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Complete lab2
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‎lab2/bfprt_select.py‎

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def median(arr: list) -> int:
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"""返回 arr 的中位数, arr 的长度不超过 5"""
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assert len(arr) <= 5
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arr.sort()
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return arr[len(arr) // 2]
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def partition(arr: list, MoM: int) -> tuple:
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"""根据 MoM 划分 arr 为三部分：L, E, G"""
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L, E, G = [], [], [] # less, equal, greater
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for num in arr:
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if num < MoM:
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L.append(num)
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elif num == MoM:
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E.append(num)
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else:
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G.append(num)
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return L, E, G
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def bfprt(arr: list) -> int:
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"""返回 arr 的中位数的中位数"""
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n = len(arr)
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if n <= 5:
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return median(arr)
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m = n // 5
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groups = [arr[i * 5:(i + 1) * 5] for i in range(m)]
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medians = [median(group) for group in groups]
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return bfprt(medians)
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def bfprt_select(arr: list, k: int) -> int:
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"""返回 arr 中第 k 小的元素"""
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# 1. 将 arr 划分为 n//5 组，每组 5 个元素
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# 2. 对每个组进行排序，找到其中位数
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# 3. 递归地调用 bfprt_select，找到这些中位数的中位数 MoM
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# 4. 以 MoM 为基准，划分 arr 为三部分：L, E, G
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# 5. 根据 k 与 L, E, G 的大小关系，递归地调用 bfprt_select
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# 6. 返回结果
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if len(arr) <= 5:
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return sorted(arr)[k] # 直接返回第 k 小的元素
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MoM = bfprt(arr)
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L, E, G = partition(arr, MoM)
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if k < len(L):
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return bfprt_select(L, k)
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elif k < len(L) + len(E):
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return E[0]
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else:
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return bfprt_select(G, k - len(L) - len(E))

‎lab2/fig/normal.png‎

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‎lab2/fig/theta.png‎

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‎lab2/fig/uniform.png‎

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‎lab2/fig/zipf.png‎

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‎lab2/gen_data.py‎

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import numpy as np
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def gen_data(data_type: str, n: int, iter_num: int) -> (list, list):
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k_list = [np.random.randint(0, n) for _ in range(iter_num)]
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if data_type == "uniform":
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return np.random.uniform(0, 1, n).tolist(), k_list
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elif data_type == "normal":
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return np.random.normal(0, 1, n).tolist(), k_list
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elif data_type == "zipf":
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return np.random.zipf(2, n).tolist(), k_list

‎lab2/lazy_select.py‎

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import random
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from math import sqrt, floor
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def rank(arr: list, x: int) -> int:
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"""返回 arr 中小于 x 的元素个数"""
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return sum(1 for num in arr if num < x)
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def min_k(sorted_arr: list, k: int) -> int:
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"""返回有序数组 sorted_arr 中第 k 小的元素"""
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return sorted_arr[k]
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def lazy_select(arr: list, k: int, theta: float = 3 / 4) -> int:
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"""拉斯维加斯算法，返回 arr 中第 k 小的元素"""
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n = len(arr)
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R_len = int(n ** theta)
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R = random.choices(arr, k=R_len) # 随机选择 n^(3/4) 个元素
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R.sort() # 此排序的时间复杂度为 O(n)
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x = (k / n) * R_len # arr 的第 k 小元素可能成为 R 的第 x 小元素
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l, h = max(floor(x - sqrt(n)), 0), min(floor(x + sqrt(n)), R_len - 1) # 考察区间 [l, h]
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L, H = min_k(R, l), min_k(R, h)
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Lp, Hp = rank(arr, L), rank(arr, H)
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P = [num for num in arr if L <= num <= H] # 将 arr 中介于 L, H 之间的元素放入 P
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if Lp <= k <= Hp and len(P) <= 4 * n ** theta + 1:
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P.sort()
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return min_k(P, k - Lp)
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else:
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if R_len < n:
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return lazy_select(arr, k, min(theta + 0.05, 1)) # 略微提高 R_len 的大小
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else:
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arr.sort()
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return min_k(arr, k)

‎lab2/main.py‎

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# This is a sample Python script.
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"""
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比较 3 种中位数选择算法的性能
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- 算法 1：排序后选择
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- 算法 2： 确定型中位数线性时间选择 (BFPRT)
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- 算法 3： 中位数选择随机算法
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# Press Shift+F10 to execute it or replace it with your code.
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# Press Double Shift to search everywhere for classes, files, tool windows, actions, and settings.
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实验内容：
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- 实现三种算法
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- 数据集自己寻找或生成
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- 运行时间比较，准确度比较
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- 扩展性比较
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- 以恰当、准确、规范的形式表述实验结果
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"""
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import time
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import numpy as np
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import matplotlib.pyplot as plt
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def print_hi(name):
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# Use a breakpoint in the code line below to debug your script.
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print(f'Hi, {name}') # Press Ctrl+F8 to toggle the breakpoint.
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from sort_select import sort_select
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from bfprt_select import bfprt_select
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from lazy_select import lazy_select
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from gen_data import gen_data
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# Press the green button in the gutter to run the script.
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if __name__ == '__main__':
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print_hi('PyCharm')
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def run_select(arr: list, k_list: list, func) -> list:
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"""测试选择算法, 返回运行结果"""
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result = []
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for k in k_list:
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result.append(func(arr, k))
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return result
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# See PyCharm help at https://www.jetbrains.com/help/pycharm/
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def test_all_on_data(arr: list, k_list: list):
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run_time = []
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start_time = time.time()
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sort_select_result = run_select(arr, k_list, sort_select)
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run_time.append(time.time() - start_time)
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start_time = time.time()
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bfprt_select_result = run_select(arr, k_list, bfprt_select)
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run_time.append(time.time() - start_time)
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start_time = time.time()
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lazy_select_result = run_select(arr, k_list, lazy_select)
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run_time.append(time.time() - start_time)
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if (sort_select_result != bfprt_select_result) or (sort_select_result != lazy_select_result):
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print("Results are not equal!")
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return run_time
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def test(data_type: str, n_list: list, iter_num: int):
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run_times = [[] for _ in range(3)] # [[sort_select], [bfprt_select], [lazy_select
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for n in n_list:
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arr, k_list = gen_data(data_type, n, iter_num)
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run_time = test_all_on_data(arr, k_list)
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for i in range(3):
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run_times[i].append(run_time[i] / iter_num)
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fig = plt.figure(dpi=400)
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ax = fig.add_subplot(111)
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ax.plot(n_list, run_times[0], label="sort_select")
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ax.plot(n_list, run_times[1], label="bfprt_select")
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ax.plot(n_list, run_times[2], label="lazy_select")
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ax.set_xlabel("Data Size")
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ax.set_ylabel("Run Time")
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ax.set_title(("Run Time of Three Select Algorithms on " + data_type + " Data").title())
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ax.legend()
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plt.show()
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def test_theta(n: int, iter_num: int):
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theta_list = np.linspace(0.5, 1, 100).tolist()
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run_times = []
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for theta in theta_list:
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arr, k_list = gen_data("uniform", n, iter_num)
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start_time = time.time()
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for k in k_list:
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lazy_select(arr, k, theta)
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run_times.append((time.time() - start_time) / iter_num)
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fig = plt.figure(dpi=400)
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ax = fig.add_subplot(111)
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ax.plot(theta_list, run_times)
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ax.set_xlabel("Theta")
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ax.set_ylabel("Run Time")
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ax.set_title("Run Time of Lazy Select Algorithm on Different Theta")
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plt.show()
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def main():
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# 测试 3 种算法的性能和扩展性
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iter_num = 3 # 测试次数
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n_list = np.linspace(10000, 100000, 20, dtype=int).tolist() # 数据规模
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data_type_list = ["uniform", "normal", "zipf"]
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for data_type in data_type_list:
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test(data_type, n_list, iter_num)
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# 测试随机算法中的关键参数 theta 对性能的影响
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n = 10000
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test_theta(n, iter_num)
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if __name__ == "__main__":
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main()

‎lab2/sort_select.py‎

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def quick_sort(arr: list) -> list:
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"""对 arr 进行快速排序"""
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if len(arr) <= 1:
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return arr
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pivot = arr[len(arr) // 2]
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left = [x for x in arr if x < pivot]
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middle = [x for x in arr if x == pivot]
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right = [x for x in arr if x > pivot]
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return quick_sort(left) + middle + quick_sort(right)
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def sort_select(arr: list, k: int) -> int:
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"""将 arr 排序后，返回其中第 k 小的元素"""
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arr = quick_sort(arr)
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return arr[k]

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