Hash Function
Posted Jozky86
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题意：
给定n个互不相同的数，找一个最小的模域，使得它们在这个模域下互不相同。n<=5e5
题解：
考虑两个数a和b，a与b模m余数相同，当且仅当|a-b|能被m整除。
这样问题就转化成找到一个最下的m，使得m不是任意一个|ai-aj|约数(不会被m整除)
n<=5e5,所以直接暴力n2肯定不行
这时就要用到FFT/NTT加速
算N个数两两之差当然用FFT，利用多项式相乘即指数相加来做，(加减用指数，相乘用系数)
刚接触FFT的同学可能不明白，减法怎么用？FFT的模板不是乘法吗？
我们这样想：FFT可以快速求两个多项式的乘积，并求出每个指数所对应的系数，两个数相乘，指数相加，也就是如果我有一个多项式的指数为x，y（对应的系数都为1），另一个多项式的指数为z，w（对应的系数都为1），相乘后会得到指数为x+z,x+w,z+y,y+w,那如果我们将z和w分别为-x，-y，那就会得到x-x，x-y，y-x，y-y，这不正是x和y任意两数的差，该差是否存在就看他们的系数是否为1
这样就在N*logN求出所有差，（注意FFT中是没有指数为负的，所有我们加入一个偏移量MAX，让MAX-x，最后得到结果再加回去就行）
回到本题，要求是求最小的m，不是任意两数差的约数。现在我们求出所有的差并标记，然后枚举这个m，如果这个m的倍数均没有被标记，输出m
所以这算是个模板题了，会使用FFT/NTT板子
代码
NTT代码
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
const int maxn=5e5+100;
const int mx=5e5;
const int mod=998244353;
namespace IO{
template<typename T>void write(T x)
{
if(x<0)
{
putchar('-');
x=-x;
}
if(x>9)
{
write(x/10);
}
putchar(x%10+'0');
}
template<typename T> void read(T &x)
{
x = 0;char ch = getchar();int f = 1;
while(!isdigit(ch)){if(ch == '-')f*=-1;ch=getchar();}
while(isdigit(ch)){x = x*10+(ch-'0');ch=getchar();}x*=f;
}
};
namespace Math{
ll w;
const int p=::mod;
struct complex{
ll real,imag;
complex(ll a=0,ll b=0){
real=a;
imag=b;
}
friend complex operator*(const complex&a,const complex&b){
complex ans;
ans.real=((a.real*b.real%p+a.imag*b.imag%p*w%p)%p+p)%p;
ans.imag=((a.real*b.imag%p+a.imag*b.real%p)%p+p)%p;
return ans;
}
};
ll x1,x2;
ll ksm(ll a,ll b,ll p){
ll ans=1;
while(b){
if(b&1) ans=(ans*a)%p;
a=(a*a)%p;
b>>=1;
}
return ans;
}
ll ksm(complex a,ll b,ll p){
complex ans(1,0);
while(b){
if(b&1) ans=ans*a;
a=a*a;
b>>=1;
}
return ans.real%p;
}
bool Cipolla(ll n,ll&x0,ll&x1){
n%=p;
if(ksm(n,(p-1)>>1,p)==p-1) return false;
ll a;
while(true){
a=rand()%p;
w=((a*a%p-n)%p+p)%p;
if(ksm(w,(p-1)>>1,p)==p-1) break;
}
complex x(a,1);
x0=(ksm(x,(p+1)>>1,p)+p)%p;
x1=(p-x0+p)%p;
return true;
}
};
namespace NTT{
#define mul(x,y) ((1ll*x*y>=mod?x*y%mod:1ll*x*y))
#define dec(x,y) (1ll*x-y<0?1ll*x-y+mod:1ll*x-y)
#define add(x,y) (1ll*x+y>=mod?1ll*x+y-mod:1ll*x+y)
#define ck(x) (x>=mod?x-mod:x)
typedef vector<int> Poly;
int ksm(int a,int n,int mod=::mod){
int res=1;
while(n){
if(n&1)res=1ll*res*a%mod;
a=1ll*a*a%mod;
n>>=1;
}
return res;
}
const int img=86583718;
const int g=3,INV=ksm(g,mod-2);
const int mx=21;
int R[maxn<<2],deer[2][mx][maxn<<2],inv[maxn<<2];
void init(const int t) {
for(int p = 1; p <= t; ++ p) {
int buf1 = ksm(g, (mod - 1) / (1 << p));
int buf0 = ksm(INV, (mod - 1) / (1 << p));
deer[0][p][0] = deer[1][p][0] = 1;
for(int i = 1; i < (1 << p); ++ i) {
deer[0][p][i] = 1ll * deer[0][p][i - 1] * buf0 % mod;
deer[1][p][i] = 1ll * deer[1][p][i - 1] * buf1 % mod;
}
}
inv[1] = 1;
for(int i = 2; i <= (1 << t); ++ i)
inv[i] = 1ll * inv[mod % i] * (mod - mod / i) % mod;
}
int NTT_init(int n) {
int lim = 1, l = 0;
while(lim < n) lim <<= 1, l ++ ;
for(int i = 0; i < lim; ++ i)
R[i] = (R[i >> 1] >> 1) | ((i & 1) << (l - 1));
return lim;
}
void ntt(Poly &A, int type, int lim) {
A.resize(lim);
for(int i = 0; i < lim; ++ i)
if(i < R[i])
swap(A[i], A[R[i]]);
for(int mid = 2, j = 1; mid <= lim; mid <<= 1, ++ j) {
int len = mid >> 1;
for(int pos = 0; pos < lim; pos += mid) {
int *wn = deer[type][j];
for(int i = pos; i < pos + len; ++ i, ++ wn) {
int tmp = 1ll * (*wn) * A[i + len] % mod;
A[i + len] = ck(A[i] - tmp + mod);
A[i] = ck(A[i] + tmp);
}
}
}
if(type == 0) {
for(int i = 0; i < lim; ++ i)
A[i] = 1ll * A[i] * inv[lim] % mod;
}
}
Poly poly_mul(Poly A, Poly B) {
int deg = A.size() + B.size() - 1;
int limit = NTT_init(deg);
Poly C(limit);
ntt(A, 1, limit);
ntt(B, 1, limit);
for(int i = 0; i < limit; ++ i)
C[i] = 1ll * A[i] * B[i] % mod;
ntt(C, 0, limit);
C.resize(deg);
return C;
}
Poly poly_inv(Poly &f, int deg) {
if(deg == 1)
return Poly(1, ksm(f[0], mod - 2));
Poly A(f.begin(), f.begin() + deg);
Poly B = poly_inv(f, (deg + 1) >> 1);
int limit = NTT_init(deg << 1);
ntt(A, 1, limit), ntt(B, 1, limit);
for(int i = 0; i < limit; ++ i)
A[i] = B[i] * (2 - 1ll * A[i] * B[i] % mod + mod) % mod;
ntt(A, 0, limit);
A.resize(deg);
return A;
}
Poly poly_idev(Poly f) {
int n = f.size();
for(int i = n - 1; i-1>=0 ; -- i) f[i] = 1ll * f[i - 1] * inv[i] % mod;
f[0] = 0;
return f;
}
Poly poly_dev(Poly f) {
int n = f.size();
for(int i = 1; i < n; ++ i) f[i - 1] = 1ll * f[i] * i % mod;
f.resize(n - 1);
return f;
}
Poly poly_ln(Poly f, int deg) {
Poly A = poly_idev(poly_mul(poly_dev(f), poly_inv(f, deg)));
return A.resize(deg), A;
}
Poly poly_exp(Poly &f, int deg) {
//cerr<<deg<<endl;
if(deg == 1)
return Poly(1, 1);
Poly B = poly_exp(f, (deg + 1) >> 1);
B.resize(deg);
Poly lnB = poly_ln(B, deg);
for(int i = 0; i < deg; ++ i)
lnB[i] = ck(f[i] - lnB[i] + mod);
int limit = NTT_init(deg << 1);
ntt(B, 1, limit), ntt(lnB, 1, limit);
for(int i = 0; i < limit; ++ i)
B[i] = 1ll * B[i] * (1 + lnB[i]) % mod;
ntt(B, 0, limit);
B.resize(deg);
return B;
}
Poly poly_pow(Poly&f,int k){
f=poly_ln(f,f.size());
for(auto&x:f)x=1ll*x*k%mod;
return poly_exp(f,f.size());
}
Poly power(Poly f,int k1,int k2,int deg){
int s=0;
while(f[s]==0&&s<f.size())++s;
if(1ll*s*k1>=deg){
return vector<int>(deg);
}
int Inv=ksm(f[s],mod-2,mod);
int Mul=ksm(f[s],k2);
deg-=s;
for(int i=0;i<deg;++i)f[i]=f[i+s];
f.resize(deg);
for(int i=0;i<deg;++i)f[i]=1ll*f[i]*Inv%mod;
auto res1=poly_ln(f,deg);
for(int i=0;i<res1.size();++i)res1[i]=1ll*res1[i]*k1%mod;
auto res2=poly_exp(res1,deg);
for(int i=0;i<deg;++i)res2[i]=1ll*res2[i]*Mul%mod;
deg+=s;
int now=s*k1;
Poly res;res.resize(deg);
for(int i=deg-1;i>=now;--i)res[i]=res2[i-now];
for(int i=now-1;i>=0;--i)res[i]=0;
return res;
}
Poly Poly_Sqrt(Poly&f,int deg){
if(deg==1)return Poly(1,1);
Poly A(f.begin(),f.begin()+deg);
Poly B=Poly_Sqrt(f,(deg+1)>>1);
Poly IB=poly_inv(B,deg);
int lim=NTT_init(deg<<1);
ntt(A,1,lim),ntt(IB,1,lim);
for(int i=0;i<lim;++i){
A[i]=1ll*A[i]*IB[i]%mod;
}
ntt(A,0,lim);
for(int i=0;i<deg;++i){
A[i]=(1ll*A[i]+B[i])%mod*inv[2]%mod;
}
A.resize(deg);
return A;
}
Poly Sqrt(Poly&f,int deg){
const int Pow=ksm(2,mod-2);
int k1=1;
if(f[0]!=1){
k1=ksm(f[0],mod-2);
for(int i=1;i<f.size();++i){
f[i]=1ll*k1*f[i]%mod;
}
ll x0,x1;
assert(Math::Cipolla(f[0],x0,x1));
k1=min(x1,x0);
f[0]=1;
}
auto Ln=poly_ln(f,deg);
for(int i=0;i<f.size();++i){
Ln[i]=1ll*Ln[i]*Pow%mod;
}
auto Exp=poly_exp(Ln,deg);
for(int i=0;i<Exp.size();++i)Exp[i]=1ll*Exp[i]*k1%mod;
return Exp;
}
Poly poly_sin(Poly&f,int deg){
Poly A(f.begin(),f.begin()+deg);
Poly B(deg),C(deg);
for(int i=0;i<deg;++i){
A[i]=1ll*A[i]*img%mod;
}
B=poly_exp(A,deg);
C=poly_inv(B,deg);
const int inv2i=ksm(img<<1,mod-2);
for(int i=0;i<deg;++i){
A[i]=1ll*(1ll*B[i]-C[i]+mod)%mod*inv2i%mod;
}
return A;
}
Poly poly_cos(Poly&f,int deg){
Poly A(f.begin(),f.begin()+deg);
Poly B(deg),C(deg);
for(int i=0;i<deg;++i){
A[i]=1ll*A[i]*img%mod;
}
B=poly_exp(A,deg);
C=poly_inv(B,deg);
const int inv2=ksm(2,mod-2);
for(int i=0;i<deg;++i){
A[i]=(1ll*B[i]+C[i])%mod*inv2%mod;
}
return A;
}
Poly poly_arcsin(Poly f,int deg){
Poly A(f.size()),B(f.size()),C(f.size());
A=poly_dev(f);
B=poly_mul(f,f);
for(int i=0;i<deg;++i){
B[i]=dec(mod,B[i]);
}
B[0]=add(B[0],1);
C=Poly_Sqrt(B,deg);
C=poly_inv(C,deg);
C=poly_mul(A,C);
C=poly_idev(C);
return C;
}
Poly poly_arctan(Poly f, int deg) {
Poly A(f.size()), B(f.size()), C(f.size());
A = poly_dev(f);
B = poly_mul(f, f);
B[0] = add(B[0], 1)Hash Function