結果
| 問題 |
No.1241 Eternal Tours
|
| コンテスト | |
| ユーザー |
 hitonanode
|
| 提出日時 | 2020-09-06 12:41:46 |
| 言語 | C++17 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
WA
|
| 実行時間 | - |
| コード長 | 15,043 bytes |
| コンパイル時間 | 2,068 ms |
| コンパイル使用メモリ | 129,500 KB |
| 最終ジャッジ日時 | 2025-01-14 07:41:00 |
|
ジャッジサーバーID （参考情報） |
judge3 / judge3 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 4 |
| other | AC * 23 WA * 17 |
ソースコード
#include <cassert>
#include <algorithm>
#include <array>
#include <chrono>
#include <iostream>
#include <set>
#include <tuple>
#include <vector>
using namespace std;
// Berlekamp-Massey 解法（与える項の数が足りず想定WA）
template <int mod>
struct ModInt
{
using lint = long long;
static int get_mod() { return mod; }
static int get_primitive_root() {
static int primitive_root = 0;
if (!primitive_root) {
primitive_root = [&](){
std::set<int> fac;
int v = mod - 1;
for (lint i = 2; i * i <= v; i++) while (v % i == 0) fac.insert(i), v /= i;
if (v > 1) fac.insert(v);
for (int g = 1; g < mod; g++) {
bool ok = true;
for (auto i : fac) if (ModInt(g).power((mod - 1) / i) == 1) { ok = false; break; }
if (ok) return g;
}
return -1;
}();
}
return primitive_root;
}
int val;
constexpr ModInt() : val(0) {}
constexpr ModInt &_setval(lint v) { val = (v >= mod ? v - mod : v); return *this; }
constexpr ModInt(lint v) { _setval(v % mod + mod); }
explicit operator bool() const { return val != 0; }
constexpr ModInt operator+(const ModInt &x) const { return ModInt()._setval((lint)val + x.val); }
constexpr ModInt operator-(const ModInt &x) const { return ModInt()._setval((lint)val - x.val + mod); }
constexpr ModInt operator*(const ModInt &x) const { return ModInt()._setval((lint)val * x.val % mod); }
constexpr ModInt operator/(const ModInt &x) const { return ModInt()._setval((lint)val * x.inv() % mod); }
constexpr ModInt operator-() const { return ModInt()._setval(mod - val); }
constexpr ModInt &operator+=(const ModInt &x) { return *this = *this + x; }
constexpr ModInt &operator-=(const ModInt &x) { return *this = *this - x; }
constexpr ModInt &operator*=(const ModInt &x) { return *this = *this * x; }
constexpr ModInt &operator/=(const ModInt &x) { return *this = *this / x; }
friend constexpr ModInt operator+(lint a, const ModInt &x) { return ModInt()._setval(a % mod + x.val); }
friend constexpr ModInt operator-(lint a, const ModInt &x) { return ModInt()._setval(a % mod - x.val + mod); }
friend constexpr ModInt operator*(lint a, const ModInt &x) { return ModInt()._setval(a % mod * x.val % mod); }
friend constexpr ModInt operator/(lint a, const ModInt &x) { return ModInt()._setval(a % mod * x.inv() % mod); }
constexpr bool operator==(const ModInt &x) const { return val == x.val; }
constexpr bool operator!=(const ModInt &x) const { return val != x.val; }
bool operator<(const ModInt &x) const { return val < x.val; } // To use std::map<ModInt, T>
friend std::istream &operator>>(std::istream &is, ModInt &x) { lint t; is >> t; x = ModInt(t); return is; }
friend std::ostream &operator<<(std::ostream &os, const ModInt &x) { os << x.val; return os; }
constexpr lint power(lint n) const {
lint ans = 1, tmp = this->val;
while (n) {
if (n & 1) ans = ans * tmp % mod;
tmp = tmp * tmp % mod;
n /= 2;
}
return ans;
}
constexpr ModInt pow(lint n) const {
return power(n);
}
constexpr lint inv() const { return this->power(mod - 2); }
constexpr ModInt operator^(lint n) const { return ModInt(this->power(n)); }
constexpr ModInt &operator^=(lint n) { return *this = *this ^ n; }
inline ModInt fac() const {
static std::vector<ModInt> facs;
int l0 = facs.size();
if (l0 > this->val) return facs[this->val];
facs.resize(this->val + 1);
for (int i = l0; i <= this->val; i++) facs[i] = (i == 0 ? ModInt(1) : facs[i - 1] * ModInt(i));
return facs[this->val];
}
ModInt doublefac() const {
lint k = (this->val + 1) / 2;
if (this->val & 1) return ModInt(k * 2).fac() / ModInt(2).power(k) / ModInt(k).fac();
else return ModInt(k).fac() * ModInt(2).power(k);
}
ModInt nCr(const ModInt &r) const {
if (this->val < r.val) return ModInt(0);
return this->fac() / ((*this - r).fac() * r.fac());
}
};
// Integer convolution for arbitrary mod
// with NTT (and Garner's algorithm) for ModInt / ModIntRuntime class.
// We skip Garner's algorithm if `skip_garner` is true or mod is in `nttprimes`.
// input: a (size: n), b (size: m)
// return: vector (size: n + m - 1)
template <typename MODINT>
std::vector<MODINT> nttconv(std::vector<MODINT> a, std::vector<MODINT> b, bool skip_garner = false);
constexpr int nttprimes[3] = {998244353, 167772161, 469762049};
// Integer FFT (Fast Fourier Transform) for ModInt class
// (Also known as Number Theoretic Transform, NTT)
// is_inverse: inverse transform
// ** Input size must be 2^n **
template <typename MODINT>
void ntt(std::vector<MODINT> &a, bool is_inverse = false)
{
int n = a.size();
if (n == 1) return;
static const int mod = MODINT::get_mod();
static const MODINT root = MODINT::get_primitive_root();
assert(__builtin_popcount(n) == 1 and (mod - 1) % n == 0);
static std::vector<MODINT> w{1}, iw{1};
for (int m = w.size(); m < n / 2; m *= 2)
{
MODINT dw = root.power((mod - 1) / (4 * m)), dwinv = 1 / dw;
w.resize(m * 2), iw.resize(m * 2);
for (int i = 0; i < m; i++) w[m + i] = w[i] * dw, iw[m + i] = iw[i] * dwinv;
}
if (!is_inverse) {
for (int m = n; m >>= 1;) {
for (int s = 0, k = 0; s < n; s += 2 * m, k++) {
for (int i = s; i < s + m; i++) {
#ifdef __clang__
a[i + m] *= w[k];
std::tie(a[i], a[i + m]) = std::make_pair(a[i] + a[i + m], a[i] - a[i + m]);
#else
MODINT x = a[i], y = a[i + m] * w[k];
a[i] = x + y, a[i + m] = x - y;
#endif
}
}
}
}
else {
for (int m = 1; m < n; m *= 2) {
for (int s = 0, k = 0; s < n; s += 2 * m, k++) {
for (int i = s; i < s + m; i++) {
#ifdef __clang__
std::tie(a[i], a[i + m]) = std::make_pair(a[i] + a[i + m], a[i] - a[i + m]);
a[i + m] *= iw[k];
#else
MODINT x = a[i], y = a[i + m];
a[i] = x + y, a[i + m] = (x - y) * iw[k];
#endif
}
}
}
int n_inv = MODINT(n).inv();
for (auto &v : a) v *= n_inv;
}
}
template <int MOD>
std::vector<ModInt<MOD>> nttconv_(const std::vector<int> &a, const std::vector<int> &b) {
int sz = a.size();
assert(a.size() == b.size() and __builtin_popcount(sz) == 1);
std::vector<ModInt<MOD>> ap(sz), bp(sz);
for (int i = 0; i < sz; i++) ap[i] = a[i], bp[i] = b[i];
if (a == b) {
ntt(ap, false);
bp = ap;
}
else {
ntt(ap, false);
ntt(bp, false);
}
for (int i = 0; i < sz; i++) ap[i] *= bp[i];
ntt(ap, true);
return ap;
}
long long extgcd_ntt_(long long a, long long b, long long &x, long long &y)
{
long long d = a;
if (b != 0) d = extgcd_ntt_(b, a % b, y, x), y -= (a / b) * x;
else x = 1, y = 0;
return d;
}
long long modinv_ntt_(long long a, long long m)
{
long long x, y;
extgcd_ntt_(a, m, x, y);
return (m + x % m) % m;
}
long long garner_ntt_(int r0, int r1, int r2, int mod)
{
using mint2 = ModInt<nttprimes[2]>;
static const long long m01 = 1LL * nttprimes[0] * nttprimes[1];
static const long long m0_inv_m1 = ModInt<nttprimes[1]>(nttprimes[0]).inv();
static const long long m01_inv_m2 = mint2(m01).inv();
int v1 = (m0_inv_m1 * (r1 + nttprimes[1] - r0)) % nttprimes[1];
auto v2 = (mint2(r2) - r0 - mint2(nttprimes[0]) * v1) * m01_inv_m2;
return (r0 + 1LL * nttprimes[0] * v1 + m01 % mod * v2.val) % mod;
}
template <typename MODINT>
std::vector<MODINT> nttconv(std::vector<MODINT> a, std::vector<MODINT> b, bool skip_garner)
{
int sz = 1, n = a.size(), m = b.size();
while (sz < n + m) sz <<= 1;
if (sz <= 16) {
std::vector<MODINT> ret(n + m - 1);
for (int i = 0; i < n; i++) {
for (int j = 0; j < m; j++) ret[i + j] += a[i] * b[j];
}
return ret;
}
int mod = MODINT::get_mod();
if (skip_garner or std::find(std::begin(nttprimes), std::end(nttprimes), mod) != std::end(nttprimes))
{
a.resize(sz), b.resize(sz);
if (a == b) { ntt(a, false); b = a; }
else ntt(a, false), ntt(b, false);
for (int i = 0; i < sz; i++) a[i] *= b[i];
ntt(a, true);
a.resize(n + m - 1);
}
else {
std::vector<int> ai(sz), bi(sz);
for (int i = 0; i < n; i++) ai[i] = a[i].val;
for (int i = 0; i < m; i++) bi[i] = b[i].val;
auto ntt0 = nttconv_<nttprimes[0]>(ai, bi);
auto ntt1 = nttconv_<nttprimes[1]>(ai, bi);
auto ntt2 = nttconv_<nttprimes[2]>(ai, bi);
a.resize(n + m - 1);
for (int i = 0; i < n + m - 1; i++) {
a[i] = garner_ntt_(ntt0[i].val, ntt1[i].val, ntt2[i].val, mod);
}
}
return a;
}
constexpr int md = 998244353;
using mint = ModInt<md>;
// Berlekamp–Massey algorithm
// <https://en.wikipedia.org/wiki/Berlekamp%E2%80%93Massey_algorithm>
// Complexity: O(N^2)
// input: S = sequence from field K
// return: L = degree of minimal polynomial,
// C_reversed = monic min. polynomial (size = L + 1, reversed order, C_reversed[0] = 1))
// Formula: convolve(S, C_reversed)[i] = 0 for i >= L
// Example:
// - [1, 2, 4, 8, 16] -> (1, [1, -2])
// - [1, 1, 2, 3, 5, 8] -> (2, [1, -1, -1])
// - [0, 0, 0, 0, 1] -> (5, [1, 0, 0, 0, 0, 998244352]) (mod 998244353)
// - [] -> (0, [1])
// - [0, 0, 0] -> (0, [1])
// - [-2] -> (1, [1, 2])
template <typename Tfield>
std::pair<int, std::vector<Tfield>> linear_recurrence(const std::vector<Tfield> &S)
{
int N = S.size();
using poly = std::vector<Tfield>;
poly C_reversed{1}, B{1};
int L = 0, m = 1;
Tfield b = 1;
// adjust: C(x) <- C(x) - (d / b) x^m B(x)
auto adjust = [](poly C, const poly &B, Tfield d, Tfield b, int m) -> poly {
C.resize(std::max(C.size(), B.size() + m));
Tfield a = d / b;
for (unsigned i = 0; i < B.size(); i++) C[i + m] -= a * B[i];
return C;
};
for (int n = 0; n < N; n++) {
Tfield d = S[n];
for (int i = 1; i <= L; i++) d += C_reversed[i] * S[n - i];
if (d == 0) m++;
else if (2 * L <= n) {
poly T = C_reversed;
C_reversed = adjust(C_reversed, B, d, b, m);
L = n + 1 - L;
B = T;
b = d;
m = 1;
}
else C_reversed = adjust(C_reversed, B, d, b, m++);
}
return std::make_pair(L, C_reversed);
}
// Calculate x^N mod f(x)
// Known as `Kitamasa method` (Fast version based on FFT)
// Input: f_reversed: monic, reversed (f_reversed[0] = 1)
// Complexity: O(K lg K lgN) (K: deg. of f)
// Example: (4, [1, -1, -1]) -> [2, 3]
// ( x^4 = (x^2 + x + 2)(x^2 - x - 1) + 3x + 2 )
// Reference: <http://misawa.github.io/others/fast_kitamasa_method.html>
// <http://sugarknri.hatenablog.com/entry/2017/11/18/233936>
template <typename Tfield>
std::vector<Tfield> monomial_mod_polynomial(long long N, const std::vector<Tfield> &f_reversed)
{
assert(!f_reversed.empty() and f_reversed[0] == 1);
int K = f_reversed.size() - 1;
if (!K) return {};
int D = 64 - __builtin_clzll(N);
std::vector<Tfield> ret(K, 0);
ret[0] = 1;
auto self_conv = [](std::vector<Tfield> x) -> std::vector<Tfield> {
return nttconv(x, x);
};
for (int d = D; d--;)
{
ret = self_conv(ret);
for (int i = 2 * K - 2; i >= K; i--)
{
for (int j = 1; j <= K; j++) ret[i - j] -= ret[i] * f_reversed[j];
}
ret.resize(K);
if ((N >> d) & 1)
{
std::vector<Tfield> c(K);
c[0] = -ret[K - 1] * f_reversed[K];
for (int i = 1; i < K; i++)
{
c[i] = ret[i - 1] - ret[K - 1] * f_reversed[K - i];
}
ret = c;
}
}
return ret;
}
template <typename T>
ostream &operator<<(ostream &os, const vector<T> &vec)
{
os << '[';
for (auto v : vec)
os << v << ',';
os << ']';
return os;
}
#define dbg(x) cerr << #x << " = " << (x) << " (L" << __LINE__ << ") " << __FILE__ << endl
int main()
{
int X, Y;
long long T;
long long a, b, c, d;
cin >> X >> Y >> T >> a >> b >> c >> d;
auto START = std::chrono::system_clock::now();
T = (T - 1) % (md - 1) + 1;
long long dist = abs(a - c) + abs(b - d);
if (dist > T)
{
puts("0");
return 0;
}
mint primitive_root = 3;
mint rx = primitive_root.pow((md - 1) / (1 << (X + 1))), rxi = rx.inv();
mint ry = primitive_root.pow((md - 1) / (1 << (Y + 1))), ryi = ry.inv();
mint rxa = rx.pow(a), rxai = rxa.inv();
mint ryb = ry.pow(b), rybi = ryb.inv();
mint rxc = rx.pow(c), rxci = rxc.inv();
mint ryd = ry.pow(d), rydi = ryd.inv();
mint rxpow = 1, rxpowi = 1, rypow, rypowi;
mint rxapow = 1, rxapowi = 1, rybpow, rybpowi;
mint rxcpow = 1, rxcpowi = 1, rydpow, rydpowi;
vector<mint> coeffs;
vector<mint> fkls;
for (int k = 0; k < 1 << X; k++)
{
rypow = 1, rypowi = 1;
rybpow = 1, rybpowi = 1;
rydpow = 1, rydpowi = 1;
for (int l = 0; l < 1 << Y; l++)
{
fkls.emplace_back(rxpow + rxpowi + rypow + rypowi + 1);
coeffs.emplace_back((rxapow - rxapowi) * (rybpow - rybpowi) * (rxcpow - rxcpowi) * (rydpow - rydpowi) * mint(1 << (X + Y + 2)).inv());
rypow *= ry, rypowi *= ryi;
rybpow *= ryb, rybpowi *= rybi;
rydpow *= ryd, rydpowi *= rydi;
}
rxpow *= rx, rxpowi *= rxi;
rxapow *= rxa, rxapowi *= rxai;
rxcpow *= rxc, rxcpowi *= rxci;
}
vector<mint> fklpow;
dbg(dist);
dbg(T);
for (auto x : fkls) fklpow.emplace_back(x.pow(dist));
vector<mint> seq;
for (long long t = dist; t - dist <= 10000; t++)
{
if (chrono::duration_cast<std::chrono::milliseconds>(std::chrono::system_clock::now() - START).count() > 2000) break;
mint tmp = 0;
for (int kl = 0; kl < 1 << (X + Y); kl++) {
tmp += coeffs[kl] * fklpow[kl];
fklpow[kl] *= fkls[kl];
}
if (t == T)
{
cout << tmp << '\n';
return 0;
}
seq.emplace_back(tmp);
}
dbg(seq.size());
auto [L, poly_reversed] = linear_recurrence(seq);
dbg(L);
auto g = monomial_mod_polynomial(T - dist, poly_reversed);
mint ret = 0;
for (int i = 0; i < int(g.size()); i++) ret += seq.at(i) * g.at(i);
cout << ret << '\n';
}