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internbootcamp/bootcamp/ckleofášandthenthlon/ckleofášandthenthlon.py

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+"""# 
+
+### 谜题描述
+Kleofáš is participating in an n-thlon - a tournament consisting of n different competitions in n different disciplines (numbered 1 through n). There are m participants in the n-thlon and each of them participates in all competitions.
+
+In each of these n competitions, the participants are given ranks from 1 to m in such a way that no two participants are given the same rank - in other words, the ranks in each competition form a permutation of numbers from 1 to m. The score of a participant in a competition is equal to his/her rank in it.
+
+The overall score of each participant is computed as the sum of that participant's scores in all competitions.
+
+The overall rank of each participant is equal to 1 + k, where k is the number of participants with strictly smaller overall score.
+
+The n-thlon is over now, but the results haven't been published yet. Kleofáš still remembers his ranks in each particular competition; however, he doesn't remember anything about how well the other participants did. Therefore, Kleofáš would like to know his expected overall rank.
+
+All competitors are equally good at each discipline, so all rankings (permutations of ranks of everyone except Kleofáš) in each competition are equiprobable.
+
+Input
+
+The first line of the input contains two space-separated integers n (1 ≤ n ≤ 100) and m (1 ≤ m ≤ 1000) — the number of competitions and the number of participants respectively.
+
+Then, n lines follow. The i-th of them contains one integer xi (1 ≤ xi ≤ m) — the rank of Kleofáš in the i-th competition.
+
+Output
+
+Output a single real number – the expected overall rank of Kleofáš. Your answer will be considered correct if its relative or absolute error doesn't exceed 10 - 9.
+
+Namely: let's assume that your answer is a, and the answer of the jury is b. The checker program will consider your answer correct, if <image>.
+
+Examples
+
+Input
+
+4 10
+2
+1
+2
+1
+
+
+Output
+
+1.0000000000000000
+
+
+Input
+
+5 5
+1
+2
+3
+4
+5
+
+
+Output
+
+2.7500000000000000
+
+
+Input
+
+3 6
+2
+4
+2
+
+
+Output
+
+1.6799999999999999
+
+Note
+
+In the first sample, Kleofáš has overall score 6. Nobody else can have overall score less than 6 (but it's possible for one other person to have overall score 6 as well), so his overall rank must be 1.
+
+Here is a reference code to solve this task. You can use this to help you genereate cases or validate the solution.
+```python
+#include <bits/stdc++.h>
+using namespace std;
+int n, m, sum;
+int a[101];
+long double f[101000], s[101000], ans;
+int main() {
+  cin >> n >> m;
+  if (m == 1) {
+    printf(\"1.0000000000000000\n\");
+    return 0;
+  }
+  f[0] = m - 1;
+  int now = 0;
+  for (int i = 1; i <= n; i++) scanf(\"%d\", &a[i]), sum += a[i];
+  for (int i = 1; i <= n; i++) {
+    for (int j = 0; j <= m * i; j++) s[j] = 0;
+    for (int j = 0; j <= m * i; j++) {
+      if (f[j] < 1e-12) continue;
+      s[j + 1] += f[j];
+      s[j + a[i]] -= f[j];
+      s[j + a[i] + 1] += f[j];
+      s[j + m + 1] -= f[j];
+    }
+    for (int j = 1; j <= m * i; j++) s[j] += s[j - 1];
+    for (int j = 0; j <= m * i; j++) f[j] = s[j] / (m - 1);
+  }
+  for (int i = 0; i < sum; i++) ans += f[i];
+  printf(\"%.16lf\n\", (double)ans + 1);
+}
+```
+
+
+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。
+"""
+
+from bootcamp import Basebootcamp
+import random
+import re
+from bootcamp import Basebootcamp
+
+class Ckleofášandthenthlonbootcamp(Basebootcamp):
+    def __init__(self, n_min=1, n_max=10, m_min=1, m_max=20):
+        self.n_min = n_min
+        self.n_max = n_max
+        self.m_min = m_min
+        self.m_max = m_max
+
+    def case_generator(self):
+        n = random.randint(self.n_min, self.n_max)
+        m = random.randint(self.m_min, self.m_max)
+        xi = [random.randint(1, m) for _ in range(n)]
+        # Ensure xi is 1 if m=1
+        if m == 1:
+            xi = [1] * n
+        return {'n': n, 'm': m, 'xi': xi}
+
+    @staticmethod
+    def prompt_func(question_case):
+        n = question_case['n']
+        m = question_case['m']
+        xi = question_case['xi']
+        xi_str = '\n'.join(map(str, xi))
+        problem = (
+            "Kleofáš participates in an n-thlon tournament with {n} competitions and {m} participants.\n"
+            "Each competition assigns unique ranks from 1 to {m}. Kleofáš's ranks are:\n{xi_str}\n"
+            "Calculate his expected overall rank. Format the answer as a float with 12+ decimal places within [answer]...[/answer]."
+        ).format(n=n, m=m, xi_str=xi_str)
+        return problem
+
+    @staticmethod
+    def extract_output(output):
+        matches = re.findall(r'\[answer\](.*?)\[/answer\]', output, re.DOTALL)
+        if not matches:
+            return None
+        last_match = matches[-1].strip()
+        try:
+            return float(last_match)
+        except ValueError:
+            return None
+
+    @classmethod
+    def _verify_correction(cls, solution, identity):
+        if solution is None:
+            return False
+        expected = cls.compute_expected_rank(identity)
+        abs_err = abs(solution - expected)
+        rel_err = abs_err / max(1e-9, abs(expected))
+        return abs_err <= 1e-9 or rel_err <= 1e-9
+
+    @classmethod
+    def compute_expected_rank(cls, identity):
+        n, m = identity['n'], identity['m']
+        xi = identity['xi']
+        sum_kl = sum(xi)
+        
+        if m == 1:
+            return 1.0
+        
+        max_possible = n * m
+        dp = [0.0] * (max_possible + 2)
+        dp[0] = m - 1.0
+        
+        for score in xi:
+            new_dp = [0.0] * (max_possible + 2)
+            for current_sum in range(len(dp)):
+                if dp[current_sum] < 1e-12:
+                    continue
+                # Left segment: 1 to score-1
+                start = current_sum + 1
+                end = current_sum + score
+                if start <= max_possible:
+                    new_dp[start] += dp[current_sum]
+                    if end <= max_possible:
+                        new_dp[end] -= dp[current_sum]
+                # Right segment: score+1 to m
+                start = current_sum + score + 1
+                end = current_sum + m + 1
+                if start <= max_possible:
+                    new_dp[start] += dp[current_sum]
+                    if end <= max_possible:
+                        new_dp[end] -= dp[current_sum]
+            
+            # Compute prefix sums and normalize
+            for j in range(1, max_possible + 1):
+                new_dp[j] += new_dp[j - 1]
+            for j in range(max_possible + 1):
+                new_dp[j] /= (m - 1)
+            dp = new_dp
+        
+        total = sum(dp[:sum_kl])
+        return total * (m - 1) + 1.0