#856196 (C++17) No.1575 Divisor Function Hard

提出ソース
結果

問題	No.1575 Divisor Function Hard
ユーザー	👑 Nachia
提出日時	2023-04-02 23:51:58
言語	C++17 (gcc 12.3.0 + boost 1.83.0)
結果	AC
実行時間	785 ms / 5,000 ms
コード長	34,200 bytes
コンパイル時間	4,732 ms
コンパイル使用メモリ	120,172 KB
実行使用メモリ	40,720 KB
最終ジャッジ日時	2023-10-25 05:15:46
合計ジャッジ時間	42,346 ms
ジャッジサーバーID （参考情報）	judge15 / judge11
このコードへのチャレンジ
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テストケース

テストケース表示
入力	結果	実行時間実行使用メモリ
testcase_00	AC	2 ms 4,348 KB
testcase_01	AC	2 ms 4,348 KB
testcase_02	AC	8 ms 4,348 KB
testcase_03	AC	2 ms 4,348 KB
testcase_04	AC	2 ms 4,348 KB
testcase_05	AC	2 ms 4,348 KB
testcase_06	AC	1 ms 4,348 KB
testcase_07	AC	2 ms 4,348 KB
testcase_08	AC	2 ms 4,348 KB
testcase_09	AC	2 ms 4,348 KB
testcase_10	AC	717 ms 40,604 KB
testcase_11	AC	576 ms 39,400 KB
testcase_12	AC	511 ms 36,624 KB
testcase_13	AC	592 ms 38,328 KB
testcase_14	AC	550 ms 39,036 KB
testcase_15	AC	454 ms 37,364 KB
testcase_16	AC	439 ms 36,592 KB
testcase_17	AC	421 ms 36,140 KB
testcase_18	AC	576 ms 38,052 KB
testcase_19	AC	730 ms 40,124 KB
testcase_20	AC	558 ms 38,672 KB
testcase_21	AC	676 ms 39,504 KB
testcase_22	AC	416 ms 36,472 KB
testcase_23	AC	554 ms 38,264 KB
testcase_24	AC	639 ms 40,356 KB
testcase_25	AC	533 ms 38,788 KB
testcase_26	AC	579 ms 38,688 KB
testcase_27	AC	509 ms 38,816 KB
testcase_28	AC	596 ms 39,644 KB
testcase_29	AC	612 ms 39,388 KB
testcase_30	AC	748 ms 40,648 KB
testcase_31	AC	739 ms 40,496 KB
testcase_32	AC	756 ms 40,708 KB
testcase_33	AC	768 ms 40,488 KB
testcase_34	AC	738 ms 40,696 KB
testcase_35	AC	756 ms 40,516 KB
testcase_36	AC	753 ms 40,492 KB
testcase_37	AC	752 ms 40,668 KB
testcase_38	AC	754 ms 40,692 KB
testcase_39	AC	758 ms 40,676 KB
testcase_40	AC	762 ms 40,504 KB
testcase_41	AC	769 ms 40,692 KB
testcase_42	AC	758 ms 40,488 KB
testcase_43	AC	738 ms 40,456 KB
testcase_44	AC	749 ms 40,488 KB
testcase_45	AC	758 ms 40,720 KB
testcase_46	AC	775 ms 40,720 KB
testcase_47	AC	785 ms 40,720 KB
testcase_48	AC	767 ms 40,720 KB
testcase_49	AC	751 ms 40,720 KB
testcase_50	AC	743 ms 40,720 KB
testcase_51	AC	761 ms 40,720 KB
testcase_52	AC	754 ms 40,720 KB
testcase_53	AC	776 ms 40,720 KB
testcase_54	AC	760 ms 40,720 KB
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ソースコード

raw source code
#line 1 "Main.cpp"
#include <iostream>
#include <string>
#include <vector>
#include <algorithm>
#include <atcoder/modint>
#line 5 "nachia\\fps\\fps-struct.hpp"
#include <cassert>
#line 3 "nachia\\math-modulo\\modulo-primitive-root.hpp"
#include <utility>

namespace nachia{

template<unsigned int MOD>
struct PrimitiveRoot{
    using u64 = unsigned long long;
    static constexpr u64 powm(u64 a, u64 i) {
        u64 res = 1, aa = a;
        while(i){
            if(i & 1) res = res * aa % MOD;
            aa = aa * aa % MOD;
            i /= 2;
        }
        return res;
    }
    static constexpr bool ExamineVal(unsigned int g){
        unsigned int t = MOD - 1;
        for(u64 d=2; d*d<=t; d++) if(t % d == 0){
            if(powm(g, (MOD - 1) / d) == 1) return false;
            while(t % d == 0) t /= d;
        }
        if(t != 1) if(powm(g, (MOD - 1) / t) == 1) return false;
        return true;
    }
    static constexpr unsigned int GetVal(){
        for(unsigned int x=2; x<MOD; x++) if(ExamineVal(x)) return x;
        return 0;
    }
    static const unsigned int val = GetVal();
};

} // namespace nachia
#line 3 "nachia\\math\\combination.hpp"

namespace nachia{

template<class Modint>
class Comb{
private:
    std::vector<Modint> F;
    std::vector<Modint> iF;
public:
    void extend(int newN){
        int prevN = (int)F.size() - 1;
        if(prevN >= newN) return;
        F.resize(newN+1);
        iF.resize(newN+1);
        for(int i=prevN+1; i<=newN; i++) F[i] = F[i-1] * Modint::raw(i);
        iF[newN] = F[newN].inv();
        for(int i=newN; i>prevN; i--) iF[i-1] = iF[i] * Modint::raw(i);
    }
    Comb(int n = 1){
        F.assign(2, Modint(1));
        iF.assign(2, Modint(1));
        extend(n);
    }
    Modint factorial(int n) const { return F[n]; }
    Modint invFactorial(int n) const { return iF[n]; }
    Modint invOf(int n) const { return iF[n] * F[n-1]; }
    Modint comb(int n, int r) const {
        if(n < 0 || n < r || r < 0) return Modint(0);
        return F[n] * iF[r] * iF[n-r];
    }
    Modint invComb(int n, int r) const {
        if(n < 0 || n < r || r < 0) return Modint(0);
        return iF[n] * F[r] * F[n-r];
    }
    Modint perm(int n, int r) const {
        if(n < 0 || n < r || r < 0) return Modint(0);
        return F[n] * iF[n-r];
    }
    Modint invPerm(int n, int r) const {
        if(n < 0 || n < r || r < 0) return Modint(0);
        return iF[n] * F[n-r];
    }
    Modint operator()(int n, int r) const { return comb(n,r); }
};

} // namespace nachia
#line 4 "nachia\\misc\\bit-operations.hpp"

namespace nachia{

int Popcount(unsigned long long c) noexcept {
#ifdef __GNUC__
    return __builtin_popcountll(c);
#else
    c = (c & (~0ull/3)) + ((c >> 1) & (~0ull/3));
    c = (c & (~0ull/5)) + ((c >> 2) & (~0ull/5));
    c = (c & (~0ull/17)) + ((c >> 4) & (~0ull/17));
    c = (c * (~0ull/257)) >> 56;
    return c;
#endif
}

// please ensure x != 0
int MsbIndex(unsigned long long x) noexcept {
#ifdef __GNUC__
    return 63 - __builtin_clzll(x);
#else
    int res = 0;
    for(int d=32; d>0; d>>=1) if(x >> d){ res |= d; x >>= d; }
    return res;
#endif
}

// please ensure x != 0
int LsbIndex(unsigned long long x) noexcept {
#ifdef __GNUC__
    return __builtin_ctzll(x);
#else
    return MsbIndex(x & -x);
#endif
}

}

#line 2 "nachia\\fps\\ntt-interface.hpp"

namespace nachia {

template<class mint>
struct NttInterface{

template<class Iter>
void Butterfly(Iter, int) const {}

template<class Iter>
void IButterfly(Iter, int) const {}

template<class Iter>
void BitReversal(Iter a, int N) const {
    for(int i=0, j=0; j<N; j++){
        if(i < j) std::swap(a[i], a[j]);
        for(int k = N>>1; k > (i^=k); k>>=1);
    }
}

};

} // namespace nachia
#line 5 "nachia\\fps\\ntt-acl.hpp"
#include <iterator>
#line 8 "nachia\\fps\\ntt-acl.hpp"
#include <array>

namespace nachia{
    
constexpr int bsf_constexpr(unsigned int n) {
    int x = 0;
    while (!(n & (1 << x))) x++;
    return x;
}

template <class mint>
struct NttFromAcl : NttInterface<mint> {

using u32 = unsigned int;
using u64 = unsigned long long;
    
static int ceil_pow2(int n) {
    int x = 0;
    while ((1U << x) < (u32)(n)) x++;
    return x;
}

struct fft_info {
    static constexpr u32 g = nachia::PrimitiveRoot<mint::mod()>::val;
    static constexpr int rank2 = bsf_constexpr(mint::mod()-1);
    std::array<mint, rank2+1> root;
    std::array<mint, rank2+1> iroot;

    std::array<mint, std::max(0, rank2-1)> rate2;
    std::array<mint, std::max(0, rank2-1)> irate2;

    std::array<mint, std::max(0, rank2-2)> rate3;
    std::array<mint, std::max(0, rank2-2)> irate3;

    fft_info(){
        root[rank2] = mint(g).pow((mint::mod() - 1) >> rank2);
        iroot[rank2] = root[rank2].inv();
        for(int i=rank2-1; i>=0; i--){
            root[i] = root[i+1] * root[i+1];
            iroot[i] = iroot[i+1] * iroot[i+1];
        }
        mint prod = 1, iprod = 1;
        for(int i=0; i<=rank2-2; i++){
            rate2[i] = root[i+2] * prod;
            irate2[i] = iroot[i+2] * iprod;
            prod *= iroot[i+2];
            iprod *= root[i+2];
        }
        prod = 1; iprod = 1;
        for(int i=0; i<=rank2-3; i++){
            rate3[i] = root[i+3] * prod;
            irate3[i] = iroot[i+3] * iprod;
            prod *= iroot[i+3];
            iprod *= root[i+3];
        }
    }
};

template<class RandomAccessIterator>
void Butterfly(RandomAccessIterator a, int n) const {
    int h = ceil_pow2(n);

    static const fft_info info;

    int len = 0;
    while(len < h){
        if(h-len == 1){
            int p = 1 << (h-len-1);
            mint rot = 1;
            for(int s=0; s<(1<<len); s++){
                int offset = s << (h-len);
                for(int i=0; i<p; i++){
                    auto l = a[i+offset];
                    auto r = a[i+offset+p] * rot;
                    a[i+offset] = l+r;
                    a[i+offset+p] = l-r;
                }
                if(s+1 != (1<<len)) rot *= info.rate2[LsbIndex(~(u32)(s))];
            }
            len++;
        } else {
            int p = 1 << (h-len-2);
            mint rot = 1, imag = info.root[2];
            for(int s=0; s<(1<<len); s++){
                mint rot2 = rot * rot;
                mint rot3 = rot2 * rot;
                int offset = s << (h-len);
                for(int i=0; i<p; i++){
                    auto mod2 = 1ULL * mint::mod() * mint::mod();
                    auto a0 = 1ULL * a[i+offset].val();
                    auto a1 = 1ULL * a[i+offset+p].val() * rot.val();
                    auto a2 = 1ULL * a[i+offset+2*p].val() * rot2.val();
                    auto a3 = 1ULL * a[i+offset+3*p].val() * rot3.val();
                    auto a1na3imag = 1ULL * mint(a1 + mod2 - a3).val() * imag.val();
                    auto na2 = mod2 - a2;
                    a[i+offset] = a0 + a2 + a1 + a3;
                    a[i+offset+1*p] = a0 + a2 + (2 * mod2 - (a1 + a3));
                    a[i+offset+2*p] = a0 + na2 + a1na3imag;
                    a[i+offset+3*p] = a0 + na2 + (mod2 - a1na3imag);
                }
                if(s+1 != (1<<len)) rot *= info.rate3[LsbIndex(~(u32)(s))];
            }
            len += 2;
        }
    }
}

template<class RandomAccessIterator>
void IButterfly(RandomAccessIterator a, int n) const {
    int h = ceil_pow2(n);

    static const fft_info info;
    constexpr int MOD = mint::mod();

    int len = h;
    while(len){
        if(len == 1){
            int p = 1 << (h-len);
            mint irot = 1;
            for(int s=0; s<(1<<(len-1)); s++){
                int offset = s << (h-len+1);
                for(int i=0; i<p; i++){
                    auto l = a[i+offset];
                    auto r = a[i+offset+p];
                    a[i+offset] = l+r;
                    a[i+offset+p] = (u64)(MOD + l.val() - r.val()) * irot.val();
                }
                if(s+1 != (1<<(len-1))) irot *= info.irate2[LsbIndex(~(u32)(s))];
            }
            len--;
        } else {
            int p = 1 << (h-len);
            mint irot = 1, iimag = info.iroot[2];
            for(int s=0; s<(1<<(len-2)); s++){
                mint irot2 = irot * irot;
                mint irot3 = irot2 * irot;
                int offset = s << (h-len+2);
                for(int i=0; i<p; i++){
                    auto a0 = 1ULL * a[i+offset+0*p].val();
                    auto a1 = 1ULL * a[i+offset+1*p].val();
                    auto a2 = 1ULL * a[i+offset+2*p].val();
                    auto a3 = 1ULL * a[i+offset+3*p].val();

                    auto a2na3iimag = 1ULL * mint((MOD + a2 - a3) * iimag.val()).val();

                    a[i+offset] = a0 + a1 + a2 + a3;
                    a[i+offset+1*p] = (a0 + (MOD - a1) + a2na3iimag) * irot.val();
                    a[i+offset+2*p] = (a0 + a1 + (MOD - a2) + (MOD - a3)) * irot2.val();
                    a[i+offset+3*p] = (a0 + (MOD - a1) + (MOD - a2na3iimag)) * irot3.val();
                }
                if(s+1 != (1<<(len-2))) irot *= info.irate3[LsbIndex(~(u32)(s))];
            }
            len -= 2;
        }
    }
}

};

} // namespace nachia
#line 11 "nachia\\fps\\fps-struct.hpp"

namespace nachia {

template<class Elem, class NttInst = NttFromAcl<Elem>>
struct FpsNtt {
public:
    using Fps = FpsNtt;
    using ElemTy = Elem;
    static constexpr unsigned int MOD = Elem::mod();
    static constexpr int CONV_THRES = 30;
    static const NttInst nttInst;
    static const unsigned int zeta = nachia::PrimitiveRoot<MOD>::GetVal();
private:
    using u32 = unsigned int;
    static Elem ZeroElem() noexcept { return Elem(0); }
    static Elem OneElem() noexcept { return Elem(1); }
    static Comb<Elem> comb;
    std::vector<Elem> a;
    int RSZ(int& sz) const { return sz = (sz < 0 ? size() : sz); }
public:

    int size() const noexcept { return a.size(); }
    Elem& operator[](int x) noexcept { return a[x]; }
    const Elem& operator[](int x) const noexcept { return a[x]; }
    Elem getCoeff(int x) const noexcept { return (0 <= x && x < size()) ? a[x] : ZeroElem(); }
    static Comb<Elem>& GetComb() { return comb; }
    static int BestNttSize(int x) noexcept { assert(x); return 1 << MsbIndex(x*2-1); }
    Fps move(){ return std::move(*this); }
    Fps& set(int i, Elem c){ a[i] = c; return *this; }

    Fps& removeLeadingZeros(){
        int newsz = size();
        while(newsz && a[newsz-1].val() == 0) newsz--;
        a.resize(newsz);
        if((int)a.capacity() / 4 > newsz) a.shrink_to_fit();
        return *this;
    }

    FpsNtt(){}
    FpsNtt(int sz) : a(sz, ZeroElem()) {}
    FpsNtt(std::vector<Elem>&& src) : a(std::move(src)) {}
    FpsNtt(const std::vector<Elem>& src) : a(src) {}
    
    Fps& ntt() {
        capSize(BestNttSize(size()));
        nttInst.Butterfly(a.begin(), size());
        return *this;
    }
    Fps& intt() {
        nttInst.IButterfly(a.begin(), a.size());
        return times(Elem::raw(size()).inv());
    }
    Fps nttDouble(Fps vanilla) const {
        int n = size();
        assert(n == (n&-n)); // n is a power of 2
        Elem q = Elem::raw(zeta).pow((Elem::mod() - 1) / (n*2));
        Elem qq = OneElem();
        for(int i=0; i<n; i++){ vanilla[i] *= qq; qq *= q; }
        vanilla.ntt();
        Fps res = clip(0, n*2);
        for(int i=0; i<n; i++) res[n+i] = vanilla[i];
        return res;
    }
    Fps nttDouble() const { return nttDouble(clip().intt().move()); }

    // Fps res(resSz);
    // for(int j=0; j<resSz-destL && j+srcL < srcR; j++) res[j+destL] = a.getCoeff(j+srcL)
    // if srcR is unspecified -> srcR = max(srcL, size());
    // if resSz is unspecified -> resSz = destL + srcR - srcL
    Fps clip(int srcL, int srcR = -1, int destL = 0, int resSz = -1) const {
        srcR = RSZ(srcR);
        if(resSz < 0) resSz = destL + srcR - srcL;
        int rj = std::min(std::min(srcR, size()) - srcL, resSz - destL);
        Fps res(resSz);
        for(int j=std::max(0, -srcL); j<rj; j++) res[j+destL] = a[j+srcL];
        return res;
    }
    Fps clip() const { return *this; }

    Fps& capSize(int l, int r) {
        if(r <= (int)size()) a.resize(r);
        if(size() <= l) a.resize(l, ZeroElem());
        return *this;
    }
    Fps& capSize(int z){ a.resize(RSZ(z), ZeroElem()); return *this; }
    Fps& times(Elem x){ for(int i=0; i<size(); i++){ a[i] *= x; } return *this; }
    Fps& timesFactorial(int z = -1){ comb.extend(RSZ(z)); for(int i=0; i<z; i++){ a[i] *= comb.factorial(i); } return *this; }
    Fps& timesInvFactorial(int z = -1){ comb.extend(RSZ(z)); for(int i=0; i<z; i++){ a[i] *= comb.invFactorial(i); } return *this; }
    Fps& clrRange(int l, int r){ for(int i=l; i<r; i++){ a[i] = ZeroElem(); } return *this; }
    Fps& negate(){ for(auto& e : a){ e = -e; } return *this; }
    Fps& mulEach(const Fps& other, int maxi = -1){
        maxi = std::min(RSZ(maxi), std::min(size(), other.size()));
        for(int i=0; i<maxi; i++) a[i] *= other[i];
        return *this;
    }
    Fps& reverse(int sz = -1){ RSZ(sz); std::reverse(a.begin(), a.begin() + sz); return *this; }

    static Fps convolution(const Fps& a, const Fps& b, int sz = -1){
        if(std::min(a.size(), b.size()) <= CONV_THRES){
            if(a.size() > b.size()) return convolution(b, a, sz);
            if(sz < 0) sz = std::max(0, a.size() + b.size() - 1);
            std::vector<Elem> res(sz);
            for(int i=0; i<a.size(); i++) for(int j=0; j<b.size() && i+j<sz; j++) res[i+j] += a[i] * b[j];
            return res;
        }
        int Z = BestNttSize(a.size() + b.size() - 1);
        return a.clip(0, Z).ntt().mulEach(b.clip(0, Z).ntt()).intt().capSize(sz).move();
    }
    Fps convolve(const Fps& r, int sz = -1) const { return convolution(*this, r, sz); }
    
    //   1
    // ----- = 1 + f + f^2 + f^3 + ...
    //  1-f
    Fps powerSum(int sz) const {
        RSZ(sz);
        if(sz == 0) return {};
        int q = std::min(sz, 32);
        Fps x = Fps(q).set(0, OneElem()).move();
        for(int i=1; i<q; i++) for(int j=1; j<=std::min(i,(int)a.size()-1); j++) x[i] += x[i-j] * a[j];
        while(x.size() < sz){
            int hN = x.size(), N = hN*2;
            Fps a = x.clip(0, N).ntt().move();
            Fps b = clip(0, N).ntt().mulEach(a).intt().clrRange(0,hN).ntt().mulEach(a).intt().move();
            for(int i=0; i<hN; i++) b[i] = x[i];
            std::swap(b, x);
        }
        return x.capSize(sz).move();
    }

    Fps inv(int sz = -1) const {
        RSZ(sz);
        Elem iA0 = a[0].inv();
        return clip(0, std::min(sz, size())).times(-iA0).set(0, ZeroElem()).powerSum(sz).times(iA0).move();
    }
    
    Fps& difference(){
        if(size() == 0) return *this;
        for(int i=0; i+1<size(); i++) a[i] = a[i+1] * Elem::raw(i+1);
        return capSize(size()-1);
    }
    Fps& integral(){
        if(size() == 0) return capSize(1);
        capSize(size()+1);
        comb.extend(size());
        for(int i=size()-1; i>=1; i--) a[i] = a[i-1] * comb.invOf(i);
        return set(0, ZeroElem());
    }
    
    Fps log(int sz = -1){
        RSZ(sz);
        assert(sz != 0);
        assert(a[0].val() == 1);
        return convolution(inv(sz), clip().difference(), sz-1).integral();
    }

    Fps exp(int sz = -1){
        RSZ(sz);
        Fps res = Fps(1).set(0, OneElem());
        while(res.size() < sz){
            auto z = res.size();
            auto tmp = res.capSize(z*2).log().set(0, -OneElem()).move();
            for(int i=0; i<z*2 && i<size(); i++) tmp[i] -= a[i];
            auto resntt = res.clip().ntt().mulEach(tmp.ntt()).intt().move();
            for(int i=z; i<z*2; i++) res[i] = -resntt[i];
        }
        return res.capSize(0, sz).move();
    }
    
    Fps pow(unsigned long long k, int sz = -1){
        int n = RSZ(sz);
        if(k == 0) return Fps(n).set(0, OneElem()).move();
        int ctz = 0;
        while(ctz<n && a[ctz].val() == 0) ctz++;
        if((unsigned long long)ctz >= (n-1) / k + 1) return Fps(n);
        Elem a0 = a[ctz];
        return clip(ctz, ctz+n-ctz*k).times(a0.inv()).log().times(Elem(k)).exp().times(a0.pow(k)).clip(0, -1, ctz*k);
    }

    auto begin(){ return a.begin(); }
    auto end(){ return a.end(); }
    auto begin() const { return a.begin(); }
    auto end() const { return a.end(); }

    std::string toString(std::string beg = "[ ", std::string delim = " ", std::string en = " ]") const {
        std::string res = beg;
        bool f = false;
        for(auto x : a){ if(f){ res += delim; } f = true; res += std::to_string(x.val()); }
        res += en;
        return res;
    }

    std::vector<Elem> getVectorMoved(){ return std::move(a); }

    Fps& operator+=(const Fps& r){
        capSize(std::max(size(), r.size()));
        for(int i=0; i<r.size(); i++) a[i] += r[i];
        return *this;
    }
    Fps& operator-=(const Fps& r){
        capSize(std::max(size(), r.size()));
        for(int i=0; i<r.size(); i++) a[i] -= r[i];
        return *this;
    }
    Fps operator+(const Fps& r) const { return (clip(0, std::max(size(), r.size())) += r).move(); }
    Fps operator-(const Fps& r) const { return (clip(0, std::max(size(), r.size())) -= r).move(); }
    Fps operator-() const { return (clip().negate()).move(); }
    Fps operator*(const Fps& r) const { return convolve(r).removeLeadingZeros().move(); }
    Fps& operator*=(const Fps& r){ return (*this) = operator*(r); }
    Fps& operator*=(Elem m){ return times(m); }
    Fps operator*(Elem m) const { return (clip() *= m).move(); }

    Elem eval(Elem x) const {
        Elem res = 0;
        for(int i=size()-1; i>=0; i--) res = res * x + a[i];
        return res;
    }
};

template<class Elem, class NttInst> Comb<Elem> FpsNtt<Elem, NttInst>::comb;
template<class Elem, class NttInst> const NttInst FpsNtt<Elem, NttInst>::nttInst;

} // namespace nachia

#line 11 "nachia\\fps\\formal-power-series-struct.hpp"

namespace nachia {

template<class Elem, class NttInst = NttFromAcl<Elem>>
struct FormalPowerSeriesNTT {
public:
    using Fps = FormalPowerSeriesNTT;
    using ElemTy = Elem;
    static constexpr unsigned int MOD = Elem::mod();
    static constexpr int CONV_THRES = 30;
    static const NttInst nttInst;
    static const unsigned int zeta = nachia::PrimitiveRoot<MOD>::GetVal();
private:
    using u32 = unsigned int;
    static Elem ZeroElem() noexcept { return Elem(0); }
    static Elem OneElem() noexcept { return Elem(1); }
    static Comb<Elem> comb;
    std::vector<Elem> a;
public:

    int size() const noexcept { return a.size(); }
    Elem& operator[](int x) noexcept { return a[x]; }
    const Elem& operator[](int x) const noexcept { return a[x]; }
    Elem getCoeff(int x) const noexcept { return (0 <= x && x < size()) ? a[x] : ZeroElem(); }
    static Comb<Elem>& GetComb() { return comb; }
    static int BestNttSize(int x) noexcept { assert(x); return 1 << MsbIndex(x*2-1); }
    Fps move(){ return std::move(*this); }
    Fps& set(int i, Elem c){ a[i] = c; return *this; }

    Fps& removeLeadingZeros(){
        int newsz = size();
        while(newsz && a[newsz-1].val() == 0) newsz--;
        a.resize(newsz);
        if((int)a.capacity() / 4 > newsz) a.shrink_to_fit();
        return *this;
    }

    FormalPowerSeriesNTT(){}
    FormalPowerSeriesNTT(int sz) : a(sz, ZeroElem()) {}
    FormalPowerSeriesNTT(std::vector<Elem>&& src) : a(std::move(src)) {}
    FormalPowerSeriesNTT(const std::vector<Elem>& src) : a(src) {}
    
    Fps& ntt() {
        capSize(BestNttSize(size()));
        nttInst.Butterfly(a.begin(), size());
        return *this;
    }
    Fps& intt() {
        nttInst.IButterfly(a.begin(), a.size());
        return times(Elem::raw(size()).inv());
    }
    Fps nttDouble(Fps vanilla) const {
        int n = size();
        assert(n == (n&-n)); // n is a power of 2
        Elem q = Elem::raw(zeta).pow((Elem::mod() - 1) / (n*2));
        Elem qq = OneElem();
        for(int i=0; i<n; i++){ vanilla[i] *= qq; qq *= q; }
        vanilla.ntt();
        Fps res = clip(0, n*2);
        for(int i=0; i<n; i++) res[n+i] = vanilla[i];
        return res;
    }
    Fps nttDouble() const { return nttDouble(clip().intt().move()); }

    // Fps res(resSz);
    // for(int j=0; j<resSz-destL && j+srcL < srcR; j++) res[j+destL] = a.getCoeff(j+srcL)
    // if srcR is unspecified -> srcR = max(srcL, size());
    // if resSz is unspecified -> resSz = destL + srcR - srcL
    Fps clip(int srcL, int srcR = -1, int destL = 0, int resSz = -1) const {
        if(srcR < 0) srcR = std::max(srcL, size());
        if(resSz < 0) resSz = destL + srcR - srcL;
        if(srcR > size()) srcR = size();
        Fps res(resSz);
        for(int j=std::max(0, -srcL); j+destL < resSz && j+srcL < srcR; j++) res[j+destL] = a[j+srcL];
        return res;
    }
    Fps clip() const { return *this; }

    Fps& capSize(int l, int r) {
        if(r <= (int)size()) a.resize(r);
        if(size() <= l) a.resize(l, ZeroElem());
        return *this;
    }
    Fps& capSize(int z){ a.resize(z, ZeroElem()); return *this; }

    Fps& times(Elem x){ for(int i=0; i<size(); i++){ a[i] *= x; } return *this; }
    Fps& clrRange(int l, int r){ for(int i=l; i<r; i++){ a[i] = ZeroElem(); } return *this; }
    Fps& negate(){ for(auto& e : a){ e = -e; } return *this; }
    Fps& mulEach(const Fps& other, size_t maxi = ~(size_t)0){
        maxi = std::min(maxi, (size_t)std::min(size(), other.size()));
        for(size_t i=0; i<maxi; i++) a[i] *= other[i];
        return *this;
    }
    Fps& reverse(int sz = -1){ std::reverse(a.begin(), a.begin() + (sz < 0 ? size() : sz)); return *this; }

    static Fps convolution(const Fps& a, const Fps& b, int sz = -1){
        if(std::min(a.size(), b.size()) <= CONV_THRES){
            if(a.size() > b.size()) return convolution(b, a, sz);
            if(sz < 0) sz = std::max(0, a.size() + b.size() - 1);
            Fps res(sz);
            for(int i=0; i<a.size(); i++) for(int j=0; j<b.size() && i+j<sz; j++) res[i+j] += a[i] * b[j];
            return res;
        }
        int Z = BestNttSize(a.size() + b.size() - 1);
        if(sz == -1) sz = Z;
        return a.clip(0, Z).ntt().mulEach(b.clip(0, Z).ntt()).intt().capSize(sz).move();
    }
    Fps convolve(const Fps& r, int sz = -1) const { return convolution(*this, r, sz); }
    
    //   1
    // ----- = 1 + f + f^2 + f^3 + ...
    //  1-f
    Fps powerSum(int sz) const {
        if(sz < 0) sz = size();
        if(sz == 0) return {};
        int q = std::min(sz, 32);
        Fps x = Fps(q).set(0, OneElem()).move();
        for(int i=1; i<q; i++) for(int j=1; j<=std::min(i,(int)a.size()-1); j++) x[i] += x[i-j] * a[j];
        while(x.size() < sz){
            int hN = x.size(), N = hN*2;
            Fps a = x.clip(0, N).ntt().move();
            Fps b = clip(0, N).ntt().mulEach(a).intt().clrRange(0,hN).ntt().mulEach(a).intt().move();
            for(int i=0; i<hN; i++) b[i] = x[i];
            std::swap(b, x);
        }
        return x.capSize(sz).move();
    }

    Fps inv(int sz = -1) const {
        if(sz < 0) sz = size();
        Elem iA0 = a[0].inv();
        return clip(0, std::min(sz, size())).times(-iA0).set(0, ZeroElem()).powerSum(sz).times(iA0).move();
    }
    
    Fps& difference(){
        if(size() == 0) return *this;
        for(int i=0; i+1<size(); i++) a[i] = a[i+1] * Elem::raw(i+1);
        return capSize(size()-1);
    }
    Fps& integral(){
        if(size() == 0) return capSize(1);
        capSize(size()+1);
        comb.extend(size());
        for(int i=size()-1; i>=1; i--) a[i] = a[i-1] * comb.invOf(i);
        return set(0, ZeroElem());
    }
    
    Fps log(int sz = -1){
        if(sz < 0) sz = size();
        assert(sz != 0);
        assert(a[0].val() == 1);
        return convolution(inv(sz), clip().difference(), sz-1).integral();
    }

    Fps exp(int sz = -1){
        if(sz < 0) sz = size();
        Fps res = Fps(1).set(0, OneElem());
        while(res.size() < sz){
            auto z = res.size();
            auto tmp = res.capSize(z*2).log().set(0, -OneElem()).move();
            for(int i=0; i<z*2 && i<size(); i++) tmp[i] -= a[i];
            auto resntt = res.clip().ntt().mulEach(tmp.ntt()).intt().move();
            for(int i=z; i<z*2; i++) res[i] = -resntt[i];
        }
        return res.capSize(0, sz).move();
    }
    
    Fps pow(unsigned long long k, int sz = -1){
        int n = sz < 0 ? size() : sz;
        if(k == 0) return Fps(n).set(0, OneElem()).move();
        int ctz = 0;
        while(ctz<n && a[ctz].val() == 0) ctz++;
        if((unsigned long long)ctz >= (n-1) / k + 1) return Fps(n);
        Elem a0 = a[ctz];
        return clip(ctz, ctz+n-ctz*k).times(a0.inv()).log().times(Elem(k)).exp().times(a0.pow(k)).clip(0, -1, ctz*k);
    }

    auto begin(){ return a.begin(); }
    auto end(){ return a.end(); }
    auto begin() const { return a.begin(); }
    auto end() const { return a.end(); }

    std::string toString(std::string beg = "[ ", std::string delim = " ", std::string en = " ]") const {
        std::string res = beg;
        bool f = false;
        for(auto x : a){ if(f){ res += delim; } f = true; res += std::to_string(x.val()); }
        res += en;
        return res;
    }

    std::vector<Elem> getVectorMoved(){ return std::move(a); }

    Fps& operator+=(const Fps& r){
        capSize(std::max(size(), r.size()));
        for(int i=0; i<r.size(); i++) a[i] += r[i];
        return *this;
    }
    Fps& operator-=(const Fps& r){
        capSize(std::max(size(), r.size()));
        for(int i=0; i<r.size(); i++) a[i] -= r[i];
        return *this;
    }
    Fps operator+(const Fps& r) const { return (clip(0, std::max(size(), r.size())) += r).move(); }
    Fps operator-(const Fps& r) const { return (clip(0, std::max(size(), r.size())) -= r).move(); }
    Fps operator-() const { return clip().negate().move(); }
    Fps operator*(const Fps& r) const { return convolve(r).removeLeadingZeros().move(); }
    Fps& operator*=(const Fps& r){ return (*this) = operator*(r); }
    Fps& operator*=(Elem m){ return times(m); }
    Fps operator*(Elem m) const { return clip().times(m).move(); }

    Elem eval(Elem x) const {
        Elem res = 0;
        for(int i=size()-1; i>=0; i--) res = res * x + a[i];
        return res;
    }
};

template<class Elem, class NttInst> Comb<Elem> FormalPowerSeriesNTT<Elem, NttInst>::comb;
template<class Elem, class NttInst> const NttInst FormalPowerSeriesNTT<Elem, NttInst>::nttInst;

} // namespace nachia

#line 2 "nachia\\fps\\ntt-setup-manager.hpp"

namespace nachia{

template<class Fps>
class FpsNttSetupManager {
    using ElemTy = typename Fps::ElemTy;
    using MyType = FpsNttSetupManager;
    Fps raw;
    mutable Fps ntt;
    static const int THRESH = 30;
    FpsNttSetupManager(Fps _raw) : raw(_raw.move()) , ntt() {}
public:
    FpsNttSetupManager() : FpsNttSetupManager(Fps()) {}
    FpsNttSetupManager(Fps _raw, Fps _ntt) : raw(_raw.move()) , ntt(_ntt.move()) {}
    const Fps& getRaw() const { return raw; }
    int size() const { return raw.size(); }
    int Least(){ return Fps::BestNttSize(raw.size()); }
    static MyType FromRaw(Fps _raw){ return FpsNttSetupManager(_raw.move()); }
    static MyType FromNtt(Fps _ntt){
        Fps x = _ntt.clip();
        return MyType(x.intt().removeLeadingZeros().move(), _ntt.move());
    }
    void doubling() const {
        if(ntt.size() == 0) ntt = raw.clip(0, Fps::BestNttSize(raw.size())).ntt().move();
        else ntt = ntt.nttDouble(raw.clip(0, ntt.size()));
    }
    Fps& ensureNtt(int sz) const {
        if(sz / 8 >= ntt.size()) ntt = raw.clip(0, sz).ntt().move();
        while(ntt.size() < sz) doubling();
        return ntt;
    }
    Fps nttClip(int sz) const { return ensureNtt(sz).clip(0,sz); }
    std::pair<Fps, Fps> destruct(){ return std::make_pair(raw.move(), ntt.move()); }
    MyType operator+(const MyType& r) const {
        Fps nntt;
        int z1 = std::min(ntt.size(), r.ntt.size());
        if(z1 >= std::max(size(), r.size())){
            nntt.capSize(std::min(ntt.size(), r.ntt.size()));
            for(int i=0; i<nntt.size(); i++) nntt[i] = ntt[i] + r.ntt[i];
        }
        return FpsNttSetupManager(raw + r.raw, nntt.move());
    }
    MyType operator*(const MyType& r) const {
        if(std::min(size(), r.size()) <= THRESH) return FromRaw(raw * r.raw);
        int sz = Fps::BestNttSize(size() + r.size() - 1);
        return FromNtt(nttClip(sz).mulEach(r.ensureNtt(sz)).move());
    }
};

} // namespace nachia
#line 4 "nachia\\fps\\product-of-many-polynomials.hpp"

namespace nachia{

template<class Fps>
Fps ProductOfManyPolynomials(std::vector<Fps> poly){
    using Modint = typename Fps::ElemTy;
    using Fps2 = FpsNttSetupManager<Fps>;
    if(poly.empty()) return std::vector<Modint>{Modint(1)};
    for(auto& p : poly) p.removeLeadingZeros();
    for(auto& p : poly) if(p.size() == 0) return Fps();
    std::vector<Fps2> poly2;
    for(auto& p : poly) poly2.push_back(Fps2::FromRaw(p.move()));
    int OFF_K = 16;
    for(int K=OFF_K; poly2.size() != 1; K*=2){
        size_t pos = poly2.size();
        for(size_t i=0; i<poly2.size(); i++) if(poly2[i].size() <= K){
            if(pos == poly2.size() || poly2[pos].size() + poly2[i].size() - 1 > K){ pos = i; continue; }
            poly2[pos] = poly2[pos] * poly2[i];
            std::swap(poly2[i--], poly2.back());
            poly2.pop_back();
        }
    }
    return poly2[0].destruct().first.move();
}

} // namespace nachia
#line 3 "nachia\\fps\\polinomial-division.hpp"

namespace nachia{

// return polynomials have no leading zeros
template<class Fps>
std::pair<Fps, Fps> PolynomialDivision(
    Fps A,
    Fps D,
    bool do_get_remainder = true
){
    D.removeLeadingZeros();
    auto dsize = D.size();
    assert(dsize != 0);
    if(A.size() == 0){ return std::make_pair(Fps(), Fps()); }
    if(A.size() < dsize){ return std::make_pair(Fps(), A.move()); }
    int n = A.size();
    int divSize = n - dsize + 1;

    D.reverse();
    A.reverse();
    auto invD = D.inv(divSize);

    auto tmp = A.clip(0, divSize).convolve(invD, divSize);
    if(!do_get_remainder || dsize == 1) return std::make_pair(tmp.reverse().move(), Fps());
    Fps tmp2 = tmp.convolve(D);
    Fps ans2(dsize - 1);
    for(int i=0; i<dsize-1; i++) ans2[i] = A[n-1-i] - tmp2[n-1-i];
    return std::make_pair(tmp.reverse().move(), ans2.removeLeadingZeros().move());
}

} // namespace nachia
#line 4 "nachia\\fps\\multipoint-evaluation.hpp"

namespace nachia{

template<class Fps>
std::vector<typename Fps::ElemTy> MultipointEvaluation(
    Fps A,
    std::vector<typename Fps::ElemTy> x
){
    using Elem = typename Fps::ElemTy;
    std::vector<int> Id;
    for(int idi=0; idi<(int)x.size(); ) Id.push_back((Id.size() & 31) ? idi++ : -1);
    size_t segtree_n = 1;
    while(segtree_n < Id.size()) segtree_n *= 2;
    std::vector<Fps> divtree(segtree_n * 2);
    for(size_t i=Id.size(); i<segtree_n; i++) divtree[segtree_n + i] = std::vector<Elem>{ 1 };
    for(size_t i=0; i<Id.size(); i++) divtree[segtree_n + i] = (Id[i] >= 0 ? std::vector<Elem>{ -x[Id[i]], 1 } : std::vector<Elem>{ 1 });
    for(int i=segtree_n-1; i>=1; i--) divtree[i] = divtree[i*2] * divtree[i*2+1];
    std::vector<Fps> remtree(segtree_n * 2);
    std::vector<bool> resolved(segtree_n * 2);
    remtree[1] = PolynomialDivision(A, divtree[1]).second;
    std::vector<Elem> res(x.size());
    for(size_t i=1; i<segtree_n*2; i++) if(divtree[i].size() > 1){
        if(resolved[i/2]){ resolved[i] = true; continue; }
        if(i != 1) remtree[i] = PolynomialDivision(remtree[i/2], divtree[i]).second;
        if(remtree[i].size() <= 32){
            size_t l = i, r = i+1; while(l < segtree_n){ l *= 2; r *= 2; }
            l -= segtree_n; r -= segtree_n;
            r = std::min(r, Id.size());
            auto& F = remtree[i];
            for(size_t j=l; j<r; j++) if(Id[j] >= 0){
                int qi = Id[j];
                Elem c = 0;
                for(int k=F.size()-1; k>=0; k--) c = c * x[qi] + F[k];
                res[qi] = c;
            }
            resolved[i] = true;
        }
    }
    return res;
}

} // namespace nachia
#line 9 "Main.cpp"
using namespace std;
using i32 = int;
using u32 = unsigned int;
using i64 = long long;
using u64 = unsigned long long;
#define rep(i,n) for(int i=0; i<(int)(n); i++)
const i64 INF = 1001001001001001001;

using Modint = atcoder::static_modint<998244353>;

std::vector<Modint> SumOfPower(std::vector<Modint> F, int maxIdx){
    using Fps = nachia::FpsNtt<Modint>;
    std::vector<Fps> polys(F.size());
    for(std::size_t i=0; i<F.size(); i++){
        polys[i] = Fps(2).set(0,1).set(1,-F[i]).move();
    }
    Fps a = nachia::ProductOfManyPolynomials<Fps>(polys).log(maxIdx+1);
    for(int i=0; i<=maxIdx; i++) a[i] *= -Modint::raw(i);
    a[0] = F.size();
    return a.getVectorMoved();
}

int main(){
    using Fps = nachia::FpsNtt<Modint>;
    int N, M, K, P, Q; cin >> N >> M >> K >> P >> Q;
    vector<int> A(N); rep(i,N) cin >> A[i];
    vector<Modint> B(M); rep(i,M){ int a; cin >> a; B[i] = a; }
    vector<Modint> C(K); rep(i,K){ int a; cin >> a; C[i] = a; }
    vector<int> U(Q); rep(i,Q) cin >> U[i];
    
    B = SumOfPower(std::move(B), P);
    C = SumOfPower(std::move(C), P);
    auto comb = Fps::GetComb();
    comb.extend(P);
    vector<Modint> X(P+1);
    rep(t,P+1) X[t] = comb(P,t) * B[t] * C[P-t];

    int maxU = *max_element(U.begin(), U.end());
    vector<Modint> W(maxU+1);
    for(int a : A) W[min(maxU,a)] += Modint::raw(1);
    for(int i=maxU-1; i>=0; i--) W[i] += W[i+1];
    rep(y,maxU+1) W[y] *= Modint::raw(y).pow(P);

    vector<Modint> mepos(maxU+1);
    rep(i,maxU+1) mepos[i] = Modint::raw(i);
    auto sumPsX = nachia::MultipointEvaluation(Fps(X), mepos);

    vector<Modint> res(maxU+1);
    for(int y=1; y<=maxU; y++) for(int d=1; d*y<=maxU; d++){
        int p = d*y;
        res[p] += sumPsX[d] * W[y];
    }
    rep(i,maxU) res[i+1] += res[i];

    for(int a : U) cout << res[a].val() << '\n';
    return 0;
}



struct ios_do_not_sync{
    ios_do_not_sync(){
        ios::sync_with_stdio(false);
        cin.tie(nullptr);
    }
} ios_do_not_sync_instance;
yukicoder

結果

テストケース

ソースコード
