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Verify/LC_binomial_coefficient.test.cpp
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- Last update: 2024-04-26 03:32:16+09:00
- Problem: https://judge.yosupo.jp/problem/binomial_coefficient
Depends on
- Binomial Coefficient for query
(Math/binomquery.hpp)
- Fast Division
(Math/fastdiv.hpp)
- Miller-Rabin
(Math/miller.hpp)
- Pollard-Rho
(Math/pollard.hpp)
- Primitive Function
(Math/primitive.hpp)
- Template/template.hpp
- Fast IO
(Utility/fastio.hpp)
- Random
(Utility/random.hpp)
Code
#define PROBLEM "https://judge.yosupo.jp/problem/binomial_coefficient"
#include "Template/template.hpp"
#include "Utility/fastio.hpp"
#include "Math/binomquery.hpp"
int main() {
int q, m;
read(q, m);
BinomialQuery buf(m);
while (q--) {
ll n, r;
read(n, r);
print(buf.nCr(n, r));
}
return 0;
}#line 1 "Verify/LC_binomial_coefficient.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/binomial_coefficient"
#line 1 "Template/template.hpp"
#include <bits/stdc++.h>
using namespace std;
#define rep(i, a, b) for (int i = (int)(a); i < (int)(b); i++)
#define rrep(i, a, b) for (int i = (int)(b-1); i >= (int)(a); i--)
#define ALL(v) (v).begin(), (v).end()
#define UNIQUE(v) sort(ALL(v)), (v).erase(unique(ALL(v)), (v).end())
#define SZ(v) (int)v.size()
#define MIN(v) *min_element(ALL(v))
#define MAX(v) *max_element(ALL(v))
#define LB(v, x) int(lower_bound(ALL(v), (x)) - (v).begin())
#define UB(v, x) int(upper_bound(ALL(v), (x)) - (v).begin())
using uint = unsigned int;
using ll = long long int;
using ull = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;
const int inf = 0x3fffffff;
const ll INF = 0x1fffffffffffffff;
template <typename T> inline bool chmax(T &a, T b) {
if (a < b) {
a = b;
return 1;
}
return 0;
}
template <typename T> inline bool chmin(T &a, T b) {
if (a > b) {
a = b;
return 1;
}
return 0;
}
template <typename T, typename U> T ceil(T x, U y) {
assert(y != 0);
if (y < 0)
x = -x, y = -y;
return (x > 0 ? (x + y - 1) / y : x / y);
}
template <typename T, typename U> T floor(T x, U y) {
assert(y != 0);
if (y < 0)
x = -x, y = -y;
return (x > 0 ? x / y : (x - y + 1) / y);
}
template <typename T> int popcnt(T x) {
return __builtin_popcountll(x);
}
template <typename T> int topbit(T x) {
return (x == 0 ? -1 : 63 - __builtin_clzll(x));
}
template <typename T> int lowbit(T x) {
return (x == 0 ? -1 : __builtin_ctzll(x));
}
#ifdef LOCAL
#define show(...) _show(0, #__VA_ARGS__, __VA_ARGS__)
#else
#define show(...) true
#endif
template <typename T> void _show(int i, T name) {
cerr << '\n';
}
template <typename T1, typename T2, typename... T3>
void _show(int i, const T1 &a, const T2 &b, const T3 &...c) {
for (; a[i] != ',' && a[i] != '\0'; i++)
cerr << a[i];
cerr << ":" << b << " ";
_show(i + 1, a, c...);
}
template <class T, class U>
ostream &operator<<(ostream &os, const pair<T, U> &p) {
os << "P(" << p.first << ", " << p.second << ")";
return os;
}
template <typename T, template <class> class C>
ostream &operator<<(ostream &os, const C<T> &v) {
os << "[";
for (auto d : v)
os << d << ", ";
os << "]";
return os;
}
#line 2 "Utility/fastio.hpp"
#include <unistd.h>
namespace fastio {
static constexpr uint32_t SZ = 1 << 17;
char ibuf[SZ];
char obuf[SZ];
char out[100];
// pointer of ibuf, obuf
uint32_t pil = 0, pir = 0, por = 0;
struct Pre {
char num[10000][4];
constexpr Pre() : num() {
for (int i = 0; i < 10000; i++) {
int n = i;
for (int j = 3; j >= 0; j--) {
num[i][j] = n % 10 | '0';
n /= 10;
}
}
}
} constexpr pre;
inline void load() {
memmove(ibuf, ibuf + pil, pir - pil);
pir = pir - pil + fread(ibuf + pir - pil, 1, SZ - pir + pil, stdin);
pil = 0;
if (pir < SZ)
ibuf[pir++] = '\n';
}
inline void flush() {
fwrite(obuf, 1, por, stdout);
por = 0;
}
void rd(char &c) {
do {
if (pil + 1 > pir)
load();
c = ibuf[pil++];
} while (isspace(c));
}
void rd(string &x) {
x.clear();
char c;
do {
if (pil + 1 > pir)
load();
c = ibuf[pil++];
} while (isspace(c));
do {
x += c;
if (pil == pir)
load();
c = ibuf[pil++];
} while (!isspace(c));
}
template <typename T> void rd_real(T &x) {
string s;
rd(s);
x = stod(s);
}
template <typename T> void rd_integer(T &x) {
if (pil + 100 > pir)
load();
char c;
do
c = ibuf[pil++];
while (c < '-');
bool minus = 0;
if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
if (c == '-') {
minus = 1, c = ibuf[pil++];
}
}
x = 0;
while ('0' <= c) {
x = x * 10 + (c & 15), c = ibuf[pil++];
}
if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
if (minus)
x = -x;
}
}
void rd(int &x) {
rd_integer(x);
}
void rd(ll &x) {
rd_integer(x);
}
void rd(i128 &x) {
rd_integer(x);
}
void rd(uint &x) {
rd_integer(x);
}
void rd(ull &x) {
rd_integer(x);
}
void rd(u128 &x) {
rd_integer(x);
}
void rd(double &x) {
rd_real(x);
}
void rd(long double &x) {
rd_real(x);
}
template <class T, class U> void rd(pair<T, U> &p) {
return rd(p.first), rd(p.second);
}
template <size_t N = 0, typename T> void rd_tuple(T &t) {
if constexpr (N < std::tuple_size<T>::value) {
auto &x = std::get<N>(t);
rd(x);
rd_tuple<N + 1>(t);
}
}
template <class... T> void rd(tuple<T...> &tpl) {
rd_tuple(tpl);
}
template <size_t N = 0, typename T> void rd(array<T, N> &x) {
for (auto &d : x)
rd(d);
}
template <class T> void rd(vector<T> &x) {
for (auto &d : x)
rd(d);
}
void read() {}
template <class H, class... T> void read(H &h, T &...t) {
rd(h), read(t...);
}
void wt(const char c) {
if (por == SZ)
flush();
obuf[por++] = c;
}
void wt(const string s) {
for (char c : s)
wt(c);
}
void wt(const char *s) {
size_t len = strlen(s);
for (size_t i = 0; i < len; i++)
wt(s[i]);
}
template <typename T> void wt_integer(T x) {
if (por > SZ - 100)
flush();
if (x < 0) {
obuf[por++] = '-', x = -x;
}
int outi;
for (outi = 96; x >= 10000; outi -= 4) {
memcpy(out + outi, pre.num[x % 10000], 4);
x /= 10000;
}
if (x >= 1000) {
memcpy(obuf + por, pre.num[x], 4);
por += 4;
} else if (x >= 100) {
memcpy(obuf + por, pre.num[x] + 1, 3);
por += 3;
} else if (x >= 10) {
int q = (x * 103) >> 10;
obuf[por] = q | '0';
obuf[por + 1] = (x - q * 10) | '0';
por += 2;
} else
obuf[por++] = x | '0';
memcpy(obuf + por, out + outi + 4, 96 - outi);
por += 96 - outi;
}
template <typename T> void wt_real(T x) {
ostringstream oss;
oss << fixed << setprecision(15) << double(x);
string s = oss.str();
wt(s);
}
void wt(int x) {
wt_integer(x);
}
void wt(ll x) {
wt_integer(x);
}
void wt(i128 x) {
wt_integer(x);
}
void wt(uint x) {
wt_integer(x);
}
void wt(ull x) {
wt_integer(x);
}
void wt(u128 x) {
wt_integer(x);
}
void wt(double x) {
wt_real(x);
}
void wt(long double x) {
wt_real(x);
}
template <class T, class U> void wt(const pair<T, U> val) {
wt(val.first);
wt(' ');
wt(val.second);
}
template <size_t N = 0, typename T> void wt_tuple(const T t) {
if constexpr (N < std::tuple_size<T>::value) {
if constexpr (N > 0) {
wt(' ');
}
const auto x = std::get<N>(t);
wt(x);
wt_tuple<N + 1>(t);
}
}
template <class... T> void wt(tuple<T...> tpl) {
wt_tuple(tpl);
}
template <class T, size_t S> void wt(const array<T, S> val) {
auto n = val.size();
for (size_t i = 0; i < n; i++) {
if (i)
wt(' ');
wt(val[i]);
}
}
template <class T> void wt(const vector<T> val) {
auto n = val.size();
for (size_t i = 0; i < n; i++) {
if (i)
wt(' ');
wt(val[i]);
}
}
void print() {
wt('\n');
}
template <class Head, class... Tail> void print(Head &&head, Tail &&...tail) {
wt(head);
if (sizeof...(Tail))
wt(' ');
print(forward<Tail>(tail)...);
}
void __attribute__((destructor)) _d() {
flush();
}
} // namespace fastio
using fastio::flush;
using fastio::print;
using fastio::read;
inline void first(bool i = true) {
print(i ? "first" : "second");
}
inline void Alice(bool i = true) {
print(i ? "Alice" : "Bob");
}
inline void yes(bool i = true) {
print(i ? "yes" : "no");
}
inline void Yes(bool i = true) {
print(i ? "Yes" : "No");
}
inline void No() {
print("No");
}
inline void YES(bool i = true) {
print(i ? "YES" : "NO");
}
inline void NO() {
print("NO");
}
inline void Yay(bool i = true) {
print(i ? "Yay!" : ":(");
}
inline void Possible(bool i = true) {
print(i ? "Possible" : "Impossible");
}
inline void POSSIBLE(bool i = true) {
print(i ? "POSSIBLE" : "IMPOSSIBLE");
}
/**
* @brief Fast IO
*/
#line 5 "Verify/LC_binomial_coefficient.test.cpp"
#line 2 "Math/fastdiv.hpp"
struct FastDiv{
using u64=uint64_t;
using u128=__uint128_t;
constexpr FastDiv():m(),s(),x(){}
constexpr FastDiv(int _m)
:m(_m),s(__lg(m-1)),x(((u128(1)<<(s+64))+m-1)/m){}
constexpr int get(){return m;}
constexpr friend u64 operator/(u64 n,const FastDiv& d){
return (u128(n)*d.x>>d.s)>>64;
}
constexpr friend int operator%(u64 n,const FastDiv& d){
return n-n/d*d.m;
}
constexpr pair<u64,int> divmod(u64 n)const{
u64 q=n/(*this);
return {q,n-q*m};
}
int m,s; u64 x;
};
/**
* @brief Fast Division
*/
#line 2 "Math/miller.hpp"
struct m64 {
using i64 = int64_t;
using u64 = uint64_t;
using u128 = __uint128_t;
static u64 mod;
static u64 r;
static u64 n2;
static u64 get_r() {
u64 ret = mod;
rep(_,0,5) ret *= 2 - mod * ret;
return ret;
}
static void set_mod(u64 m) {
assert(m < (1LL << 62));
assert((m & 1) == 1);
mod = m;
n2 = -u128(m) % m;
r = get_r();
assert(r * mod == 1);
}
static u64 get_mod() { return mod; }
u64 a;
m64() : a(0) {}
m64(const int64_t &b) : a(reduce((u128(b) + mod) * n2)){};
static u64 reduce(const u128 &b) {
return (b + u128(u64(b) * u64(-r)) * mod) >> 64;
}
u64 get() const {
u64 ret = reduce(a);
return ret >= mod ? ret - mod : ret;
}
m64 &operator*=(const m64 &b) {
a = reduce(u128(a) * b.a);
return *this;
}
m64 operator*(const m64 &b) const { return m64(*this) *= b; }
bool operator==(const m64 &b) const {
return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a);
}
bool operator!=(const m64 &b) const {
return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a);
}
m64 pow(u128 n) const {
m64 ret(1), mul(*this);
while (n > 0) {
if (n & 1) ret *= mul;
mul *= mul;
n >>= 1;
}
return ret;
}
};
typename m64::u64 m64::mod, m64::r, m64::n2;
bool Miller(ll n){
if(n<2 or (n&1)==0)return (n==2);
m64::set_mod(n);
ll d=n-1; while((d&1)==0)d>>=1;
vector<ll> seeds;
if(n<(1<<30))seeds={2, 7, 61};
else seeds={2, 325, 9375, 28178, 450775, 9780504};
for(auto& x:seeds){
if(n<=x)break;
ll t=d;
m64 y=m64(x).pow(t);
while(t!=n-1 and y!=1 and y!=n-1){
y*=y;
t<<=1;
}
if(y!=n-1 and (t&1)==0)return 0;
} return 1;
}
/**
* @brief Miller-Rabin
*/
#line 2 "Utility/random.hpp"
namespace Random {
mt19937_64 randgen(chrono::steady_clock::now().time_since_epoch().count());
using u64 = unsigned long long;
u64 get() {
return randgen();
}
template <typename T> T get(T L) { // [0,L]
return get() % (L + 1);
}
template <typename T> T get(T L, T R) { // [L,R]
return get(R - L) + L;
}
double uniform() {
return double(get(1000000000)) / 1000000000;
}
string str(int n) {
string ret;
rep(i, 0, n) ret += get('a', 'z');
return ret;
}
template <typename Iter> void shuffle(Iter first, Iter last) {
if (first == last)
return;
int len = 1;
for (auto it = first + 1; it != last; it++) {
len++;
int j = get(0, len - 1);
if (j != len - 1)
iter_swap(it, first + j);
}
}
template <typename T> vector<T> select(int n, T L, T R) { // [L,R]
if (n * 2 >= R - L + 1) {
vector<T> ret(R - L + 1);
iota(ALL(ret), L);
shuffle(ALL(ret));
ret.resize(n);
return ret;
} else {
unordered_set<T> used;
vector<T> ret;
while (SZ(used) < n) {
T x = get(L, R);
if (!used.count(x)) {
used.insert(x);
ret.push_back(x);
}
}
return ret;
}
}
void relabel(int n, vector<pair<int, int>> &es) {
shuffle(ALL(es));
vector<int> ord(n);
iota(ALL(ord), 0);
shuffle(ALL(ord));
for (auto &[u, v] : es)
u = ord[u], v = ord[v];
}
template <bool directed, bool simple> vector<pair<int, int>> genGraph(int n) {
vector<pair<int, int>> cand, es;
rep(u, 0, n) rep(v, 0, n) {
if (simple and u == v)
continue;
if (!directed and u > v)
continue;
cand.push_back({u, v});
}
int m = get(SZ(cand));
vector<int> ord;
if (simple)
ord = select(m, 0, SZ(cand) - 1);
else {
rep(_, 0, m) ord.push_back(get(SZ(cand) - 1));
}
for (auto &i : ord)
es.push_back(cand[i]);
relabel(n, es);
return es;
}
vector<pair<int, int>> genTree(int n) {
vector<pair<int, int>> es;
rep(i, 1, n) es.push_back({get(i - 1), i});
relabel(n, es);
return es;
}
}; // namespace Random
/**
* @brief Random
*/
#line 4 "Math/pollard.hpp"
vector<ll> Pollard(ll n) {
if (n <= 1)
return {};
if (Miller(n))
return {n};
if ((n & 1) == 0) {
vector<ll> v = Pollard(n >> 1);
v.push_back(2);
return v;
}
for (ll x = 2, y = 2, d;;) {
ll c = Random::get(2LL, n - 1);
do {
x = (__int128_t(x) * x + c) % n;
y = (__int128_t(y) * y + c) % n;
y = (__int128_t(y) * y + c) % n;
d = __gcd(x - y + n, n);
} while (d == 1);
if (d < n) {
vector<ll> lb = Pollard(d), rb = Pollard(n / d);
lb.insert(lb.end(), ALL(rb));
return lb;
}
}
}
/**
* @brief Pollard-Rho
*/
#line 4 "Math/primitive.hpp"
ll mpow(ll a, ll t, ll m) {
ll res = 1;
FastDiv im(m);
while (t) {
if (t & 1)
res = __int128_t(res) * a % im;
a = __int128_t(a) * a % im;
t >>= 1;
}
return res;
}
ll minv(ll a, ll m) {
ll b = m, u = 1, v = 0;
while (b) {
ll t = a / b;
a -= t * b;
swap(a, b);
u -= t * v;
swap(u, v);
}
u = (u % m + m) % m;
return u;
}
ll getPrimitiveRoot(ll p) {
vector<ll> ps = Pollard(p - 1);
sort(ALL(ps));
rep(x, 1, inf) {
for (auto &q : ps) {
if (mpow(x, (p - 1) / q, p) == 1)
goto fail;
}
return x;
fail:;
}
assert(0);
}
ll extgcd(ll a, ll b, ll &p, ll &q) {
if (b == 0) {
p = 1;
q = 0;
return a;
}
ll d = extgcd(b, a % b, q, p);
q -= a / b * p;
return d;
}
pair<ll, ll> crt(const vector<ll> &vs, const vector<ll> &ms) {
ll V = vs[0], M = ms[0];
rep(i, 1, vs.size()) {
ll p, q, v = vs[i], m = ms[i];
if (M < m)
swap(M, m), swap(V, v);
ll d = extgcd(M, m, p, q);
if ((v - V) % d != 0)
return {0, -1};
ll md = m / d, tmp = (v - V) / d % md * p % md;
V += M * tmp;
M *= md;
}
V = (V % M + M) % M;
return {V, M};
}
ll ModLog(ll a, ll b, ll p) {
ll g = 1;
for (ll t = p; t; t >>= 1)
g = g * a % p;
g = __gcd(g, p);
ll t = 1, c = 0;
for (; t % g; c++) {
if (t == b)
return c;
t = t * a % p;
}
if (b % g)
return -1;
t /= g, b /= g;
ll n = p / g, h = 0, gs = 1;
for (; h * h < n; h++)
gs = gs * a % n;
unordered_map<ll, ll> bs;
for (ll s = 0, e = b; s < h; bs[e] = ++s)
e = e * a % n;
for (ll s = 0, e = t; s < n;) {
e = e * gs % n, s += h;
if (bs.count(e)) {
return c + s - bs[e];
}
}
return -1;
}
ll mod_root(ll k, ll a, ll m) {
if (a == 0)
return k ? 0 : -1;
if (m == 2)
return a & 1;
k %= m - 1;
ll g = gcd(k, m - 1);
if (mpow(a, (m - 1) / g, m) != 1)
return -1;
a = mpow(a, minv(k / g, (m - 1) / g), m);
FastDiv im(m);
auto _subroot = [&](ll p, int e, ll a) -> ll { // x^(p^e)==a(mod m)
ll q = m - 1;
int s = 0;
while (q % p == 0) {
q /= p;
s++;
}
int d = s - e;
ll pe = mpow(p, e, m),
res = mpow(a, ((pe - 1) * minv(q, pe) % pe * q + 1) / pe, m), c = 1;
while (mpow(c, (m - 1) / p, m) == 1)
c++;
c = mpow(c, q, m);
map<ll, ll> mp;
ll v = 1, block = sqrt(d * p) + 1,
bs = mpow(c, mpow(p, s - 1, m - 1) * block % (m - 1), m);
rep(i, 0, block + 1) mp[v] = i, v = v * bs % im;
ll gs = minv(mpow(c, mpow(p, s - 1, m - 1), m), m);
rep(i, 0, d) {
ll err = a * minv(mpow(res, pe, m), m) % im;
ll pos = mpow(err, mpow(p, d - 1 - i, m - 1), m);
rep(j, 0, block + 1) {
if (mp.count(pos)) {
res = res *
mpow(c,
(block * mp[pos] + j) * mpow(p, i, m - 1) %
(m - 1),
m) %
im;
break;
}
pos = pos * gs % im;
}
}
return res;
};
for (ll d = 2; d * d <= g; d++)
if (g % d == 0) {
int sz = 0;
while (g % d == 0) {
g /= d;
sz++;
}
a = _subroot(d, sz, a);
}
if (g > 1)
a = _subroot(g, 1, a);
return a;
}
ull floor_root(ull a, ull k) {
if (a <= 1 or k == 1)
return a;
if (k >= 64)
return 1;
if (k == 2)
return sqrtl(a);
constexpr ull LIM = -1;
if (a == LIM)
a--;
auto mul = [&](ull &x, const ull &y) {
if (x <= LIM / y)
x *= y;
else
x = LIM;
};
auto pw = [&](ull x, ull t) -> ull {
ull y = 1;
while (t) {
if (t & 1)
mul(y, x);
mul(x, x);
t >>= 1;
}
return y;
};
ull ret = (k == 3 ? cbrt(a) - 1 : pow(a, nextafter(1 / double(k), 0)));
while (pw(ret + 1, k) <= a)
ret++;
return ret;
}
/**
* @brief Primitive Function
*/
#line 4 "Math/binomquery.hpp"
struct BinomialQuery{
struct X{ // for m=p^q
int p,q,m,delta;
vector<int> fact,ifac;
FastDiv im,ip;
X(){}
X(int _p,int _q):p(_p),q(_q),ip(p){
m=1;
while(_q--)m*=p;
im=FastDiv(m);
fact.resize(m);
ifac.resize(m);
fact[0]=fact[1]=ifac[0]=ifac[1]=1;
rep(i,2,m){
if(i%ip==0)fact[i]=fact[i-1];
else fact[i]=(ll(fact[i-1])*i)%im;
}
ifac[m-1]=minv(fact[m-1],m);
for(int i=m-2;i>1;i--){
if(i%ip==p-1)ifac[i]=ifac[i+1];
else ifac[i]=(ll(ifac[i+1])*(i+1))%im;
}
delta=(p==2 and q>=3?0:1);
}
int nCr(ll n,ll r){
if(n<0 or r<0 or n<r)return 0;
ll s=n-r,ret=1;
int e0=0,em=0;
for(int i=1;n;i++){
ret=(ret*fact[n%m])%im;
ret=(ret*ifac[r%m])%im;
ret=(ret*ifac[s%m])%im;
n=n/ip,r=r/ip,s=s/ip;
int add=n-r-s;
e0+=add;
if(e0>=q)return 0;
if(i>=q)em^=add;
}
if(delta and em&1)ret=m-ret;
ret=(ret*mpow(p,e0,m))%im;
return ret;
}
};
int mod;
vector<ll> ms;
vector<X> Xs;
BinomialQuery(int M):mod(M){
for(int p=2;p*p<=M;p++)if(M%p==0){
int e=0,base=1;
while(M%p==0)M/=p,e++,base*=p;
ms.push_back(base);
Xs.push_back(X(p,e));
}
if(M!=1){
ms.push_back(M);
Xs.push_back(X(M,1));
}
}
int nCr(ll n,ll r){
if(mod==1)return 0;
vector<ll> ret;
for(auto& buf:Xs)ret.push_back(buf.nCr(n,r));
return crt(ret,ms).first;
}
};
/**
* @brief Binomial Coefficient for query
*/
#line 7 "Verify/LC_binomial_coefficient.test.cpp"
int main() {
int q, m;
read(q, m);
BinomialQuery buf(m);
while (q--) {
ll n, r;
read(n, r);
print(buf.nCr(n, r));
}
return 0;
}