畳み込み
(math/convolution.hpp)

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• Last update: 2022-10-20 00:25:59+09:00
• Include: #include "math/convolution.hpp"

概要

• 数論変換を用いた二数列の畳み込みを行う。
• 法 $m$ について、 $m - 1$ は結果の列長以上となる最小の 2 冪で割り切れる必要がある。 $998244353 = 2^{23} \times 119 + 1$ がよく用いられる。

詳細

• <typename mint> vector<mint> convolution(vector<mint> a, vector<mint> b)
  $a, b$を畳み込んだ数列を返す。$a, b$それぞれの列長を $n, m$ とすると、結果の列長は $n + m - 1$ となる。$O((n + m)\log(n + m))$時間。

Depends on

• 数論変換 (math/ntt.hpp)
• template.hpp

Required by

• math/bostan_mori.hpp
• 形式的冪級数 (math/fps.hpp)
• math/kth_of_lrs.hpp
• Subset Convolution (math/subset_convolution.hpp)

Verified with

• test/judge.yosupo.jp/Convolution.0.test.cpp
• test/judge.yosupo.jp/Exp_of_Formal_Power_Series.0.test.cpp
• test/judge.yosupo.jp/Inv_of_Formal_Power_Series.0.test.cpp
• test/judge.yosupo.jp/Kth_Term_of_Linearly_Recurrent_Sequence.0.test.cpp
• test/judge.yosupo.jp/Log_of_Formal_Power_Series.0.test.cpp
• test/judge.yosupo.jp/Pow_of_Formal_Power_Series.0.test.cpp
• test/judge.yosupo.jp/Subset_Convolution.0.test.cpp

Code

#pragma once

#include "../template.hpp"
#include "ntt.hpp"

template <typename V> vector<V> convolution_naive(vector<V> a, vector<V> b) {
    ll na = a.size(), nb = b.size();
    vector<V> c(na + nb - 1);
    if (na < nb) swap(a, b), swap(na, nb);
    for (ll i : rep(na)) {
        for (ll j : rep(nb)) c[i + j] += a[i] * b[j];
    }
    return c;
}

template <typename mint> vector<mint> convolution_ntt(vector<mint> a, vector<mint> b) {
    ll _n = a.size() + b.size() - 1, n;
    for (n = 1; n < _n; n <<= 1) {}
    a.resize(n), b.resize(n);
    ntt(a), ntt(b);
    for (ll i : rep(n)) a[i] *= b[i];
    intt(a);
    a.resize(_n);
    return a;
}

template <typename mint> vector<mint> convolution(const vector<mint> &a, const vector<mint> &b) {
    if (min(a.size(), b.size()) <= 60) {
        return convolution_naive(a, b);
    } else {
        return convolution_ntt(a, b);
    }
}

#line 2 "math/convolution.hpp"

#line 2 "template.hpp"

#include <bits/stdc++.h>
using namespace std;

#define all(a) begin(a), end(a)
#define rall(a) rbegin(a), rend(a)
#define uniq(a) (a).erase(unique(all(a)), (a).end())
#define t0 first
#define t1 second
using ll = long long;
using ull = unsigned long long;
using pll = pair<ll, ll>;
using vll = vector<ll>;
constexpr double pi = 3.14159265358979323846;
constexpr ll dy[9] = {0, 1, 0, -1, 1, 1, -1, -1, 0};
constexpr ll dx[9] = {1, 0, -1, 0, 1, -1, -1, 1, 0};
constexpr ll sign(ll a) { return (a > 0) - (a < 0); }
constexpr ll fdiv(ll a, ll b) { return a / b - ((a ^ b) < 0 && a % b); }
constexpr ll cdiv(ll a, ll b) { return -fdiv(-a, b); }
constexpr ll pw(ll n) { return 1ll << n; }
constexpr ll flg(ll n) { return 63 - __builtin_clzll(n); }
constexpr ll clg(ll n) { return n == 1 ? 0 : flg(n - 1) + 1; }
constexpr ll safemod(ll x, ll mod) { return (x % mod + mod) % mod; }
template <typename T> using priority_queue_rev = priority_queue<T, vector<T>, greater<T>>;
template <typename T> constexpr T sq(const T &a) { return a * a; }
template <typename T, typename U> constexpr bool chmax(T &a, const U &b) { return a < b ? a = b, true : false; }
template <typename T, typename U> constexpr bool chmin(T &a, const U &b) { return a > b ? a = b, true : false; }
template <typename T, typename U> ostream &operator<<(ostream &os, const pair<T, U> &a) {
    os << "(" << a.first << ", " << a.second << ")";
    return os;
}
template <typename T, typename U, typename V> ostream &operator<<(ostream &os, const tuple<T, U, V> &a) {
    os << "(" << get<0>(a) << ", " << get<1>(a) << ", " << get<2>(a) << ")";
    return os;
}
template <typename T> ostream &operator<<(ostream &os, const vector<T> &a) {
    os << "(";
    for (auto itr = a.begin(); itr != a.end(); ++itr) os << *itr << (next(itr) != a.end() ? ", " : "");
    os << ")";
    return os;
}
template <typename T> ostream &operator<<(ostream &os, const set<T> &a) {
    os << "(";
    for (auto itr = a.begin(); itr != a.end(); ++itr) os << *itr << (next(itr) != a.end() ? ", " : "");
    os << ")";
    return os;
}
template <typename T> ostream &operator<<(ostream &os, const multiset<T> &a) {
    os << "(";
    for (auto itr = a.begin(); itr != a.end(); ++itr) os << *itr << (next(itr) != a.end() ? ", " : "");
    os << ")";
    return os;
}
template <typename T, typename U> ostream &operator<<(ostream &os, const map<T, U> &a) {
    os << "(";
    for (auto itr = a.begin(); itr != a.end(); ++itr) os << *itr << (next(itr) != a.end() ? ", " : "");
    os << ")";
    return os;
}
#ifdef ONLINE_JUDGE
#define dump(...) (void(0))
#else
void debug() { cerr << endl; }
template <typename Head, typename... Tail> void debug(Head &&head, Tail &&... tail) {
    cerr << head;
    if (sizeof...(Tail)) cerr << ", ";
    debug(tail...);
}
#define dump(...) cerr << __LINE__ << ": " << #__VA_ARGS__ << " = ", debug(__VA_ARGS__)
#endif
struct rep {
    struct itr {
        ll v;
        itr(ll v) : v(v) {}
        void operator++() { ++v; }
        ll operator*() const { return v; }
        bool operator!=(itr i) const { return v < *i; }
    };
    ll l, r;
    rep(ll l, ll r) : l(l), r(r) {}
    rep(ll r) : rep(0, r) {}
    itr begin() const { return l; };
    itr end() const { return r; };
};
struct per {
    struct itr {
        ll v;
        itr(ll v) : v(v) {}
        void operator++() { --v; }
        ll operator*() const { return v; }
        bool operator!=(itr i) const { return v > *i; }
    };
    ll l, r;
    per(ll l, ll r) : l(l), r(r) {}
    per(ll r) : per(0, r) {}
    itr begin() const { return r - 1; };
    itr end() const { return l - 1; };
};
struct io_setup {
    static constexpr int PREC = 20;
    io_setup() {
        cout << fixed << setprecision(PREC);
        cerr << fixed << setprecision(PREC);
    };
} iOS;
#line 2 "math/ntt.hpp"

#line 4 "math/ntt.hpp"

template <typename mint> void ntt(vector<mint> &a, bool inv = false) {
    ll n = a.size(), m = n >> 1;
    mint root = 2;
    while (root.pow((mint::mod() - 1) >> 1) == 1) root += 1;
    mint wn = root.pow((mint::mod() - 1) / n);
    if (inv) wn = wn.inv();
    vector<mint> b(n);
    for (ll i = 1; i < n; i <<= 1, wn *= wn, swap(a, b)) {
        mint wj = 1;
        for (ll j = 0; j < m; j += i, wj *= wn) {
            for (ll k : rep(i)) {
                b[0 + (j << 1) + k] = (a[0 + j + k] + a[m + j + k]);
                b[i + (j << 1) + k] = (a[0 + j + k] - a[m + j + k]) * wj;
            }
        }
    }
    if (inv) {
        mint ninv = mint(n).inv();
        for (mint &ai : a) ai *= ninv;
    }
}
template <typename mint> void intt(vector<mint> &a) { ntt(a, true); }
#line 5 "math/convolution.hpp"

template <typename V> vector<V> convolution_naive(vector<V> a, vector<V> b) {
    ll na = a.size(), nb = b.size();
    vector<V> c(na + nb - 1);
    if (na < nb) swap(a, b), swap(na, nb);
    for (ll i : rep(na)) {
        for (ll j : rep(nb)) c[i + j] += a[i] * b[j];
    }
    return c;
}

template <typename mint> vector<mint> convolution_ntt(vector<mint> a, vector<mint> b) {
    ll _n = a.size() + b.size() - 1, n;
    for (n = 1; n < _n; n <<= 1) {}
    a.resize(n), b.resize(n);
    ntt(a), ntt(b);
    for (ll i : rep(n)) a[i] *= b[i];
    intt(a);
    a.resize(_n);
    return a;
}

template <typename mint> vector<mint> convolution(const vector<mint> &a, const vector<mint> &b) {
    if (min(a.size(), b.size()) <= 60) {
        return convolution_naive(a, b);
    } else {
        return convolution_ntt(a, b);
    }
}

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