前言
English version of this editorial is provided after the sample code.
题意简述
本题为交互题，你需要猜 \(t\) 个 \([0,2024]\) 间的非负整数 \(x_1,x_2,\ldots,x_t\)，可以询问最多 \(15\) 次，每次询问形如“给定一个大小为 \(N(1\le N\le 2025)\) 的集合 \(S\) 满足 \(S\) 的每个元素都是 \([0,2024]\) 间的非负整数，交互库回答每个 \(x_i\) 是否在该集合内”。
注意，你的询问需要一次性全部给出，之后交互库再进行回答。但是，对于每个 \(x_i\)，交互库会在最多一个询问上给出错误的回复，而你需要准确地找出每个 \(x_i\)。
题目解法
如果交互库不会给出错误的回复，那么解决问题是简单的：考虑二进制拆位，对于每一位，询问值域中所有该位为 \(1\) 的数构成的集合，如果目标数存在于该集合内，代表目标数的该位为 \(1\)，否则为 \(0\)。可以在 \(\lceil\log_2V\rceil\) 次询问内完成。
如果交互库会在最多一个询问上给出错误的回复呢？考虑给 \([0,2024]\) 间每个整数一个独特的“编号”：设整数 \(i\) 的编号为 \(a_i\)，如果 \(\forall i\ne j,\mathrm{popcount}(a_i\oplus a_j)\ge 3\)（其中 \(\mathrm{popcount}\) 表示一个整数二进制下 \(1\) 的个数，\(\oplus\) 表示按位异或），那么使用上面的询问方法询问这些编号，一定会得出正确的答案。具体确定答案的方法为，假设询问中得到了一个数 \(x\)，那么只要找到唯一一个 \(a_i\) 满足 \(\mathrm{popcount}(a_i\oplus x)\le 1\) 的 \(a_i\)，\(i\) 即为答案——可以证明这种编号构造方法能唯一确定答案。
现在的问题即为构造一组这样的 \(a_i\)。我们发现可以从 \(0\) 开始给每个数分配编号，将所有不合法的编号插入一个 std::set，之后找之中没出现过的最小非负整数，将其作为编号分配给下一个整数。用上面的方法进行暴力枚举，发现存在一组构造使得 \(a_i\le 32330\)，可以在 \(15\) 次询问内找到答案。
示例代码（C++17）
#include<bits/stdc++.h>
using namespace std;
const int N=2024;
int main(){
ios::sync_with_stdio(false);
vector<int> a; set<int> s;
for(int i=0,x=0;i<=N;i++){
a.emplace_back(x),s.emplace(x);
for(int j=0;j<15;j++)
for(int k=0;k<15;k++)
s.emplace(x^(1<<j)^(j==k?0:(1<<k)));
while(s.find(x)!=s.end())x++;
}
cerr<<a.back()<<endl;
for(int i=0;i<=N;i++)
for(int j=i+1;j<=N;j++)
assert(__builtin_popcount(a[i]^a[j])>=3);
int t; cin>>t;
vector<vector<int> > v(15);
cout<<"15\n";
for(int i=0;i<15;i++){
for(int j=0;j<=N;j++)
if(a[j]>>i&1)
v[i].emplace_back(j);
for(int j:v[i])cout<<j<<' ';
cout<<endl;
}
while(t--){
int c=0,r=-1;
for(int i=0;i<15;i++){
int x; cin>>x;
if(x)c|=1<<i;
}
for(int i=0;i<=N;i++)
if(__builtin_popcount(a[i]^c)<=1)r=i;
cout<<r<<endl;
}
return 0;
}
English Version
Translated by ChatGPT
Problem
This is an interactive problem, where you need to guess $ t $ non-negative integers $ x_1, x_2, \ldots, x_t $ within the range \([0, 2024]\). You can ask at most \(15\) queries, and each query takes the form: "Given a set $ S $ of size $ N(1\le N\le 2025)$, where every element of $ S $ is a non-negative integer within \([0, 2024]\), does the interactor return whether each $ x_i $ is in this set or not?"
Note that you must provide all the queries at once, after which the interactor will respond. However, for each $ x_i $, the system will give incorrect feedback for at most one query, and your goal is to accurately identify each $ x_i $.
Solution
If the interactor never gave incorrect feedback, the solution would be straightforward: Consider the binary representation of each number. For each bit position, query a set consisting of all numbers in the range whose value in that bit position is \(1\). If the target number belongs to that set, then the corresponding bit is \(1\); otherwise, it is \(0\). This could be done in $ \lceil \log_2 V \rceil $ queries.
However, with the interactor potentially giving an incorrect response for up to one query, we need to take a different approach. We assign a unique "code" to each integer in the range \([0, 2024]\). Let the code for integer $ i $ be $ a_i $, and ensure that for all $ i \neq j $, we have $ \mathrm{popcount}(a_i \oplus a_j) \geq 3 $, where $ \mathrm{popcount} $ denotes the number of \(1\)s in the binary representation of an integer, and $ \oplus $ represents the bitwise XOR operation. By querying these codes in the method described earlier, we can guarantee a correct result.
To determine the answer, suppose we obtain a result $ x $ from the queries. We need to find the unique $ a_i $ such that $ \mathrm{popcount}(a_i \oplus x) \leq 1 $, which will give us the correct $ i $. This method ensures that we can uniquely identify the correct answer.
The problem now becomes constructing such a set of $ a_i $. We can assign codes starting from \(0\), and for each code, insert invalid codes into a std::set. Then, for each subsequent number, assign the smallest non-negative integer that has not yet been used as a code. Using brute force exploration, we find that there exists a set of codes such that $ a_i \leq 32330 $, which allows us to find the answer within \(15\) queries.