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215 lines
6.3 KiB
Python
Executable file
215 lines
6.3 KiB
Python
Executable file
"""#
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### 谜题描述
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You are given two arrays a and b of positive integers, with length n and m respectively.
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Let c be an n × m matrix, where c_{i,j} = a_i ⋅ b_j.
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You need to find a subrectangle of the matrix c such that the sum of its elements is at most x, and its area (the total number of elements) is the largest possible.
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Formally, you need to find the largest number s such that it is possible to choose integers x_1, x_2, y_1, y_2 subject to 1 ≤ x_1 ≤ x_2 ≤ n, 1 ≤ y_1 ≤ y_2 ≤ m, (x_2 - x_1 + 1) × (y_2 - y_1 + 1) = s, and $$$∑_{i=x_1}^{x_2}{∑_{j=y_1}^{y_2}{c_{i,j}}} ≤ x.$$$
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Input
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The first line contains two integers n and m (1 ≤ n, m ≤ 2000).
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The second line contains n integers a_1, a_2, …, a_n (1 ≤ a_i ≤ 2000).
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The third line contains m integers b_1, b_2, …, b_m (1 ≤ b_i ≤ 2000).
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The fourth line contains a single integer x (1 ≤ x ≤ 2 ⋅ 10^{9}).
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Output
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If it is possible to choose four integers x_1, x_2, y_1, y_2 such that 1 ≤ x_1 ≤ x_2 ≤ n, 1 ≤ y_1 ≤ y_2 ≤ m, and ∑_{i=x_1}^{x_2}{∑_{j=y_1}^{y_2}{c_{i,j}}} ≤ x, output the largest value of (x_2 - x_1 + 1) × (y_2 - y_1 + 1) among all such quadruplets, otherwise output 0.
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Examples
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Input
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3 3
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1 2 3
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1 2 3
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9
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Output
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4
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Input
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5 1
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5 4 2 4 5
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2
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5
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Output
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1
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Note
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Matrix from the first sample and the chosen subrectangle (of blue color):
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<image>
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Matrix from the second sample and the chosen subrectangle (of blue color):
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<image>
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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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n,m=map(int,raw_input().split())
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a=list(map(int,raw_input().split()))
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b=list(map(int,raw_input().split()))
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x=int(raw_input())
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pref_a=[0]
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for i in a:
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pref_a.append(pref_a[-1]+i)
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pref_b=[0]
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for i in b:
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pref_b.append(pref_b[-1]+i)
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min_a=[float('inf')]*(n+1)
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min_b=[float('inf')]*(m+1)
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for i in range(1,n+1):
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for j in range(i,n+1):
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min_a[j-i+1]=min(min_a[j-i+1],pref_a[j]-pref_a[i-1])
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for i in range(1,m+1):
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for j in range(i,m+1):
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min_b[j-i+1]=min(min_b[j-i+1],pref_b[j]-pref_b[i-1])
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ans=0
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for i in range(1,n+1):
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for j in range(1,m+1):
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if min_a[i]*min_b[j]<=x:
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ans=max(ans,i*j)
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print(ans)
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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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from bootcamp import Basebootcamp
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def calculate_min_prefix_sums(arr):
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n = len(arr)
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prefix = [0] * (n + 1)
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for i in range(n):
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prefix[i+1] = prefix[i] + arr[i]
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min_sums = [float('inf')] * (n + 1)
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for k in range(1, n+1):
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min_sum = min(prefix[i+k] - prefix[i] for i in range(n - k + 1))
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min_sums[k] = min_sum
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return min_sums
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def calculate_max_area(n, m, a, b, x):
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min_a = calculate_min_prefix_sums(a)
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min_b = calculate_min_prefix_sums(b)
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max_area = 0
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for i in range(1, n+1):
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for j in range(1, m+1):
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if min_a[i] * min_b[j] <= x:
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max_area = max(max_area, i * j)
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return max_area
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class Cmaximumsubrectanglebootcamp(Basebootcamp):
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def __init__(self, max_n=2000, max_m=2000, max_val=2000):
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"""
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参数说明:
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max_n: a数组的最大长度（符合题目约束）
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max_m: b数组的最大长度（符合题目约束）
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max_val: 数组元素的最大值（符合题目约束）
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"""
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super().__init__()
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self.max_n = max_n
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self.max_m = max_m
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self.max_val = max_val
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def case_generator(self):
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# 生成随机长度（1 ≤ n, m ≤ 2000）
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n = random.randint(1, self.max_n)
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m = random.randint(1, self.max_m)
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# 生成随机数组（元素范围1~2000）
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a = [random.randint(1, self.max_val) for _ in range(n)]
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b = [random.randint(1, self.max_val) for _ in range(m)]
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# 计算最小子数组和
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min_a = calculate_min_prefix_sums(a)
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min_b = calculate_min_prefix_sums(b)
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# 收集所有可能的乘积
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products = []
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for i in range(1, n+1):
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for j in range(1, m+1):
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products.append(min_a[i] * min_b[j])
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# 生成x的策略（确保覆盖所有边界情况）
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if not products: # 理论上不可能发生
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x = 1
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else:
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min_product = min(products)
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max_product = max(products)
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# 生成模式选择（按概率分布）
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mode = random.choices(
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population=[0, 1, 2, 3],
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weights=[0.3, 0.3, 0.2, 0.2], # 增加无解情况的概率
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k=1
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)[0]
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if mode == 0: # 正常范围解
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x = random.randint(min_product, max_product)
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elif mode == 1: # 无解情况
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x = random.randint(1, min_product-1) if min_product > 1 else 0
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elif mode == 2: # 超大值覆盖所有解
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x = max_product * random.randint(1, 100)
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else: # 极小值特殊情况
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x = 1 if random.random() < 0.5 else 0
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# 确保x符合题目约束（1 ≤ x ≤ 2e9）
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x = max(1, min(x, 2*10**9))
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return {
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'n': n,
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'm': m,
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'a': a.copy(),
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'b': b.copy(),
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'x': x
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}
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@staticmethod
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def prompt_func(question_case) -> str:
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return f"""给定两个正整数数组：
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数组a（长度={question_case['n']}）：{question_case['a']}
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数组b（长度={question_case['m']}）：{question_case['b']}
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定义矩阵c，其中每个元素c[i][j] = a[i] × b[j]。请找到c中元素和不超过{question_case['x']}的最大矩形区域（元素个数最多）。如果不存在这样的区域，输出0。
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答案必须为整数且用[ANSWER]标签包裹，例如：[ANSWER]0[/ANSWER] 或 [ANSWER]42[/ANSWER]
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"""
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@staticmethod
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def extract_output(output):
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import re
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matches = re.findall(r'\[ANSWER\]\s*(\d+)\s*\[/ANSWER\]', output)
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try:
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return int(matches[-1]) if matches else None
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except:
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return None
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@classmethod
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def _verify_correction(cls, solution, identity):
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return solution == calculate_max_area(
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identity['n'], identity['m'],
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identity['a'], identity['b'],
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identity['x']
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)
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