Hash Function 2021牛客暑期多校训练营1 数论 + NTT

Hash Function

solution

任意选择该集合中的两个数，a，b，由于a%mod<>b%mod任意选择该集合中的两个数，a，b，由于a\%mod<>b\%mod任意选择该集合中的两个数，a，b，由于a%mod<>b%mod

推出(a−b)%mod<>0.推出(a-b)\%mod<>0.推出(a−b)%mod<>0.

因此只需要得到所有(c=a−b)的值，将c的因子，即(c%d==0)的d全部标记即可。因此只需要得到所有(c=a-b)的值，将c的因子，即(c\%d==0)的d全部标记即可。因此只需要得到所有(c=a−b)的值，将c的因子，即(c%d==0)的d全部标记即可。

由于数据范围0−5e5，n2会T，用NTT来加速.由于数据范围0-5e5，n^2会T，用NTT来加速.由于数据范围0−5e5，n2会T，用NTT来加速.

A=∑ai，B=∑a5e5−iA=\sum a_i，B=\sum a_{5e5 - i}A=∑ai，B=∑a5e5−i
C=A∗BC=A*BC=A∗B

遍历C，标记所有元素的因子。之后找到一个未被标记的最小元素即可.遍历C，标记所有元素的因子。之后找到一个未被标记的最小元素即可.遍历C，标记所有元素的因子。之后找到一个未被标记的最小元素即可.

code

/*SiberianSquirrel*//*CuteKiloFish*/
#include <bits/stdc++.h>
using namespace std;
#define gcd(a,b) __gcd(a,b)
#define Polynomial vector<ll>
#define Inv(x) quick_pow(x, mod - 2)
#define DEBUG(x, y) cout << x << ": " << y << '\n';
using ld = long double;
using ll = long long;
using ull = unsigned long long;
const ll mod = 998244353, mod_g = 3, img = 86583718;
//const ll mod = 1004535809, mod_g = 3;
const int N = int(1e5 + 10);
template<typename T> inline T read() {T x = 0, f = 1;char ch = getchar();while(ch < '0' || ch > '9') { if(ch == '-') f = -1; ch = getchar(); }while(ch >= '0' && ch <= '9') { x = x * 10 + ch - '0'; ch = getchar(); }return x * f;
}
template<typename T> inline T print(T x) {if(x < 0) { putchar('-'); x =- x; }if(x > 9) print(x / 10);putchar(x % 10 + '0');
}void print(Polynomial &a, int len){for(int i = 0; i < len; ++ i)cout << a[i] << ' ';cout << '\n';
}
void exgcd(int a,int b,int &x,int &y){if(!b) {x = 1; y = 0;return;}exgcd(b, a % b, y, x);y -= x * (a / b);
}
int quick_pow(int ans, int p, int res = 1) {for(; p; p >>= 1, ans = 1ll * ans * ans % mod)if(p & 1) res = 1ll * res * ans % mod;return res % mod;
}Polynomial R;
//二进制向上取整，为方便NTT变换准备。
inline int Binary_Rounding(const int &n) {int len = 1; while(len < n) len <<= 1;return len;
}
//预处理R数组，准备变换，在每次NTT之前理论都要调用此函数。
inline int Prepare_Transformation(int n){int L = 0, len;for(len = 1; len < n; len <<= 1) L++;R.clear(); R.resize(len);for(int i = 0; i < len; ++ i)R[i] = (R[i>>1]>>1)|((i&1)<<(L-1));return len;
}
void NTT(Polynomial &a, int f){int n = a.size();for(int i = 0; i < n; ++ i)if(i < R[i])swap(a[i], a[R[i]]);for(int i = 1; i < n; i <<= 1)for(int j = 0, gn = quick_pow(mod_g,(mod - 1) / (i<<1)); j < n; j += (i<<1))for(int k = 0, g = 1, x, y; k < i; ++ k, g = 1ll * g * gn % mod)x = a[j + k], y = 1ll * g * a[i + j + k] % mod,a[j + k] = (x + y) % mod, a[i + j + k] = (x - y + mod) % mod;if(f == -1){reverse(a.begin() + 1, a.end());int inv = Inv(n);for(int i = 0; i < n; ++ i) a[i] = 1ll * a[i] * inv % mod;}
}inline Polynomial operator +(const Polynomial &a, const int &b){int sizea = a.size(); Polynomial ret = a; ret.resize(sizea);for(int i = 0; i < sizea; ++ i)ret[i] = (1ll * a[i] + b + mod) % mod;return ret;
}
inline Polynomial operator -(const Polynomial &a, const int &b){int sizea = a.size(); Polynomial ret = a; ret.resize(sizea);for(int i = 0; i < sizea; ++ i)ret[i] = (1ll * a[i] - b + mod) % mod;return ret;
}
inline Polynomial operator *(const Polynomial &a, const int &b){int sizea = a.size(); Polynomial ret = a; ret.resize(sizea);for(int i = 0; i < sizea; ++ i) ret[i] = (1ll * a[i] * b % mod + mod) % mod;return ret;
}
inline Polynomial operator +(const Polynomial &a, const Polynomial &b){int sizea = a.size(), sizeb = b.size(), size = max(sizea, sizeb);Polynomial ret = a; ret.resize(size);for(int i = 0; i < sizeb; ++ i) ret[i] = (1ll * ret[i] + b[i]) % mod;return ret;
}
inline Polynomial operator -(const Polynomial &a, const Polynomial &b){int sizea = a.size(), sizeb = b.size(), size = max(sizea, sizeb);Polynomial ret = a; ret.resize(size);for(int i = 0; i < sizeb; ++ i) ret[i] = (1ll * ret[i] - b[i] + mod) % mod;return ret;
}
inline Polynomial Inverse(const Polynomial &a){Polynomial ret, inv_a;ret.resize(1);ret[0] = Inv(a[0]); int ed = a.size();for(int len = 2; len <= ed; len <<= 1){int n = Prepare_Transformation(len << 1);inv_a = a; inv_a.resize(n); ret.resize(n);for(int i = len; i < n; ++ i) inv_a[i] = 0;NTT(inv_a, 1); NTT(ret, 1);for(int i = 0; i < n; ++ i)ret[i] = 1ll * (2ll - 1ll * inv_a[i] * ret[i] % mod + mod) % mod * ret[i] % mod;NTT(ret, -1);for(int i = len; i < n; ++ i) ret[i] = 0;}ret.resize(ed);return ret;
}
inline Polynomial operator *(const Polynomial &a, const Polynomial &b){Polynomial lsa = a, lsb = b, ret;int n = lsa.size(), m = lsb.size();n = Prepare_Transformation(n + m);lsa.resize(n); lsb.resize(n); ret.resize(n);NTT(lsa,1); NTT(lsb,1);for(int i = 0; i < n; ++ i) ret[i] = 1ll * lsa[i] * lsb[i] % mod;NTT(ret,-1);return ret;
}
inline Polynomial operator /(const Polynomial &a, const Polynomial &b){Polynomial ret = a, ls = b;reverse(ret.begin(), ret.end());reverse(ls.begin(), ls.end());ls.resize(Binary_Rounding(a.size() + b.size()));ls = Inverse(ls);ls.resize(a.size() + b.size());ret = ret * ls; ret.resize(a.size() - b.size() + 1);reverse(ret.begin(), ret.end());return ret;
}
inline Polynomial operator %(const Polynomial &a, const Polynomial &b){Polynomial ret = a / b;ret = ret * b; ret.resize(a.size() + b.size());ret = a - ret; ret.resize(a.size() + b.size());return ret;
}
inline Polynomial Derivation(const Polynomial &a){int size = a.size(); Polynomial ret; ret.resize(size);for(int i = 1; i < size; ++ i) ret[i - 1] = 1ll * i * a[i] % mod;ret[size - 1] = 0;return ret;
}
inline Polynomial Integral(const Polynomial &a){int size = a.size(); Polynomial ret; ret.resize(size);for(int i = 1; i < size; ++ i) ret[i] = 1ll * Inv(i) * a[i - 1] % mod;ret[0] = 0;return ret;
}
inline Polynomial Composition_Inverse(const Polynomial &a){int n = a.size();Polynomial ret, Cinv = a, Pow;Cinv.resize(n); ret.resize(n); Pow.resize(n); Pow[0] = 1;for(int i = 0; i < n - 1; ++ i) Cinv[i] = Cinv[i + 1]; Cinv[n - 1] = 0;Cinv = Inverse(Cinv);for(int i = 1; i < n; ++ i){Pow = Pow * Cinv; Pow.resize(n);ret[i] = 1ll * Pow[i - 1] * Inv(i) % mod;}return ret;
}
inline Polynomial Logarithmic(const Polynomial &a){Polynomial ln_a = Derivation(a) * Inverse(a);ln_a.resize(a.size());return Integral(ln_a);
}
inline Polynomial Exponential(const Polynomial &a, int Constant = 1){Polynomial ret, D; int ed = a.size();ret.resize(1); ret[0] = Constant;for(int len = 2; len <= ed; len <<= 1){D = Logarithmic(ret); D.resize(len);D[0] = (1ll * a[0] + 1ll - D[0] + mod) % mod;for(int i = 1; i < len; ++i) D[i] = (1ll * a[i] - D[i] + mod) % mod;int n = Prepare_Transformation(len<<1);ret.resize(n); D.resize(n);NTT(ret, 1); NTT(D,1);for(int i = 0; i < n; ++ i) ret[i] = 1ll * ret[i] * D[i] % mod;NTT(ret, -1);for(int i = len; i < (len<<1); ++ i) ret[i] = D[i] = 0;}ret.resize(ed);return ret;
}ll fac[int(6e5 + 10)] = {1}, ifac[int(6e5 + 10)] = {1};
ll C(int n, int m) {if(m == 0 || n == m) return 1;if(n < m) return 0;return 1ll * fac[n] * ifac[m] % mod * ifac[n - m] % mod;
}bool vis[500000];inline void solve() {int n; cin >> n;Polynomial a(500010), b(500010);for(int i = 0; i < n; ++ i) {int x; cin >> x;a[x] = 1;b[500000 - x] = 1;}Polynomial c = a * b;for(int i = 1; i <= 500000; ++ i) {if(c[500000 + i] != 0) {int x = i;for(int i = 1; i * i <= x; ++ i) if(x % i == 0) vis[i] = vis[x / i] = 1;}}for(int i = 1; ; ++ i) {if(!vis[i]) {cout << i << '\n';break;}}
}signed main() {ios_base::sync_with_stdio(false);cin.tie(0);cout.tie(0);
#ifdef ACM_LOCALfreopen("input", "r", stdin);
//    freopen("output", "w", stdout);signed test_index_for_debug = 1;char acm_local_for_debug = 0;do {if (acm_local_for_debug == '$') exit(0);if (test_index_for_debug > 20)throw runtime_error("Check the stdin!!!");auto start_clock_for_debug = clock();solve();auto end_clock_for_debug = clock();cout << "Test " << test_index_for_debug << " successful" << endl;cerr << "Test " << test_index_for_debug++ << " Run Time: "<< double(end_clock_for_debug - start_clock_for_debug) / CLOCKS_PER_SEC << "s" << endl;cout << "--------------------------------------------------" << endl;} while (cin >> acm_local_for_debug && cin.putback(acm_local_for_debug));
#elsesolve();
#endifreturn 0;
}