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208 lines
6.6 KiB
Python
Executable file
208 lines
6.6 KiB
Python
Executable file
"""#
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### 谜题描述
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Allen and Bessie are playing a simple number game. They both know a function f: \{0, 1\}^n → R, i. e. the function takes n binary arguments and returns a real value. At the start of the game, the variables x_1, x_2, ..., x_n are all set to -1. Each round, with equal probability, one of Allen or Bessie gets to make a move. A move consists of picking an i such that x_i = -1 and either setting x_i → 0 or x_i → 1.
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After n rounds all variables are set, and the game value resolves to f(x_1, x_2, ..., x_n). Allen wants to maximize the game value, and Bessie wants to minimize it.
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Your goal is to help Allen and Bessie find the expected game value! They will play r+1 times though, so between each game, exactly one value of f changes. In other words, between rounds i and i+1 for 1 ≤ i ≤ r, f(z_1, ..., z_n) → g_i for some (z_1, ..., z_n) ∈ \{0, 1\}^n. You are to find the expected game value in the beginning and after each change.
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Input
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The first line contains two integers n and r (1 ≤ n ≤ 18, 0 ≤ r ≤ 2^{18}).
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The next line contains 2^n integers c_0, c_1, ..., c_{2^n-1} (0 ≤ c_i ≤ 10^9), denoting the initial values of f. More specifically, f(x_0, x_1, ..., x_{n-1}) = c_x, if x = \overline{x_{n-1} … x_0} in binary.
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Each of the next r lines contains two integers z and g (0 ≤ z ≤ 2^n - 1, 0 ≤ g ≤ 10^9). If z = \overline{z_{n-1} ... z_0} in binary, then this means to set f(z_0, ..., z_{n-1}) → g.
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Output
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Print r+1 lines, the i-th of which denotes the value of the game f during the i-th round. Your answer must have absolute or relative error within 10^{-6}.
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Formally, let your answer be a, and the jury's answer be b. Your answer is considered correct if \frac{|a - b|}{max{(1, |b|)}} ≤ 10^{-6}.
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Examples
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Input
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2 2
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0 1 2 3
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2 5
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0 4
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Output
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1.500000
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2.250000
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3.250000
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Input
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1 0
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2 3
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Output
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2.500000
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Input
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2 0
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1 1 1 1
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Output
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1.000000
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Note
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Consider the second test case. If Allen goes first, he will set x_1 → 1, so the final value will be 3. If Bessie goes first, then she will set x_1 → 0 so the final value will be 2. Thus the answer is 2.5.
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In the third test case, the game value will always be 1 regardless of Allen and Bessie's play.
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Here is a reference code to solve this task. You can use this to help you genereate cases or validate the solution.
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```python
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from fractions import *
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from math import *
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from itertools import *
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from fractions import*
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import string
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import copy
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import random
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import bisect
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from decimal import *
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from collections import deque
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def id_generator(size=20, chars=string.digits):
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return ''.join(random.choice(chars) for _ in range(size))
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def mp():
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return map(int,str(raw_input()).split())
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n,r=mp()
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l=list(mp())
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x=pow(2,n)
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s=sum(l)
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print(s/float(x))
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for i in range(r):
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a,b=mp()
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s=s-l[a]
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s+=b
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l[a]=b
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print(s/float(x))
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```
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请完成上述谜题的训练场环境类实现，包括所有必要的方法。
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"""
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from bootcamp import Basebootcamp
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import random
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import re
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from math import isclose
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from bootcamp import Basebootcamp
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class Dgamebootcamp(Basebootcamp):
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def __init__(self, **params):
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self.n = params.get('n', 2)
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self.r = params.get('r', 0)
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# 验证n和r的范围
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if self.n < 1 or self.n > 18:
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raise ValueError("n must be between 1 and 18")
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if self.r < 0 or self.r > (2 ** 18):
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raise ValueError("r must be between 0 and 2^18")
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def case_generator(self):
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n = self.n
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r = self.r
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size = 2 ** n
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# 生成初始c数组，考虑边界情况
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initial_c = [random.randint(0, 10**9) for _ in range(size)]
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# 确保至少有一次修改，如果r>0
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updates = []
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for _ in range(r):
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z = random.randint(0, size - 1)
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g = random.randint(0, 10**9)
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updates.append((z, g))
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case = {
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'n': n,
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'initial_c': initial_c,
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'r': r,
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'updates': updates
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}
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return case
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@staticmethod
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def prompt_func(question_case):
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n = question_case['n']
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initial_c = question_case['initial_c']
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r = question_case['r']
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prompt = (
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"Allen和Bessie正在玩一个数字游戏。游戏规则如下：\n"
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"游戏由一个函数f决定，f接受n个二进制变量，返回一个实数值。"
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"初始时，所有变量设为-1，每轮随机选择一个玩家设置一个变量为0或1。"
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"游戏目标是计算在每一轮修改后的期望值，输出r+1个结果，每个结果保留六位小数。\n"
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"初始函数值为："
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)
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prompt += f"{initial_c}\n"
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prompt += "每次修改操作如下：\n"
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for i, (z, g) in enumerate(question_case['updates'], 1):
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prompt += f"第{i}次修改：将f的值{z}修改为{g}\n"
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prompt += (
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"请输出每次修改后的期望值，每个值放在[answer]标签内，"
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"保留六位小数，例如：[answer]1.500000[/answer]。"
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)
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return prompt
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@staticmethod
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def extract_output(output):
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pattern = r'\[answer\](.*?)\[\/answer\]'
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matches = re.findall(pattern, output)
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if not matches:
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return None
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solutions = []
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for m in matches:
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m_clean = m.strip()
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# 处理可能的格式问题，例如多余的空格或换行符
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if m_clean == "":
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continue
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try:
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num = float(m_clean)
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solutions.append(num)
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except ValueError:
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pass
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if not solutions:
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return None
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return solutions
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@classmethod
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def _verify_correction(cls, solution, identity):
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if isinstance(solution, list):
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solutions = solution
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else:
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solutions = [solution]
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n = identity['n']
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size = 2 ** n
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initial_c = identity['initial_c']
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updates = identity['updates']
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r = identity['r']
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correct = []
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current_sum = sum(initial_c)
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correct_value = current_sum / size
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correct.append(correct_value)
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current_c = initial_c.copy()
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for z, g in updates:
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delta = g - current_c[z]
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current_sum += delta
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current_c[z] = g
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correct_value = current_sum / size
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correct.append(correct_value)
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if len(solutions) != len(correct):
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return False
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for s, c in zip(solutions, correct):
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if not isclose(s, c, rel_tol=1e-6, abs_tol=1e-6):
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return False
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return True
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