Library

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:heavy_check_mark: test/math/Catalan_Convolution.test.cpp

Depends on

Code

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

#include "../../math/catalan_convolution.hpp"

#include "../../convolution/ntt.hpp"
#include "../../math/catalan_number.hpp"
#include "../../modint/modint.hpp"
#include "../../template/template.hpp"

namespace ebi {

using mint = modint998244353;

void main_() {
    const int n = 100000;
    int t = 10;
    std::vector<mint> c(n);
    rep(i, 0, n) c[i] = catalan_number<mint>(i);
    std::vector<mint> a = {1};
    rep(i, 1, t) {
        a = convolution(a, c);
        a.resize(n);
        rep(j, 0, n) assert(a[j] == catalan_convolution<mint>(i, j));
    }
    return;
}

}  // namespace ebi

int main() {
    ebi::fast_io();
    int t = 1;
    // std::cin >> t;
    while (t--) {
        ebi::main_();
    }
    int a, b;
    std::cin >> a >> b;
    std::cout << a + b << '\n';
    return 0;
}
#line 1 "test/math/Catalan_Convolution.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"

#line 2 "math/catalan_convolution.hpp"

#line 2 "math/binomial.hpp"

#include <bit>
#include <cassert>
#include <iostream>
#include <ranges>
#include <vector>

#line 2 "modint/base.hpp"

#include <concepts>
#line 5 "modint/base.hpp"
#include <utility>

namespace ebi {

template <class T>
concept Modint = requires(T a, T b) {
    a + b;
    a - b;
    a * b;
    a / b;
    a.inv();
    a.val();
    a.pow(std::declval<long long>());
    T::mod();
};

template <Modint mint> std::istream &operator>>(std::istream &os, mint &a) {
    long long x;
    os >> x;
    a = x;
    return os;
}

template <Modint mint>
std::ostream &operator<<(std::ostream &os, const mint &a) {
    return os << a.val();
}

}  // namespace ebi
#line 10 "math/binomial.hpp"

namespace ebi {

template <Modint mint> struct Binomial {
  private:
    static void extend(int len = -1) {
        int sz = (int)fact.size();
        if (len < 0)
            len = 2 * sz;
        else if (len <= sz)
            return;
        else
            len = std::max(2 * sz, (int)std::bit_ceil(std::uint32_t(len)));
        len = std::min(len, mint::mod());
        assert(sz <= len);
        fact.resize(len);
        inv_fact.resize(len);
        for (int i : std::views::iota(sz, len)) {
            fact[i] = fact[i - 1] * i;
        }
        inv_fact[len - 1] = fact[len - 1].inv();
        for (int i : std::views::iota(sz, len) | std::views::reverse) {
            inv_fact[i - 1] = inv_fact[i] * i;
        }
    }

  public:
    Binomial() = default;

    Binomial(int n) {
        extend(n + 1);
    }

    static mint f(int n) {
        if (n >= (int)fact.size()) [[unlikely]] {
            extend(n + 1);
        }
        return fact[n];
    }

    static mint inv_f(int n) {
        if (n >= (int)fact.size()) [[unlikely]] {
            extend(n + 1);
        }
        return inv_fact[n];
    }

    static mint c(int n, int r) {
        if (r < 0 || n < r) return 0;
        return f(n) * inv_f(r) * inv_f(n - r);
    }

    static mint p(int n, int r) {
        if (r < 0 || n < r) return 0;
        return f(n) * inv_f(n - r);
    }

    static mint inv(int n) {
        return inv_f(n) * f(n - 1);
    }

    static void reserve(int n) {
        extend(n + 1);
    }

  private:
    static std::vector<mint> fact, inv_fact;
};

template <Modint mint>
std::vector<mint> Binomial<mint>::fact = std::vector<mint>(2, 1);

template <Modint mint>
std::vector<mint> Binomial<mint>::inv_fact = std::vector<mint>(2, 1);

}  // namespace ebi
#line 5 "math/catalan_convolution.hpp"

namespace ebi {

template <Modint mint> mint catalan_convolution(int pow, int n) {
    return Binomial<mint>::c(n + n + pow - 1, n) * pow *
           Binomial<mint>::inv(n + pow);
}

}  // namespace ebi
#line 4 "test/math/Catalan_Convolution.test.cpp"

#line 2 "convolution/ntt.hpp"

#include <algorithm>
#include <array>
#line 8 "convolution/ntt.hpp"

#line 2 "math/internal_math.hpp"

#line 4 "math/internal_math.hpp"

namespace ebi {

namespace internal {

constexpr int primitive_root_constexpr(int m) {
    if (m == 2) return 1;
    if (m == 167772161) return 3;
    if (m == 469762049) return 3;
    if (m == 754974721) return 11;
    if (m == 998244353) return 3;
    if (m == 880803841) return 26;
    if (m == 924844033) return 5;
    return -1;
}
template <int m> constexpr int primitive_root = primitive_root_constexpr(m);

}  // namespace internal

}  // namespace ebi
#line 11 "convolution/ntt.hpp"

namespace ebi {

namespace internal {

template <Modint mint, int g = internal::primitive_root<mint::mod()>>
struct ntt_info {
    static constexpr int rank2 =
        std::countr_zero((unsigned int)(mint::mod() - 1));

    std::array<mint, rank2 + 1> root, inv_root;

    ntt_info() {
        root[rank2] = mint(g).pow((mint::mod() - 1) >> rank2);
        inv_root[rank2] = root[rank2].inv();
        for (int i = rank2 - 1; i >= 0; i--) {
            root[i] = root[i + 1] * root[i + 1];
            inv_root[i] = inv_root[i + 1] * inv_root[i + 1];
        }
    }
};

template <Modint mint> void butterfly(std::vector<mint>& a) {
    static const ntt_info<mint> info;
    int n = int(a.size());
    int bit_size = std::countr_zero(a.size());
    assert(n == (int)std::bit_ceil(a.size()));

    // bit reverse
    for (int i = 0, j = 1; j < n - 1; j++) {
        for (int k = n >> 1; k > (i ^= k); k >>= 1)
            ;
        if (j < i) {
            std::swap(a[i], a[j]);
        }
    }

    for (int bit = 0; bit < bit_size; bit++) {
        for (int i = 0; i < n / (1 << (bit + 1)); i++) {
            mint zeta1 = 1;
            mint zeta2 = info.root[1];
            for (int j = 0; j < (1 << bit); j++) {
                int idx = i * (1 << (bit + 1)) + j;
                int jdx = idx + (1 << bit);
                mint p1 = a[idx];
                mint p2 = a[jdx];
                a[idx] = p1 + zeta1 * p2;
                a[jdx] = p1 + zeta2 * p2;
                zeta1 *= info.root[bit + 1];
                zeta2 *= info.root[bit + 1];
            }
        }
    }
}

template <Modint mint> void butterfly_inv(std::vector<mint>& a) {
    static const ntt_info<mint> info;
    int n = int(a.size());
    int bit_size = std::countr_zero(a.size());
    assert(n == (int)std::bit_ceil(a.size()));

    // bit reverse
    for (int i = 0, j = 1; j < n - 1; j++) {
        for (int k = n >> 1; k > (i ^= k); k >>= 1)
            ;
        if (j < i) {
            std::swap(a[i], a[j]);
        }
    }

    for (int bit = 0; bit < bit_size; bit++) {
        for (int i = 0; i < n / (1 << (bit + 1)); i++) {
            mint zeta1 = 1;
            mint zeta2 = info.inv_root[1];
            for (int j = 0; j < (1 << bit); j++) {
                int idx = i * (1 << (bit + 1)) + j;
                int jdx = idx + (1 << bit);
                mint p1 = a[idx];
                mint p2 = a[jdx];
                a[idx] = p1 + zeta1 * p2;
                a[jdx] = p1 + zeta2 * p2;
                zeta1 *= info.inv_root[bit + 1];
                zeta2 *= info.inv_root[bit + 1];
            }
        }
    }
    mint inv_n = mint(n).inv();
    for (int i = 0; i < n; i++) {
        a[i] *= inv_n;
    }
}

}  // namespace internal

template <Modint mint>
std::vector<mint> convolution_naive(const std::vector<mint>& f,
                                    const std::vector<mint>& g) {
    if (f.empty() || g.empty()) return {};
    int n = int(f.size()), m = int(g.size());
    std::vector<mint> c(n + m - 1);
    if (n < m) {
        for (int j = 0; j < m; j++) {
            for (int i = 0; i < n; i++) {
                c[i + j] += f[i] * g[j];
            }
        }
    } else {
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) {
                c[i + j] += f[i] * g[j];
            }
        }
    }
    return c;
}

template <Modint mint>
std::vector<mint> convolution(const std::vector<mint>& f,
                              const std::vector<mint>& g) {
    if (f.empty() || g.empty()) return {};
    if (std::min(f.size(), g.size()) < 60) return convolution_naive(f, g);
    int n = std::bit_ceil(f.size() + g.size() - 1);
    std::vector<mint> a(n), b(n);
    std::copy(f.begin(), f.end(), a.begin());
    std::copy(g.begin(), g.end(), b.begin());
    internal::butterfly(a);
    internal::butterfly(b);
    for (int i = 0; i < n; i++) {
        a[i] *= b[i];
    }
    internal::butterfly_inv(a);
    a.resize(f.size() + g.size() - 1);
    return a;
}

}  // namespace ebi
#line 2 "math/catalan_number.hpp"

#line 5 "math/catalan_number.hpp"

namespace ebi {

template <Modint mint> mint catalan_number(int n) {
    return Binomial<mint>::c(2 * n, n) * Binomial<mint>::inv(n + 1);
}

}  // namespace ebi
#line 2 "modint/modint.hpp"

#line 5 "modint/modint.hpp"

#line 7 "modint/modint.hpp"

namespace ebi {

template <int m> struct static_modint {
  private:
    using modint = static_modint;

  public:
    static constexpr int mod() {
        return m;
    }

    static constexpr modint raw(int v) {
        modint x;
        x._v = v;
        return x;
    }

    constexpr static_modint() : _v(0) {}

    constexpr static_modint(long long v) {
        v %= (long long)umod();
        if (v < 0) v += (long long)umod();
        _v = (unsigned int)v;
    }

    constexpr unsigned int val() const {
        return _v;
    }

    constexpr unsigned int value() const {
        return val();
    }

    constexpr modint &operator++() {
        _v++;
        if (_v == umod()) _v = 0;
        return *this;
    }
    constexpr modint &operator--() {
        if (_v == 0) _v = umod();
        _v--;
        return *this;
    }

    constexpr modint operator++(int) {
        modint res = *this;
        ++*this;
        return res;
    }
    constexpr modint operator--(int) {
        modint res = *this;
        --*this;
        return res;
    }

    constexpr modint &operator+=(const modint &rhs) {
        _v += rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    constexpr modint &operator-=(const modint &rhs) {
        _v -= rhs._v;
        if (_v >= umod()) _v += umod();
        return *this;
    }
    constexpr modint &operator*=(const modint &rhs) {
        unsigned long long x = _v;
        x *= rhs._v;
        _v = (unsigned int)(x % (unsigned long long)umod());
        return *this;
    }
    constexpr modint &operator/=(const modint &rhs) {
        return *this = *this * rhs.inv();
    }

    constexpr modint operator+() const {
        return *this;
    }
    constexpr modint operator-() const {
        return modint() - *this;
    }

    constexpr modint pow(long long n) const {
        assert(0 <= n);
        modint x = *this, res = 1;
        while (n) {
            if (n & 1) res *= x;
            x *= x;
            n >>= 1;
        }
        return res;
    }
    constexpr modint inv() const {
        assert(_v);
        return pow(umod() - 2);
    }

    friend modint operator+(const modint &lhs, const modint &rhs) {
        return modint(lhs) += rhs;
    }
    friend modint operator-(const modint &lhs, const modint &rhs) {
        return modint(lhs) -= rhs;
    }
    friend modint operator*(const modint &lhs, const modint &rhs) {
        return modint(lhs) *= rhs;
    }

    friend modint operator/(const modint &lhs, const modint &rhs) {
        return modint(lhs) /= rhs;
    }
    friend bool operator==(const modint &lhs, const modint &rhs) {
        return lhs.val() == rhs.val();
    }
    friend bool operator!=(const modint &lhs, const modint &rhs) {
        return !(lhs == rhs);
    }

  private:
    unsigned int _v = 0;

    static constexpr unsigned int umod() {
        return m;
    }
};

using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;

}  // namespace ebi
#line 1 "template/template.hpp"
#include <bits/stdc++.h>

#define rep(i, a, n) for (int i = (int)(a); i < (int)(n); i++)
#define rrep(i, a, n) for (int i = ((int)(n)-1); i >= (int)(a); i--)
#define Rep(i, a, n) for (i64 i = (i64)(a); i < (i64)(n); i++)
#define RRep(i, a, n) for (i64 i = ((i64)(n)-i64(1)); i >= (i64)(a); i--)
#define all(v) (v).begin(), (v).end()
#define rall(v) (v).rbegin(), (v).rend()

#line 2 "template/debug_template.hpp"

#line 4 "template/debug_template.hpp"

namespace ebi {

#ifdef LOCAL
#define debug(...)                                                      \
    std::cerr << "LINE: " << __LINE__ << "  [" << #__VA_ARGS__ << "]:", \
        debug_out(__VA_ARGS__)
#else
#define debug(...)
#endif

void debug_out() {
    std::cerr << std::endl;
}

template <typename Head, typename... Tail> void debug_out(Head h, Tail... t) {
    std::cerr << " " << h;
    if (sizeof...(t) > 0) std::cerr << " :";
    debug_out(t...);
}

}  // namespace ebi
#line 2 "template/int_alias.hpp"

#line 4 "template/int_alias.hpp"

namespace ebi {

using ld = long double;
using std::size_t;
using i8 = std::int8_t;
using u8 = std::uint8_t;
using i16 = std::int16_t;
using u16 = std::uint16_t;
using i32 = std::int32_t;
using u32 = std::uint32_t;
using i64 = std::int64_t;
using u64 = std::uint64_t;
using i128 = __int128_t;
using u128 = __uint128_t;

}  // namespace ebi
#line 2 "template/io.hpp"

#line 5 "template/io.hpp"
#include <optional>
#line 7 "template/io.hpp"

namespace ebi {

template <typename T1, typename T2>
std::ostream &operator<<(std::ostream &os, const std::pair<T1, T2> &pa) {
    return os << pa.first << " " << pa.second;
}

template <typename T1, typename T2>
std::istream &operator>>(std::istream &os, std::pair<T1, T2> &pa) {
    return os >> pa.first >> pa.second;
}

template <typename T>
std::ostream &operator<<(std::ostream &os, const std::vector<T> &vec) {
    for (std::size_t i = 0; i < vec.size(); i++)
        os << vec[i] << (i + 1 == vec.size() ? "" : " ");
    return os;
}

template <typename T>
std::istream &operator>>(std::istream &os, std::vector<T> &vec) {
    for (T &e : vec) std::cin >> e;
    return os;
}

template <typename T>
std::ostream &operator<<(std::ostream &os, const std::optional<T> &opt) {
    if (opt) {
        os << opt.value();
    } else {
        os << "invalid value";
    }
    return os;
}

void fast_io() {
    std::cout << std::fixed << std::setprecision(15);
    std::cin.tie(nullptr);
    std::ios::sync_with_stdio(false);
}

}  // namespace ebi
#line 2 "template/utility.hpp"

#line 5 "template/utility.hpp"

#line 2 "graph/base.hpp"

#line 7 "graph/base.hpp"

#line 2 "data_structure/simple_csr.hpp"

#line 6 "data_structure/simple_csr.hpp"

namespace ebi {

template <class E> struct simple_csr {
    simple_csr() = default;

    simple_csr(int n, const std::vector<std::pair<int, E>>& elements)
        : start(n + 1, 0), elist(elements.size()) {
        for (auto e : elements) {
            start[e.first + 1]++;
        }
        for (auto i : std::views::iota(0, n)) {
            start[i + 1] += start[i];
        }
        auto counter = start;
        for (auto [i, e] : elements) {
            elist[counter[i]++] = e;
        }
    }

    simple_csr(const std::vector<std::vector<E>>& es)
        : start(es.size() + 1, 0) {
        int n = es.size();
        for (auto i : std::views::iota(0, n)) {
            start[i + 1] = (int)es[i].size() + start[i];
        }
        elist.resize(start.back());
        for (auto i : std::views::iota(0, n)) {
            std::copy(es[i].begin(), es[i].end(), elist.begin() + start[i]);
        }
    }

    int size() const {
        return (int)start.size() - 1;
    }

    const auto operator[](int i) const {
        return std::ranges::subrange(elist.begin() + start[i],
                                     elist.begin() + start[i + 1]);
    }
    auto operator[](int i) {
        return std::ranges::subrange(elist.begin() + start[i],
                                     elist.begin() + start[i + 1]);
    }

    const auto operator()(int i, int l, int r) const {
        return std::ranges::subrange(elist.begin() + start[i] + l,
                                     elist.begin() + start[i + 1] + r);
    }
    auto operator()(int i, int l, int r) {
        return std::ranges::subrange(elist.begin() + start[i] + l,
                                     elist.begin() + start[i + 1] + r);
    }

  private:
    std::vector<int> start;
    std::vector<E> elist;
};

}  // namespace ebi
#line 9 "graph/base.hpp"

namespace ebi {

template <class T> struct Edge {
    int from, to;
    T cost;
    int id;
};

template <class E> struct Graph {
    using cost_type = E;
    using edge_type = Edge<cost_type>;

    Graph(int n_) : n(n_) {}

    Graph() = default;

    void add_edge(int u, int v, cost_type c) {
        buff.emplace_back(u, edge_type{u, v, c, m});
        edges.emplace_back(edge_type{u, v, c, m++});
    }

    void add_undirected_edge(int u, int v, cost_type c) {
        buff.emplace_back(u, edge_type{u, v, c, m});
        buff.emplace_back(v, edge_type{v, u, c, m});
        edges.emplace_back(edge_type{u, v, c, m});
        m++;
    }

    void read_tree(int offset = 1, bool is_weighted = false) {
        read_graph(n - 1, offset, false, is_weighted);
    }

    void read_parents(int offset = 1) {
        for (auto i : std::views::iota(1, n)) {
            int p;
            std::cin >> p;
            p -= offset;
            add_undirected_edge(p, i, 1);
        }
        build();
    }

    void read_graph(int e, int offset = 1, bool is_directed = false,
                    bool is_weighted = false) {
        for (int i = 0; i < e; i++) {
            int u, v;
            std::cin >> u >> v;
            u -= offset;
            v -= offset;
            if (is_weighted) {
                cost_type c;
                std::cin >> c;
                if (is_directed) {
                    add_edge(u, v, c);
                } else {
                    add_undirected_edge(u, v, c);
                }
            } else {
                if (is_directed) {
                    add_edge(u, v, 1);
                } else {
                    add_undirected_edge(u, v, 1);
                }
            }
        }
        build();
    }

    void build() {
        assert(!prepared);
        csr = simple_csr<edge_type>(n, buff);
        buff.clear();
        prepared = true;
    }

    int size() const {
        return n;
    }

    int node_number() const {
        return n;
    }

    int edge_number() const {
        return m;
    }

    edge_type get_edge(int i) const {
        return edges[i];
    }

    std::vector<edge_type> get_edges() const {
        return edges;
    }

    const auto operator[](int i) const {
        return csr[i];
    }
    auto operator[](int i) {
        return csr[i];
    }

  private:
    int n, m = 0;

    std::vector<std::pair<int,edge_type>> buff;

    std::vector<edge_type> edges;
    simple_csr<edge_type> csr;
    bool prepared = false;
};

}  // namespace ebi
#line 8 "template/utility.hpp"

namespace ebi {

template <class T> inline bool chmin(T &a, T b) {
    if (a > b) {
        a = b;
        return true;
    }
    return false;
}

template <class T> inline bool chmax(T &a, T b) {
    if (a < b) {
        a = b;
        return true;
    }
    return false;
}

template <class T> T safe_ceil(T a, T b) {
    if (a % b == 0)
        return a / b;
    else if (a >= 0)
        return (a / b) + 1;
    else
        return -((-a) / b);
}

template <class T> T safe_floor(T a, T b) {
    if (a % b == 0)
        return a / b;
    else if (a >= 0)
        return a / b;
    else
        return -((-a) / b) - 1;
}

constexpr i64 LNF = std::numeric_limits<i64>::max() / 4;

constexpr int INF = std::numeric_limits<int>::max() / 2;

const std::vector<int> dy = {1, 0, -1, 0, 1, 1, -1, -1};
const std::vector<int> dx = {0, 1, 0, -1, 1, -1, 1, -1};

}  // namespace ebi
#line 9 "test/math/Catalan_Convolution.test.cpp"

namespace ebi {

using mint = modint998244353;

void main_() {
    const int n = 100000;
    int t = 10;
    std::vector<mint> c(n);
    rep(i, 0, n) c[i] = catalan_number<mint>(i);
    std::vector<mint> a = {1};
    rep(i, 1, t) {
        a = convolution(a, c);
        a.resize(n);
        rep(j, 0, n) assert(a[j] == catalan_convolution<mint>(i, j));
    }
    return;
}

}  // namespace ebi

int main() {
    ebi::fast_io();
    int t = 1;
    // std::cin >> t;
    while (t--) {
        ebi::main_();
    }
    int a, b;
    std::cin >> a >> b;
    std::cout << a + b << '\n';
    return 0;
}
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