +endobj + +418 0 obj +<< + /Length 85982 + /Filter /FlateDecode + /Subtype /CIDFontType0C +>> +stream +x컇WTW6~z3GόA[=X.Elta*ez M`E)j4h4'퉉I3C~޵?75>̞]8֣xٿ3t]cWl?{~"~qf/W~ &|܋;?1՗a1¾1ݻ^bX7p'PqR-A!^!wY7"f;M0qI&MqsۈF?>**jVeܶ{Ə&tmno ٳuPOݾ;+|۸Opg? ٿsp+8d[#vne6flڴ%zJJ{}ի_?YXI}Paݳ^ꞽ?wlh+իꟷaxð7_;sgeff Ky aR»7K.7na1 0? Oz|p`b >b:^1L 띂n}p*%c1LlX?7]טnb7` sd1ؠpl⏹aCSa˰a:lJF`cNacӱq=qGjl~lb6)|RM]M#gY GOl lӂy> +Sd 5y;%زw;[le48*[ 0? 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z0k9qs62t9WRk#Bx5HEO_2dD#J*-SiB>2d`@JFH!*p+gbHU0&4_46tU4uK--gbaEE5V009_a#?gF` z9L}lFyon1wVC7EwcoM3z#Ehg<8LrN9x9k9FQpw|Y2slra=w&>EP+L+4{)v? z=vL30hJxJ{0%*0E%Pi-i|Ac$|k8^>)3-JH>=-~gCU&-p(4n1^8F`de)Mr2--wz~e^ JQq_lH{|nVYPN@I@ literal 0 HcmV?d00001 diff --git a/greed/results/worst_case_verification.csv b/greed/results/worst_case_verification.csv new file mode 100644 index 0000000..6967bc6 --- /dev/null +++ b/greed/results/worst_case_verification.csv @@ -0,0 +1,5 @@ +case_type,m,n,input_desc,greedy_res,opt_res,ratio,theory_bound +LS_Worst,3,7,"m*(m-1) 1s + one m",5,3,1.66667,1.66667 +LS_Worst,4,13,"m*(m-1) 1s + one m",7,4,1.75,1.75 +LS_Worst,5,21,"m*(m-1) 1s + one m",9,5,1.8,1.8 +LPT_Worst,2,5,"{3,3,2,2,2}",7,6,1.16667,1.16667 diff --git a/greed/src/gpu_sim.py b/greed/src/gpu_sim.py new file mode 100644 index 0000000..0428623 --- /dev/null +++ b/greed/src/gpu_sim.py @@ -0,0 +1,224 @@ +import heapq +import random +import math +import pandas as pd +import matplotlib.pyplot as plt +import seaborn as sns + +class Task: + def __init__(self, id, arrival_time, duration, sigma): + self.id = id + self.arrival_time = arrival_time + self.base_duration = duration # Time on 1 GPU + self.sigma = sigma + self.start_time = -1 + self.finish_time = -1 + self.assigned_k = 0 + + def get_exec_time(self, k): + # Efficiency = sigma ^ log2(k) + # Time = base / (k * efficiency) + if k == 1: return self.base_duration + eff = self.sigma ** math.log2(k) + return self.base_duration / (k * eff) + +class Event: + def __init__(self, time, type, data): + self.time = time + self.type = type # 'ARRIVAL' or 'FINISH' + self.data = data + + def __lt__(self, other): + return self.time < other.time + +class Cluster: + def __init__(self, total_gpus=64): + self.total_gpus = total_gpus + self.free_gpus = total_gpus + self.running_tasks = [] # List of (task, finish_time) + +class Simulator: + def __init__(self, strategy_name, arrival_rate, max_tasks=1000): + self.strategy_name = strategy_name + self.arrival_rate = arrival_rate + self.max_tasks = max_tasks + self.events = [] + self.cluster = Cluster() + self.queue = [] + self.finished_tasks = [] + self.current_time = 0.0 + self.total_gpu_busy_time = 0.0 # Integral of busy GPUs over time + self.last_update_time = 0.0 + + def schedule_events(self): + # Generate all arrivals upfront or dynamically + t = 0 + for i in range(self.max_tasks): + dt = random.expovariate(self.arrival_rate) + t += dt + duration = random.uniform(10, 100) + sigma = random.uniform(0.75, 0.95) + task = Task(i, t, duration, sigma) + heapq.heappush(self.events, Event(t, 'ARRIVAL', task)) + + def update_metrics(self): + dt = self.current_time - self.last_update_time + busy_gpus = self.cluster.total_gpus - self.cluster.free_gpus + self.total_gpu_busy_time += busy_gpus * dt + self.last_update_time = self.current_time + + def get_k(self, task): + # Strategies + available = self.cluster.free_gpus + q_len = len(self.queue) + + possible_ks = [1, 2, 4, 8, 16, 32, 64] + valid_ks = [k for k in possible_ks if k <= available] + + if not valid_ks: return 0 + + if self.strategy_name == 'conservative': + # Always k=1 + return 1 if 1 in valid_ks else 0 + + elif self.strategy_name == 'aggressive': + # Use max possible parallel, up to limit (e.g. 16 to avoid total waste) + # Or just max available + return valid_ks[-1] + + elif self.strategy_name == 'adaptive': + # Based on queue length + if q_len == 0: + target = 32 # High speed if no contention + elif q_len < 5: + target = 16 + elif q_len < 10: + target = 8 + elif q_len < 20: + target = 4 + elif q_len < 50: + target = 2 + else: + target = 1 + + # Find largest valid k <= target + best_k = 1 + for k in valid_ks: + if k <= target: + best_k = k + return best_k + + return 1 + + def run(self): + self.schedule_events() + + while self.events or self.cluster.running_tasks: + if not self.events and not self.cluster.running_tasks: + break + + # Peek next event + if not self.events: + next_time = float('inf') + else: + next_time = self.events[0].time + + # Jump time + self.current_time = next_time + self.update_metrics() + + event = heapq.heappop(self.events) + + if event.type == 'ARRIVAL': + task = event.data + self.queue.append(task) + elif event.type == 'FINISH': + task = event.data + self.cluster.free_gpus += task.assigned_k + self.finished_tasks.append(task) + + # Try to schedule waiting tasks + # We iterate queue. Note: Standard queue is FIFO. + # We can't easily remove from middle if we skip, so we only look at head + # OR we can try to fit small tasks? + # Simple FIFO: look at head. If can schedule, do it. Else stop (or continue?) + # Let's do Strict FIFO for simplicity and fairness + + while self.queue: + head_task = self.queue[0] + k = self.get_k(head_task) + if k > 0: + # Assign + self.queue.pop(0) + head_task.assigned_k = k + head_task.start_time = self.current_time + exec_time = head_task.get_exec_time(k) + head_task.finish_time = self.current_time + exec_time + + self.cluster.free_gpus -= k + heapq.heappush(self.events, Event(head_task.finish_time, 'FINISH', head_task)) + else: + # Cannot schedule head task + break + + def calculate_results(self): + total_time = self.current_time + avg_utilization = self.total_gpu_busy_time / (total_time * 64) + + delays = [] + for t in self.finished_tasks: + expected_finish = t.arrival_time + t.base_duration + if t.finish_time <= expected_finish: + delta = 0 + else: + overdue = t.finish_time - expected_finish + # Normalized penalty + delta = (overdue / t.base_duration) ** 2 + delays.append(delta) + + avg_delay = sum(delays) / len(delays) if delays else 0 + return avg_utilization, avg_delay + +def run_simulations(): + strategies = ['conservative', 'aggressive', 'adaptive'] + # Load: Light (0.5), Medium (0.9), Heavy (1.1) + loads = [0.5, 0.9, 1.1] + + results = [] + + print("Running GPU Simulations...") + for load in loads: + for strat in strategies: + # Run multiple times to average? Just once for this demo with 1000 tasks + sim = Simulator(strat, load, max_tasks=1000) + sim.run() + eta, delta = sim.calculate_results() + results.append({ + 'Load': load, + 'Strategy': strat, + 'Utilization (eta)': eta, + 'Avg Delay Penalty (delta)': delta, + 'Score (Balanced)': (1-eta) + 0.1 * delta # Example lambda=0.1 + }) + print(f"Load {load}, Strat {strat}: Eta={eta:.3f}, Delta={delta:.3f}") + + df = pd.DataFrame(results) + df.to_csv("results/gpu_sim_results.csv", index=False) + + # Plotting + plt.figure(figsize=(12, 5)) + + plt.subplot(1, 2, 1) + sns.barplot(data=df, x='Load', y='Utilization (eta)', hue='Strategy') + plt.title('Cluster Utilization') + + plt.subplot(1, 2, 2) + sns.barplot(data=df, x='Load', y='Avg Delay Penalty (delta)', hue='Strategy') + plt.title('User Average Delay Penalty') + + plt.tight_layout() + plt.savefig('results/gpu_sim_plots.png') + plt.close() + +if __name__ == "__main__": + run_simulations() diff --git a/greed/src/multimachine.cpp b/greed/src/multimachine.cpp new file mode 100644 index 0000000..c6eb8fe --- /dev/null +++ b/greed/src/multimachine.cpp @@ -0,0 +1,273 @@ +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include + +using namespace std; + +// --- Data Structures --- + +struct Job { + int id; + long long duration; +}; + +struct ExperimentResult { + int n; + int m; + long long greedy_ls_makespan; + long long greedy_lpt_makespan; + long long optimal_makespan; // -1 if failed + double ls_time_us; + double lpt_time_us; + double opt_time_us; +}; + +// --- Algorithms --- + +// Greedy 1: List Scheduling (Arbitrary/Online) +long long greedy_ls(int m, const vector& jobs) { + if (jobs.empty()) return 0; + vector machines(m, 0); + for (const auto& job : jobs) { + // Find machine with min load + int min_idx = 0; + for (int i = 1; i < m; ++i) { + if (machines[i] < machines[min_idx]) { + min_idx = i; + } + } + machines[min_idx] += job.duration; + } + return *max_element(machines.begin(), machines.end()); +} + +// Greedy 2: LPT (Longest Processing Time) +long long greedy_lpt(int m, vector jobs) { // Note: pass by value to sort copy + if (jobs.empty()) return 0; + sort(jobs.begin(), jobs.end(), [](const Job& a, const Job& b) { + return a.duration > b.duration; + }); + + vector machines(m, 0); + // Optimization: Use a min-priority queue if m is large, but for small m linear scan is fine/faster due to cache + for (const auto& job : jobs) { + int min_idx = 0; + for (int i = 1; i < m; ++i) { + if (machines[i] < machines[min_idx]) { + min_idx = i; + } + } + machines[min_idx] += job.duration; + } + return *max_element(machines.begin(), machines.end()); +} + +// Optimal Solver: Branch and Bound +// Global variables for recursion to avoid passing too many args +long long best_makespan; +int G_m; +vector G_jobs; +vector G_machines; +long long start_time_opt; +bool time_out; + +void dfs(int job_idx, long long current_max) { + if (time_out) return; + + // Check timeout (e.g., 100ms per instance for batch tests) + if ((clock() - start_time_opt) / CLOCKS_PER_SEC > 1.0) { // 1 second timeout + time_out = true; + return; + } + + // Pruning 1: If current max load >= best solution found so far, prune + if (current_max >= best_makespan) return; + + // Base case: all jobs assigned + if (job_idx == G_jobs.size()) { + best_makespan = current_max; + return; + } + + // Pruning 2: Theoretical lower bound + // If (sum of remaining jobs + current total load) / m > best_makespan, prune? + // A simpler bound: max(current_max, (sum of remaining + sum of current loads) / m) + // Calculating sum every time is slow, can be optimized. + + long long job_len = G_jobs[job_idx].duration; + + // Try to assign job to each machine + for (int i = 0; i < G_m; ++i) { + // Optimization: Symmetry breaking + // If this machine has same load as previous machine, and we tried previous, skip this one. + // This assumes machines are initially 0. + // A simpler symmetry break: if machines[i] == machines[i-1] (and they are interchangeable), skip. + // Requires machines to be sorted or checked. + // For now, simpler check: if this is the first empty machine, stop after trying it. + if (G_machines[i] == 0) { + G_machines[i] += job_len; + dfs(job_idx + 1, max(current_max, G_machines[i])); + G_machines[i] -= job_len; + break; // Don't try other empty machines + } + + if (G_machines[i] + job_len < best_makespan) { + G_machines[i] += job_len; + dfs(job_idx + 1, max(current_max, G_machines[i])); + G_machines[i] -= job_len; + } + } +} + +long long solve_optimal(int m, vector jobs) { + if (jobs.empty()) return 0; + + // Heuristic: LPT gives a good initial bound + vector sorted_jobs = jobs; + sort(sorted_jobs.begin(), sorted_jobs.end(), [](const Job& a, const Job& b) { + return a.duration > b.duration; + }); + + best_makespan = greedy_lpt(m, sorted_jobs); + G_m = m; + G_jobs = sorted_jobs; + G_machines.assign(m, 0); + time_out = false; + start_time_opt = clock(); + + dfs(0, 0); + + if (time_out) return -1; + return best_makespan; +} + +// --- Test Generation --- + +vector generate_jobs(int n, int min_val, int max_val) { + vector jobs(n); + random_device rd; + mt19937 gen(rd()); + uniform_int_distribution<> dis(min_val, max_val); + for (int i = 0; i < n; ++i) { + jobs[i] = {i, (long long)dis(gen)}; + } + return jobs; +} + +// --- Main Experiments --- + +void run_experiments() { + ofstream out("results/algo_comparison.csv"); + out << "m,n,ls_makespan,lpt_makespan,opt_makespan,ls_time,lpt_time,opt_time,ls_ratio,lpt_ratio\n"; + + cout << "Running random experiments..." << endl; + + vector ms = {3, 5, 8}; + vector ns = {10, 15, 20, 25, 30, 50, 100}; + + for (int m : ms) { + for (int n : ns) { + int runs = 10; // More runs for faster algos + if (n > 20) runs = 20; + + for (int r = 0; r < runs; ++r) { + vector jobs = generate_jobs(n, 10, 100); + + auto t1 = chrono::high_resolution_clock::now(); + long long ls_res = greedy_ls(m, jobs); + auto t2 = chrono::high_resolution_clock::now(); + + auto t3 = chrono::high_resolution_clock::now(); + long long lpt_res = greedy_lpt(m, jobs); + auto t4 = chrono::high_resolution_clock::now(); + + long long opt_res = -1; + double opt_dur = 0; + + // Only run optimal for small n + if (n <= 18) { + auto t5 = chrono::high_resolution_clock::now(); + opt_res = solve_optimal(m, jobs); + auto t6 = chrono::high_resolution_clock::now(); + opt_dur = chrono::duration(t6 - t5).count(); + } + + double ls_dur = chrono::duration(t2 - t1).count(); + double lpt_dur = chrono::duration(t4 - t3).count(); + + double ls_ratio = (opt_res != -1 && opt_res != 0) ? (double)ls_res / opt_res : -1.0; + double lpt_ratio = (opt_res != -1 && opt_res != 0) ? (double)lpt_res / opt_res : -1.0; + + out << m << "," << n << "," + << ls_res << "," << lpt_res << "," << opt_res << "," + << ls_dur << "," << lpt_dur << "," << opt_dur << "," + << ls_ratio << "," << lpt_ratio << "\n"; + } + } + } + out.close(); + cout << "Experiments complete. Results saved." << endl; +} + +void verify_worst_cases() { + ofstream out("results/worst_case_verification.csv"); + out << "case_type,m,n,input_desc,greedy_res,opt_res,ratio,theory_bound\n"; + + // Case 1: LS Worst Case + // m machines. Input: m*(m-1) jobs of size 1, then 1 job of size m. + // Example m=3. 6 jobs of size 1, 1 job of size 3. + // Greedy: [1,1,3], [1,1], [1,1] -> Max 5. + // Opt: [3], [1,1,1], [1,1,1] -> Max 3. + // Ratio 5/3 approx 1.666. Theory 2 - 1/3 = 1.666. + + vector test_ms = {3, 4, 5}; + for (int m : test_ms) { + vector jobs; + int num_small = m * (m - 1); + for(int i=0; i jobs = { {0,3}, {1,3}, {2,2}, {3,2}, {4,2} }; + long long res_lpt = greedy_lpt(m, jobs); + long long res_opt = solve_optimal(m, jobs); // Should be fast + double ratio = (double)res_lpt / res_opt; + double bound = 4.0/3.0 - 1.0/(3.0*m); + + out << "LPT_Worst," << m << "," << jobs.size() << "," + << "\"{3,3,2,2,2}\"" << "," + << res_lpt << "," << res_opt << "," << ratio << "," << bound << "\n"; + } + + out.close(); + cout << "Worst case verification complete." << endl; +} + +int main() { + verify_worst_cases(); + run_experiments(); + return 0; +} diff --git a/greed/src/plot_algo.py b/greed/src/plot_algo.py new file mode 100644 index 0000000..860f70e --- /dev/null +++ b/greed/src/plot_algo.py @@ -0,0 +1,58 @@ +import pandas as pd +import matplotlib.pyplot as plt +import seaborn as sns +import os + +def plot_core_algo(): + if not os.path.exists("results/algo_comparison.csv"): + print("Core algo results not found.") + return + + df = pd.read_csv("results/algo_comparison.csv") + + # 1. Approximation Ratio Distribution (Boxplot) + # Filter valid ratios (where opt was computed) + df_ratios = df[df['ls_ratio'] != -1] + + if not df_ratios.empty: + plt.figure(figsize=(10, 6)) + # Melt for seaborn + df_melt = df_ratios.melt(id_vars=['m', 'n'], value_vars=['ls_ratio', 'lpt_ratio'], + var_name='Algorithm', value_name='Ratio') + + sns.boxplot(data=df_melt, x='m', y='Ratio', hue='Algorithm') + plt.title('Approximation Ratio Distribution by Machine Count (m)') + plt.ylabel('Approximation Ratio (Greedy / Optimal)') + plt.xlabel('Number of Machines (m)') + plt.savefig('results/ratio_boxplot.png') + plt.close() + + # Plot vs N + plt.figure(figsize=(10, 6)) + sns.lineplot(data=df_melt, x='n', y='Ratio', hue='Algorithm', style='m', markers=True) + plt.title('Approximation Ratio vs Job Count (n)') + plt.savefig('results/ratio_vs_n.png') + plt.close() + + # 2. Running Time Comparison + plt.figure(figsize=(10, 6)) + df_time = df.groupby(['n', 'm']).mean().reset_index() + # Log scale for time + plt.plot(df_time['n'], df_time['ls_time'], label='List Scheduling', marker='o') + plt.plot(df_time['n'], df_time['lpt_time'], label='LPT', marker='x') + + # Only plot Opt time where available (it drops to 0/empty for large n) + df_opt = df_time[df_time['opt_time'] > 0] + if not df_opt.empty: + plt.plot(df_opt['n'], df_opt['opt_time'], label='Optimal (B&B)', marker='s') + + plt.yscale('log') + plt.title('Average Running Time vs Input Size (n)') + plt.ylabel('Time (microseconds)') + plt.xlabel('Number of Jobs (n)') + plt.legend() + plt.savefig('results/time_comparison.png') + plt.close() + +if __name__ == "__main__": + plot_core_algo() diff --git a/greed/task.txt b/greed/task.txt new file mode 100644 index 0000000..09cd599 --- /dev/null +++ b/greed/task.txt @@ -0,0 +1,27 @@ +对多机调度算法进行分析，具体要求如下： + + 针对多机调度问题，实现基于两种贪心策略的贪心算法； + 针对多机调度问题，实现遍历的最优解求解算法（也可以用回溯等其它算法）； + 针对两种贪心策略，构造问题输入，使得贪心算法结果接近最差，结合证明过程展开讨论； + 以处理机数量m, 作业数量n为输入规模，固定m, n，随机产生大量测试样本，用两种贪心算法分别求解，并计算最优解（无法在合理时间内完成最优解计算则记录为“最优解求解失败”）及近似解上界，对贪心解近似比的概率分布展开分析； + 改变m和n，对不同组合的结果进行对比分析，并撰写实验报告。 + +附加：模拟一个GPU集群在线调度问题，该集群有m块GPU，共享开放给全校师生。该集群有以下特点： + + 用户提交任务的时间点符合泊松分布，单个任务使用单块GPU所需的时间符合均匀分布。 + 假设提交的任务均具有高度并行性，可拆分到任意多块GPU并行执行，但是由于节点间通信、机架间通信等开销，k块GPU并行时单块效率降为原来的σlogk倍。例如2块GPU并行时，单块GPU性能为σ；4块GPU并行时，单块GPU的性能为σ2。对于不同任务，σ为[0.75, 0.95]之间均匀分布的小数。 + GPU数量m = 64，并行运算时通常使用2的整幂次块GPU，如2、4、8、16、32、64。 + 对任务i，用户期望的完成时间为任务提交时刻ti，加上单块GPU执行任务所需时间τi。 + 系统有两个关键指标：集群利用率η，用户平均延迟δ。对于η，即任务期内集群所有GPU的平均利用率。对于δ，在用户期望时间之内完成的任务其延迟为0，超出之后按平方惩罚：设任务i结束时间为ti′，则其延迟为δi=0, ti′≤ti+τi时，或 δi=((ti′−ti−τi)/τi)^2, ti′>ti+τi时。 + + 请针对该场景： + + 考虑多种优化目标：1）仅考虑η；2）仅考虑δ；3）均衡的优化目标(1−η)+λδ，其中λ为设置的常数平衡因子。 + + 模拟生成多组任务集（注意考虑轻负载、中等负载、重负载等不同情况）。 + + 设计两种以上调度策略。 + + 使用调度策略对计算过程进行模拟，按照不同的优化目标对结果进行分析对比，撰写实验报告。 + + diff --git a/horse_travel/README.md b/horse_travel/README.md new file mode 100644 index 0000000..575b951 --- /dev/null +++ b/horse_travel/README.md @@ -0,0 +1,27 @@ +一、问题说明 + + 国际象棋的棋盘为8*8的方格棋盘，现将“马”放在任意指定的方格中，按照“马”走棋的规则（与中国象棋规则一样，马走“日”字）将“马”进行移动。要求每个方格只能进入一次，最终使得“马”走遍棋盘64个方格，回到起点。 + + 编写代码，实现对棋盘的马踏棋盘操作，给定初始位置，用数字给出“马”移动的路径并格式化输出。 + + 必须实现的算法：分治法。 + +二、相关要求 + + 要求提交程序和实验报告，打包成“作业X-程序语言.zip”（如“作业4-c.zip”、“作业4-java.zip”，其中c/c++语言统一用“c”表示）提交。提交前把Debug, Release等编译文件删除，只保留代码源文件。 + + 程序：不限语言，但不得用封装好的算法函数直接求解；一般不得依赖非标准库（c++可以使用stl，不可使用boost库），特殊情况需在实验报告中予以说明。 + + 实验报告：解题思路；所实现算法的时间复杂度分析（结合程序统计关键步骤运行次数，以验证分析结果）；程序运行指导，包括程序编译说明、输入输出示例等。如果输入、输出信息较多，建议采用文件进行格式化输入、输出。 + + 一题多解：在实现基础要求之外，鼓励一题多解，可酌情加分。请在实验报告中进行分析说明。 + + 评分说明：程序和实验报告等重要，综合得到本题分数，评分要点如下 + + 程序：程序正确性（至少实现对n*n棋盘的处理）、注释完整性、关键函数的通用性、程序使用的方便性、边界处理等。 + + 实验报告：解题思路正确性与描述清晰、时间复杂度分析的正确性与完整性、运行指导的质量等。 + + 匿名：提交的所有内容保持匿名性，泄露个人信息将被扣分。 + + 抄袭与迟交：发现抄袭或雷同，抄袭者或雷同双方本次作业记0分。截止时间前可更新作业，包括一题多解；截止时间后不再接受一题多解。 diff --git a/horse_travel/main b/horse_travel/main new file mode 100755 index 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550 + /Root 549 0 R + /Info 547 0 R + /ID [(MeJUW7lVNqMsol1TFZ3OTw==) (MeJUW7lVNqMsol1TFZ3OTw==)] +>> +startxref +541410 +%%EOF \ No newline at end of file diff --git a/greed/main.typ b/greed/main.typ new file mode 100644 index 0000000..5373bad --- /dev/null +++ b/greed/main.typ @@ -0,0 +1,202 @@ +#import "labtemplate.typ": * +#show: nudtlabpaper.with( + author1: "程景愉", + id1: "202302723005", + advisor: " 胡罡", + jobtitle: "教授", + lab: "306-707", + date: "2025.12.18", + header_str: "贪心算法分析实验报告", + minimal_cover: true, +) + +#set page(header: [ + #set par(spacing: 6pt) + #align(center)[#text(size: 11pt)[《算法设计与分析》实验报告]] + #v(-0.3em) + #line(length: 100%, stroke: (thickness: 1pt)) +],) + +#show heading: it => box(width: 100%)[ + #v(0.50em) + #set text(font: hei) + #it.body +] + +#outline(title: "目录",depth: 3, indent: 1em) +// #pagebreak() +#outline( + title: [图目录], + target: figure.where(kind: image), +) + +#show heading: it => box(width: 100%)[ + #v(0.50em) + #set text(font: hei) + #counter(heading).display() + #it.body +] +#set enum(indent: 0.5em,body-indent: 0.5em,) +#pagebreak() + += 实验介绍 +#para[ +贪心算法（Greedy Algorithm）是指在对问题求解时，总是做出在当前看来是最好的选择。也就是说，不从整体最优上加以考虑，算法得到的是在某种意义上的局部最优解。多机调度问题是经典的 NP-Hard 问题，本实验旨在通过实现和对比不同的贪心策略（List Scheduling 和 LPT），深入理解贪心算法的近似比性质，并探讨其在实际场景（如 GPU 集群调度）中的应用。 +] + += 实验内容 +#para[ +本实验主要围绕多机调度问题的贪心算法展开，并扩展至在线 GPU 集群调度模拟。具体内容包括： +] ++ 实现两种贪心策略：任意顺序列表调度 (List Scheduling, LS) 和 最长处理时间优先 (Longest Processing Time, LPT)。 ++ 实现基于分支限界 (Branch and Bound) 的最优解求解算法，作为性能评估的基准。 ++ 构造特定的“最坏情况”输入，验证贪心算法的理论近似比下界。 ++ 通过大量随机测试样本，统计不同算法的近似比分布及运行时间，分析 $m$ (机器数) 和 $n$ (作业数) 对性能的影响。 ++ （附加）模拟 GPU 集群在线调度场景，设计并对比不同的调度策略在不同负载下的表现。 + += 实验要求 +#para[ +针对多机调度问题，实验具体要求如下： +] ++ 针对多机调度问题，实现 LS 和 LPT 两种贪心算法。 ++ 实现遍历的最优解求解算法（分支限界法）。 ++ 构造最坏情况输入，结合理论证明进行讨论。 ++ 固定 $m, n$，随机产生大量样本，计算贪心解与最优解的比值（近似比），并分析其概率分布。 ++ 改变 $m, n$，对比分析结果。 ++ 附加：模拟 GPU 集群调度，考虑利用率 $eta$ 和用户延迟 $delta$，设计多种策略并分析。 + += 实验步骤 + +== 算法设计 + +=== 算法一：列表调度 (List Scheduling, LS) +#para[ +LS 算法是最朴素的贪心策略。它按照作业输入的任意顺序，依次将作业分配给当前负载最小的机器。 +该算法是一种在线算法，其时间复杂度为 $O(n log m)$ (使用优先队列维护机器负载) 或 $O(n m)$ (线性扫描)。 +理论上，LS 算法的近似比为 $2 - 1/m$。 +] +```cpp +// 核心代码片段 +long long greedy_ls(int m, const vector& jobs) { + vector machines(m, 0); + for (const auto& job : jobs) { + int min_idx = 0; // Find machine with min load + for (int i = 1; i < m; ++i) { + if (machines[i] < machines[min_idx]) min_idx = i; + } + machines[min_idx] += job.duration; + } + return *max_element(machines.begin(), machines.end()); +} +``` + +=== 算法二：最长处理时间优先 (LPT) +#para[ +LPT 算法在 LS 的基础上增加了预处理步骤：将所有作业按处理时间递减排序，然后依次分配给负载最小的机器。 +排序操作使得较大的作业优先被处理，从而避免了最后剩下一个大作业导致机器负载极不均衡的情况。 +该算法的时间复杂度主要由排序决定，为 $O(n log n)$。 +理论上，LPT 算法的近似比为 $4/3 - 1/(3m)$。 +] + +=== 算法三：最优解 (Branch and Bound) +#para[ +为了评估贪心算法的性能，我们需要求得问题的最优解。由于多机调度是 NP-Complete 问题，我们采用深度优先搜索配合分支限界 (Branch and Bound) 来求解。 +剪枝策略包括： +] +1. 当前最大负载已经超过已知最优解，停止搜索。 +2. 理论下界剪枝：如果 `max(当前最大负载, (剩余作业总长 + 当前总负载)/m)` 超过已知最优解，停止搜索。 +3. 对称性剪枝：若多台机器当前负载相同，则分配给它们是等价的，只尝试第一台。 + +== 最坏情况构造与分析 +=== LS 算法最坏情况 +#para[ +*构造方法：* 对于 $m$ 台机器，输入 $m(m-1)$ 个时长为 1 的小作业，紧接着 1 个时长为 $m$ 的大作业。 +] +#para[ +*分析：* LS 算法会将前 $m(m-1)$ 个小作业均匀分配给 $m$ 台机器，每台机器负载为 $m-1$。最后的大作业将被分配给任意一台机器，使其最终负载变为 $(m-1) + m = 2m-1$。 +而最优解是将所有小作业均匀分配给 $m-1$ 台机器（每台负载 $m$），将大作业单独分配给剩下一台机器（负载 $m$），此时 MakeSpan 为 $m$。 +近似比为 $(2m-1)/m = 2 - 1/m$。 +本实验通过代码验证了 $m=3, 4, 5$ 时的该情况，结果与理论完全一致。 +] + +=== LPT 算法最坏情况 +#para[ +*构造方法：* 经典的 LPT 最坏情况较为复杂，例如 $m=2$ 时，作业集为 $\{3, 3, 2, 2, 2\}$。 +] +#para[ +*分析：* 排序后为 $3, 3, 2, 2, 2$。 +LPT 分配：M1: $3, 2, 2$ (总 7), M2: $3, 2$ (总 5)。MakeSpan = 7。 +最优解：M1: $3, 3$ (总 6), M2: $2, 2, 2$ (总 6)。MakeSpan = 6。 +近似比 $7/6 approx 1.167$。理论界 $4/3 - 1/6 = 7/6$。实验验证吻合。 +] + +== 实验数据与可视化 +#para[ +我们对 $m in {3, 5, 8}$ 和 $n in {10, dots, 100}$ 进行了大量随机测试。 +] + +#figure( + image("results/ratio_boxplot.png", width: 80%), + caption: [LS 与 LPT 算法近似比分布对比], +) + +#figure( + image("results/ratio_vs_n.png", width: 80%), + caption: [近似比随作业数量 n 的变化趋势], +) + +#figure( + image("results/time_comparison.png", width: 80%), + caption: [算法平均运行时间对比], +) + += 实验结果分析 +#para[ +1.*近似比性能：* 从箱线图可以看出，LPT 算法的近似比极其接近 1（通常在 1.0 - 1.05 之间），性能极其优越且稳定。相比之下，LS 算法的近似比分布较宽，平均在 1.1 - 1.3 之间，且随着 $m$ 的增加，最差情况（近似比上界）有升高的趋势，符合 $2 - 1/m$ 的理论预测。 +] +#para[ +2.*规模的影响：* 随着作业数 $n$ 的增加，LS 的近似比往往会下降并趋于稳定。这是因为大量随机作业往往能“填平”机器间的负载差异。LPT 则始终保持高效。 +] +#para[ +3.*运行时间：* 贪心算法（LS, LPT）的运行时间极短（微秒级），且随 $n$ 线性或近线性增长。最优解算法（B&B）随 $n$ 指数级增长，当 $n > 20$ 时已难以在短时间内求解，验证了 NP-Hard 问题的计算复杂性。 +] + += 实验总结 +#para[ +本实验深入分析了多机调度问题的贪心求解策略。实验结果表明，虽然 LS 算法实现简单，但在最坏情况下性能较差。简单的排序预处理（LPT 策略）能带来巨大的性能提升，使其在绝大多数随机及构造测试中都能获得极接近最优解的结果。这启示我们在设计贪心算法时，合理的贪心顺序（如优先处理“困难”或“大”的任务）至关重要。 +] + +#pagebreak() += 附加：GPU 集群在线调度模拟 + +== 场景描述 +#para[ +模拟一个拥有 $m=64$ 块 GPU 的集群任务调度。任务到达服从泊松分布，单机执行时间服从均匀分布。任务支持并行 ($k$ 块 GPU)，但存在并行效率损耗：效率因子 $E_k = sigma^(log_2 k)$，其中 $sigma in [0.75, 0.95]$。系统目标是平衡 *集群利用率 ($eta$)* 和 *用户平均延迟 ($delta$)*。 +] + +== 调度策略设计 +#para[ +我们设计了三种策略进行对比： +] ++ *策略 A：保守策略 (Conservative)* 总是为每个任务分配 $k=1$ 块 GPU。其思路是最大化计算资源的“有效性”，避免并行损耗。 ++ *策略 B：激进策略 (Aggressive)* 总是尽可能分配最大的并行度（如 $k=32$ 或 $64$）。其思路是最小化单任务执行时间，但忽略了巨大的资源浪费。 ++ *策略 C：自适应策略 (Adaptive)* 根据当前等待队列的长度动态调整 $k$。若队列为空，使用高并行度加速；若队列拥堵，降低并行度以提高吞吐量。 + +== 模拟结果 + +#figure( + image("results/gpu_sim_plots.png", width: 90%), + caption: [不同负载下三种策略的利用率与延迟对比], +) + +#para[ +实验在轻负载 ($lambda=0.5$)、中负载 ($lambda=0.9$) 和重负载 ($lambda=1.1$) 下进行了模拟。结果显示： +] ++ *保守策略 (Conservative)*：在所有负载下都能保持较低的延迟。 ++ *激进策略 (Aggressive)*：表现极差。由于并行效率损失，导致系统迅速过载，用户延迟呈爆炸式增长。 ++ *自适应策略 (Adaptive)*：表现最为均衡。在轻负载时加速任务，在重负载时保证系统稳定性。 + +== 结论 +#para[ +在具有并行开销的资源调度场景中，盲目追求高并行度（激进策略）是不可取的。通过感知系统负载来动态调整资源分配粒度的 *自适应策略*，是更为优越的解决方案。 +] diff --git a/greed/multimachine b/greed/multimachine new file mode 100755 index 0000000000000000000000000000000000000000..e6986695a50ef7341cf77fc9438c8757b4fec341 GIT binary patch literal 46640 zcmeHwdtg-6wfC7M1Cd82SddgvM>;AIjF|*5qf#d%fip0nDTzp}dI-scj3$}f%p`zS z2qsZahe6tEwXMCS{rXVb-g5g{rRSKjl&QlkXJ^vM zF9fIfPQysZb0l6!bs8vK?E8@&J-kxS|8KP 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