firiexp's Library

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:heavy_check_mark: test/yosupo_binomial_coefficient.test.cpp

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Code

#define PROBLEM "https://judge.yosupo.jp/problem/binomial_coefficient"

#include <cstdint>
#include <map>
#include <numeric>
#include <vector>
using namespace std;
using ll = long long;
using uint = unsigned;
using ull = unsigned long long;

#include <cstdio>
#include <cstring>
#include <string>
#include <type_traits>

#include "../util/fastio.cpp"
#include "../math/CRT.cpp"
#include "../math/prime/primefactor.cpp"
#include "../math/binom_mod_prime_power.cpp"

ll brute(ll n, ll k, ll mod) {
    if (k < 0 || k > n) return 0;
    vector<vector<ll>> dp(n + 1, vector<ll>(k + 1));
    dp[0][0] = 1 % mod;
    for (ll i = 0; i < n; ++i) {
        for (ll j = 0; j <= min(i, k); ++j) {
            dp[i + 1][j] += dp[i][j];
            dp[i + 1][j] %= mod;
            if (j != k) {
                dp[i + 1][j + 1] += dp[i][j];
                dp[i + 1][j + 1] %= mod;
            }
        }
    }
    return dp[n][k];
}

int main() {
    {
        map<ll, BinomModPrimePower> cache;
        for (int mod = 2; mod <= 120; ++mod) {
            auto pf = prime_factor<ll>(mod);
            vector<pair<ll, int>> fac;
            for (ll p : pf) {
                if (fac.empty() || fac.back().first != p) fac.emplace_back(p, 1);
                else fac.back().second++;
            }
            for (int n = 0; n <= 20; ++n) {
                for (int k = 0; k <= n; ++k) {
                    vector<pair<ll, ll>> rem;
                    for (auto [p, e] : fac) {
                        ll pe = 1;
                        for (int i = 0; i < e; ++i) pe *= p;
                        auto it = cache.find(pe);
                        if (it == cache.end()) it = cache.emplace(pe, BinomModPrimePower(p, e)).first;
                        rem.emplace_back(it->second.C(n, k), pe);
                    }
                    if (CRT(rem).first != brute(n, k, mod)) return 1;
                }
            }
        }
    }

    Scanner sc;
    Printer pr;
    int t;
    ll mod;
    sc.read(t, mod);
    auto pf = prime_factor<ll>(mod);
    vector<pair<ll, int>> fac;
    for (ll p : pf) {
        if (fac.empty() || fac.back().first != p) fac.emplace_back(p, 1);
        else fac.back().second++;
    }
    vector<BinomModPrimePower> binoms;
    vector<ll> mods;
    for (auto [p, e] : fac) {
        binoms.emplace_back(p, e);
        mods.emplace_back(binoms.back().modulus());
    }
    while (t--) {
        ll n, k;
        sc.read(n, k);
        vector<pair<ll, ll>> rem;
        for (int i = 0; i < (int)binoms.size(); ++i) {
            rem.emplace_back(binoms[i].C(n, k), mods[i]);
        }
        pr.println(CRT(rem).first);
    }
    return 0;
}
#line 1 "test/yosupo_binomial_coefficient.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/binomial_coefficient"

#include <cstdint>
#include <map>
#include <numeric>
#include <vector>
using namespace std;
using ll = long long;
using uint = unsigned;
using ull = unsigned long long;

#include <cstdio>
#include <cstring>
#include <string>
#include <type_traits>

#line 1 "util/fastio.cpp"
using namespace std;

extern "C" int fileno(FILE *);
extern "C" int isatty(int);

template<class T, class = void>
struct is_fastio_range : false_type {};

template<class T>
struct is_fastio_range<T, void_t<decltype(declval<T &>().begin()), decltype(declval<T &>().end())>> : true_type {};

template<class T, class = void>
struct has_fastio_value : false_type {};

template<class T>
struct has_fastio_value<T, void_t<decltype(declval<const T &>().value())>> : true_type {};

struct FastIoDigitTable {
    char num[40000];

    constexpr FastIoDigitTable() : num() {
        for (int i = 0; i < 10000; ++i) {
            int x = i;
            for (int j = 3; j >= 0; --j) {
                num[i * 4 + j] = char('0' + x % 10);
                x /= 10;
            }
        }
    }
};

struct Scanner {
    static constexpr int BUFSIZE = 1 << 17;
    static constexpr int OFFSET = 64;
    char buf[BUFSIZE + 1];
    int idx, size;
    bool interactive;

    Scanner() : idx(0), size(0), interactive(isatty(fileno(stdin))) {}

    inline void load() {
        int len = size - idx;
        memmove(buf, buf + idx, len);
        if (interactive) {
            if (fgets(buf + len, BUFSIZE + 1 - len, stdin)) size = len + (int)strlen(buf + len);
            else size = len;
        } else {
            size = len + (int)fread(buf + len, 1, BUFSIZE - len, stdin);
        }
        idx = 0;
        buf[size] = 0;
    }

    inline void ensure() {
        if (idx + OFFSET > size) load();
    }

    inline void ensure_interactive() {
        if (idx == size) load();
    }

    inline char skip() {
        if (interactive) {
            ensure_interactive();
            while (buf[idx] && buf[idx] <= ' ') {
                ++idx;
                ensure_interactive();
            }
            return buf[idx++];
        }
        ensure();
        while (buf[idx] && buf[idx] <= ' ') {
            ++idx;
            ensure();
        }
        return buf[idx++];
    }

    template<class T, typename enable_if<is_integral<T>::value, int>::type = 0>
    void read(T &x) {
        if (interactive) {
            char c = skip();
            bool neg = false;
            if constexpr (is_signed<T>::value) {
                if (c == '-') {
                    neg = true;
                    ensure_interactive();
                    c = buf[idx++];
                }
            }
            x = 0;
            while (c >= '0') {
                x = x * 10 + (c & 15);
                ensure_interactive();
                c = buf[idx++];
            }
            if constexpr (is_signed<T>::value) {
                if (neg) x = -x;
            }
            return;
        }
        char c = skip();
        bool neg = false;
        if constexpr (is_signed<T>::value) {
            if (c == '-') {
                neg = true;
                c = buf[idx++];
            }
        }
        x = 0;
        while (c >= '0') {
            x = x * 10 + (c & 15);
            c = buf[idx++];
        }
        if constexpr (is_signed<T>::value) {
            if (neg) x = -x;
        }
    }

    template<class T, typename enable_if<!is_integral<T>::value && !is_fastio_range<T>::value && !is_same<typename decay<T>::type, string>::value && has_fastio_value<T>::value, int>::type = 0>
    void read(T &x) {
        long long v;
        read(v);
        x = T(v);
    }

    template<class Head, class Next, class... Tail>
    void read(Head &head, Next &next, Tail &...tail) {
        read(head);
        read(next, tail...);
    }

    template<class T, class U>
    void read(pair<T, U> &p) {
        read(p.first, p.second);
    }

    template<class T, typename enable_if<is_fastio_range<T>::value && !is_same<typename decay<T>::type, string>::value, int>::type = 0>
    void read(T &a) {
        for (auto &x : a) read(x);
    }

    void read(char &c) {
        c = skip();
    }

    void read(string &s) {
        s.clear();
        if (interactive) {
            ensure_interactive();
            while (buf[idx] && buf[idx] <= ' ') {
                ++idx;
                ensure_interactive();
            }
            while (true) {
                int start = idx;
                while (idx < size && buf[idx] > ' ') ++idx;
                s.append(buf + start, idx - start);
                if (idx < size) break;
                load();
                if (size == 0) break;
            }
            if (idx < size) ++idx;
            return;
        }
        ensure();
        while (buf[idx] && buf[idx] <= ' ') {
            ++idx;
            ensure();
        }
        while (true) {
            int start = idx;
            while (idx < size && buf[idx] > ' ') ++idx;
            s.append(buf + start, idx - start);
            if (idx < size) break;
            load();
        }
        if (idx < size) ++idx;
    }
};

struct Printer {
    static constexpr int BUFSIZE = 1 << 17;
    static constexpr int OFFSET = 64;
    char buf[BUFSIZE];
    int idx;
    bool interactive;
    inline static constexpr FastIoDigitTable table{};

    Printer() : idx(0), interactive(isatty(fileno(stdout))) {}
    ~Printer() { flush(); }

    inline void flush() {
        if (idx) {
            fwrite(buf, 1, idx, stdout);
            idx = 0;
        }
    }

    inline void pc(char c) {
        if (idx > BUFSIZE - OFFSET) flush();
        buf[idx++] = c;
        if (interactive && c == '\n') flush();
    }

    inline void print_range(const char *s, size_t n) {
        size_t pos = 0;
        while (pos < n) {
            if (idx == BUFSIZE) flush();
            size_t chunk = min(n - pos, (size_t)(BUFSIZE - idx));
            memcpy(buf + idx, s + pos, chunk);
            idx += (int)chunk;
            pos += chunk;
        }
    }

    void print(const char *s) {
        print_range(s, strlen(s));
    }

    void print(const string &s) {
        print_range(s.data(), s.size());
    }

    void print(char c) {
        pc(c);
    }

    void print(bool b) {
        pc(char('0' + (b ? 1 : 0)));
    }

    template<class T, typename enable_if<is_integral<T>::value && !is_same<T, bool>::value, int>::type = 0>
    void print(T x) {
        if (idx > BUFSIZE - 100) flush();
        using U = typename make_unsigned<T>::type;
        U y;
        if constexpr (is_signed<T>::value) {
            if (x < 0) {
                buf[idx++] = '-';
                y = U(0) - static_cast<U>(x);
            } else {
                y = static_cast<U>(x);
            }
        } else {
            y = x;
        }
        if (y == 0) {
            buf[idx++] = '0';
            return;
        }
        static constexpr int TMP_SIZE = sizeof(U) * 10 / 4;
        char tmp[TMP_SIZE];
        int pos = TMP_SIZE;
        while (y >= 10000) {
            pos -= 4;
            memcpy(tmp + pos, table.num + (y % 10000) * 4, 4);
            y /= 10000;
        }
        if (y >= 1000) {
            memcpy(buf + idx, table.num + (y << 2), 4);
            idx += 4;
        } else if (y >= 100) {
            memcpy(buf + idx, table.num + (y << 2) + 1, 3);
            idx += 3;
        } else if (y >= 10) {
            unsigned q = (unsigned(y) * 205) >> 11;
            buf[idx] = char('0' + q);
            buf[idx + 1] = char('0' + (unsigned(y) - q * 10));
            idx += 2;
        } else {
            buf[idx++] = char('0' + y);
        }
        memcpy(buf + idx, tmp + pos, TMP_SIZE - pos);
        idx += TMP_SIZE - pos;
    }

    template<class T, typename enable_if<!is_integral<T>::value && !is_fastio_range<T>::value && !is_same<typename decay<T>::type, string>::value && has_fastio_value<T>::value, int>::type = 0>
    void print(const T &x) {
        print(x.value());
    }

    template<class T, typename enable_if<is_fastio_range<T>::value && !is_same<typename decay<T>::type, string>::value, int>::type = 0>
    void print(const T &a) {
        bool first = true;
        for (auto &&x : a) {
            if (!first) pc(' ');
            first = false;
            print(x);
        }
    }

    template<class T>
    void println(const T &x) {
        print(x);
        pc('\n');
    }

    template<class Head, class... Tail>
    void println(const Head &head, const Tail &...tail) {
        print(head);
        ((pc(' '), print(tail)), ...);
        pc('\n');
    }

    void println() {
        pc('\n');
    }
};

template<class T>
Scanner &operator>>(Scanner &in, T &x) {
    in.read(x);
    return in;
}

template<class T>
Printer &operator<<(Printer &out, const T &x) {
    out.print(x);
    return out;
}

/**
 * @brief 高速入出力(Fast IO)
 */
#line 1 "math/extgcd.cpp"
template<typename T>
T extgcd(T a, T b, T &x ,T &y){
    for (T u = y = 1, v = x = 0; a; ) {
        ll q = b/a;
        swap(x -= q*u, u);
        swap(y -= q*v, v);
        swap(b -= q*a, a);
    }
    return b;
}
 

/**
 * @brief 拡張ユークリッド互除法(Extended GCD)
 */
#line 2 "math/CRT.cpp"

pair<ll, ll> CRT(const vector<pair<ll, ll>> &a){
    ll R = 0, M = 1;
    for (auto &&i : a) {
        ll r = (i.first+i.second)%i.second, m = i.second;
        if(m < M) swap(r, R), swap(m, M);
        if(M%m == 0){
            if(R % m != r) return {};
            continue;
        }
        ll p, q;
        ll g = extgcd(M, m, p, q); // p = inv(M') mod m'
        ll mm = m/g;
        if((r-R)%g) return {0, 0};
        ll x = (r-R)/g % mm * p % mm;
        R += x*M;
        M *= mm;
        if(R < 0) R += M;
    }
    return {R, M};
}

/**
 * @brief 中国剰余定理(Chinese Remainder Theorem)
 */
#line 1 "math/prime/primefactor.cpp"
template<typename T>
struct ExactDiv {
    T t, i, val;
    ExactDiv() {}
    ExactDiv(T n) : t(T(-1) / n), i(mul_inv(n)) , val(n) {};
    T mul_inv(T n) {
        T x = n;
        for (int i = 0; i < 5; ++i) x *= 2 - n * x;
        return x;
    }
    bool divide(T n) const {
        if(val == 2) return !(n & 1);
        return n * this->i <= this->t;
    }
};

vector<ExactDiv<uint>> get_prime(int n){
    if(n <= 1) return vector<ExactDiv<uint>>();
    vector<bool> is_prime(n+1, true);
    vector<ExactDiv<uint>> prime;
    is_prime[0] = is_prime[1] = false;
    for (int i = 2; i <= n; ++i) {
        if(is_prime[i]) prime.emplace_back(i);
        for (auto &&j : prime){
            if(i*j.val > n) break;
            is_prime[i*j.val] = false;
            if(j.divide(i)) break;
        }
    }
    return prime;
}
const auto primes = get_prime(32000);

template<class T>
vector<T> prime_factor(T n){
    vector<T> res;
    for (auto &&i : primes) {
        while (i.divide(n)){
            res.emplace_back(i.val);
            n /= i.val;
        }
    }
    if(n != 1) res.emplace_back(n);
    return res;
}

/**
 * @brief 素因数分解(試し割り)
 */
#line 1 "math/modinv.cpp"
template<typename T>  
T mod_inv(T x, T M){  
   T u = 1, t = 1, v = 0, s = 0, m = M;  
   while (x) { T q = m/x; swap(s -= q*u, u); swap(t -= q*v, v); swap(m -= q*x, x); }  
   if(s < 0) s += M;  
   return s;  
}
#line 2 "math/binom_mod_prime_power.cpp"

struct BinomModPrimePower {
    ll p, mod;
    int q;
    ll block_prod;
    vector<ll> ppow;
    vector<int> prod;

    explicit BinomModPrimePower(ll prime, int exponent) : p(prime), mod(1), q(exponent), ppow(exponent + 1, 1) {
        for (int i = 0; i < q; ++i) {
            mod *= p;
            ppow[i + 1] = mod;
        }
        block_prod = (p == 2 && q >= 3 ? 1 : mod - 1);
        prod.assign(mod + 1, 1);
        for (int i = 1; i <= mod; ++i) {
            prod[i] = prod[i - 1];
            if (i % p != 0) prod[i] = (ull)prod[i] * i % mod;
        }
    }

    pair<ll, ll> factorial(ll n) const {
        ll x = 1, e = 0;
        while (n) {
            if (block_prod != 1 && (n / mod) & 1) x = mod - x;
            x = (ull)x * prod[n % mod] % mod;
            n /= p;
            e += n;
        }
        return {x, e};
    }

    ll C(ll n, ll k) const {
        if (k < 0 || k > n) return 0;
        auto [a, ea] = factorial(n);
        auto [b, eb] = factorial(k);
        auto [c, ec] = factorial(n - k);
        ll e = ea - eb - ec;
        if (e >= q) return 0;
        ll x = (ull)b * c % mod;
        return (ull)a * mod_inv(x, mod) % mod * ppow[e] % mod;
    }

    ll modulus() const {
        return mod;
    }
};

/**
 * @brief 二項係数(mod p^q)
 */
#line 21 "test/yosupo_binomial_coefficient.test.cpp"

ll brute(ll n, ll k, ll mod) {
    if (k < 0 || k > n) return 0;
    vector<vector<ll>> dp(n + 1, vector<ll>(k + 1));
    dp[0][0] = 1 % mod;
    for (ll i = 0; i < n; ++i) {
        for (ll j = 0; j <= min(i, k); ++j) {
            dp[i + 1][j] += dp[i][j];
            dp[i + 1][j] %= mod;
            if (j != k) {
                dp[i + 1][j + 1] += dp[i][j];
                dp[i + 1][j + 1] %= mod;
            }
        }
    }
    return dp[n][k];
}

int main() {
    {
        map<ll, BinomModPrimePower> cache;
        for (int mod = 2; mod <= 120; ++mod) {
            auto pf = prime_factor<ll>(mod);
            vector<pair<ll, int>> fac;
            for (ll p : pf) {
                if (fac.empty() || fac.back().first != p) fac.emplace_back(p, 1);
                else fac.back().second++;
            }
            for (int n = 0; n <= 20; ++n) {
                for (int k = 0; k <= n; ++k) {
                    vector<pair<ll, ll>> rem;
                    for (auto [p, e] : fac) {
                        ll pe = 1;
                        for (int i = 0; i < e; ++i) pe *= p;
                        auto it = cache.find(pe);
                        if (it == cache.end()) it = cache.emplace(pe, BinomModPrimePower(p, e)).first;
                        rem.emplace_back(it->second.C(n, k), pe);
                    }
                    if (CRT(rem).first != brute(n, k, mod)) return 1;
                }
            }
        }
    }

    Scanner sc;
    Printer pr;
    int t;
    ll mod;
    sc.read(t, mod);
    auto pf = prime_factor<ll>(mod);
    vector<pair<ll, int>> fac;
    for (ll p : pf) {
        if (fac.empty() || fac.back().first != p) fac.emplace_back(p, 1);
        else fac.back().second++;
    }
    vector<BinomModPrimePower> binoms;
    vector<ll> mods;
    for (auto [p, e] : fac) {
        binoms.emplace_back(p, e);
        mods.emplace_back(binoms.back().modulus());
    }
    while (t--) {
        ll n, k;
        sc.read(n, k);
        vector<pair<ll, ll>> rem;
        for (int i = 0; i < (int)binoms.size(); ++i) {
            rem.emplace_back(binoms[i].C(n, k), mods[i]);
        }
        pr.println(CRT(rem).first);
    }
    return 0;
}
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