[CF_1282D]Guess the Root

2024-03-16 23:32

文章标签 cf guess 1282d

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Guess the Root

题解

~~拉格朗日板题~~

按理说只要n+1个点就可以表示出一个n阶的多项式，对于求法很容易想到高斯消元。

但高斯消元法太麻烦了，于是，我们便开始了拉格朗日插值法。因为它更方便。

拉格朗日插值法可以通过点值求出原式，即 $f\left(k \right )=\sum_{i=0}^{n}y_{i}\prod _{i\neq j}\frac{k-x_{j}}{x_{i}-x_{j}}$ 。

将所有的点带入后就成了一个关于 $k$ 的多项式，从 $0$ 到 $10^6+3$ 枚举 $k$ 即可。

源码

~~一个忘了在外面预处理inv的蒟蒻~~

#pragma GCC optimize(2)
#pragma GCC optimize(3)
#pragma GCC optimize("Ofast")
#pragma GCC optimize("inline")
#pragma GCC optimize("-fgcse")
#pragma GCC optimize("-fgcse-lm")
#pragma GCC optimize("-fipa-sra")
#pragma GCC optimize("-ftree-pre")
#pragma GCC optimize("-ftree-vrp")
#pragma GCC optimize("-fpeephole2")
#pragma GCC optimize("-ffast-math")
#pragma GCC optimize("-fsched-spec")
#pragma GCC optimize("unroll-loops")
#pragma GCC optimize("-falign-jumps")
#pragma GCC optimize("-falign-loops")
#pragma GCC optimize("-falign-labels")
#pragma GCC optimize("-fdevirtualize")
#pragma GCC optimize("-fcaller-saves")
#pragma GCC optimize("-fcrossjumping")
#pragma GCC optimize("-fthread-jumps")
#pragma GCC optimize("-funroll-loops")
#pragma GCC optimize("-fwhole-program")
#pragma GCC optimize("-freorder-blocks")
#pragma GCC optimize("-fschedule-insns")
#pragma GCC optimize("inline-functions")
#pragma GCC optimize("-ftree-tail-merge")
#pragma GCC optimize("-fschedule-insns2")
#pragma GCC optimize("-fstrict-aliasing")
#pragma GCC optimize("-fstrict-overflow")
#pragma GCC optimize("-falign-functions")
#pragma GCC optimize("-fcse-skip-blocks")
#pragma GCC optimize("-fcse-follow-jumps")
#pragma GCC optimize("-fsched-interblock")
#pragma GCC optimize("-fpartial-inlining")
#pragma GCC optimize("no-stack-protector")
#pragma GCC optimize("-freorder-functions")
#pragma GCC optimize("-findirect-inlining")
#pragma GCC optimize("-fhoist-adjacent-loads")
#pragma GCC optimize("-frerun-cse-after-loop")
#pragma GCC optimize("inline-small-functions")
#pragma GCC optimize("-finline-small-functions")
#pragma GCC optimize("-ftree-switch-conversion")
#pragma GCC optimize("-foptimize-sibling-calls")
#pragma GCC optimize("-fexpensive-optimizations")
#pragma GCC optimize("-funsafe-loop-optimizations")
#pragma GCC optimize("inline-functions-called-once")
#pragma GCC optimize("-fdelete-null-pointer-checks")
#include<cstdio>
#include<cstring>
#include<cmath>
#include<iostream>
#include<algorithm>
#include<vector>
#include<queue>
#include<map>
using namespace std;
typedef long long LL;
#define int LL
const int mo=1e6+3;
int y[15];
int qkpow(int a,int s){int t=1;while(s){if(s&1)t=t*a%mo;a=a*a%mo;s>>=1;}return t;
}
signed main(){for(int i=0;i<=10;i++){cout<<"? "<<i<<endl;cin>>y[i];if(!y[i]){cout<<"! "<<i<<endl;return 0;	}for(int j=0;j<=10;j++)if(i!=j)y[i]=y[i]*qkpow(i-j,mo-2)%mo;}for(int i=11;i<mo;i++){int S=0;for(int j=0;j<=10;j++){int sum=y[j];for(int k=0;k<=10;k++){if(k==j)continue;sum=sum*(i-k)%mo%mo;}S=(S+sum)%mo;}if(!S){cout<<"! "<<i<<endl;return 0;}}cout<<"! -1"<<endl;return 0;
}