QOJ.ac
QOJ
| ID | Problem | Submitter | Result | Time | Memory | Language | File size | Submit time | Judge time |
|---|---|---|---|---|---|---|---|---|---|
| #257483 | #7754. Rolling For Days | ucup-team296 | WA | 1ms | 3484kb | C++20 | 15.1kb | 2023-11-19 07:53:20 | 2023-11-19 07:53:20 |
Judging History
answer
#include <bits/stdc++.h>
/**
* Author: Niyaz Nigmatullin
*
*/
using namespace std;
string to_string(string s) {
return '"' + s + '"';
}
string to_string(const char* s) {
return to_string((string) s);
}
string to_string(bool b) {
return (b ? "true" : "false");
}
template <typename A, typename B>
string to_string(pair<A, B> p) {
return "(" + to_string(p.first) + ", " + to_string(p.second) + ")";
}
template <typename A>
string to_string(A v) {
bool first = true;
string res = "{";
for (const auto &x : v) {
if (!first) {
res += ", ";
}
first = false;
res += to_string(x);
}
res += "}";
return res;
}
void debug_out() { cerr << endl; }
template <typename Head, typename... Tail>
void debug_out(Head H, Tail... T) {
cerr << " " << to_string(H);
debug_out(T...);
}
#ifdef LOCAL
#define debug(...) cerr << "[" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__)
#else
#define debug(...) 42
#endif
template <typename T>
T inverse(T a, T m) {
assert(a != 0);
T u = 0, v = 1;
while (a != 0) {
T t = m / a;
m -= t * a; swap(a, m);
u -= t * v; swap(u, v);
}
assert(m == 1);
return u;
}
template <typename T>
class Modular {
public:
using Type = typename decay<decltype(T::value)>::type;
constexpr Modular() : value() {}
template <typename U>
Modular(const U& x) {
value = normalize(x);
}
template <typename U>
static Type normalize(const U& x) {
Type v;
if (-mod() <= x && x < mod()) v = static_cast<Type>(x);
else v = static_cast<Type>(x % mod());
if (v < 0) v += mod();
return v;
}
const Type& operator()() const { return value; }
template <typename U>
explicit operator U() const { return static_cast<U>(value); }
constexpr static Type mod() { return T::value; }
Modular& operator+=(const Modular& other) { if ((value += other.value) >= mod()) value -= mod(); return *this; }
Modular& operator-=(const Modular& other) { if ((value -= other.value) < 0) value += mod(); return *this; }
template <typename U> Modular& operator+=(const U& other) { return *this += Modular(other); }
template <typename U> Modular& operator-=(const U& other) { return *this -= Modular(other); }
Modular& operator++() { return *this += 1; }
Modular& operator--() { return *this -= 1; }
Modular operator++(int) { Modular result(*this); *this += 1; return result; }
Modular operator--(int) { Modular result(*this); *this -= 1; return result; }
Modular operator-() const { return Modular(-value); }
template <typename U = T>
typename enable_if<is_same<typename Modular<U>::Type, int>::value, Modular>::type& operator*=(const Modular& rhs) {
#ifdef _WIN32
uint64_t x = static_cast<int64_t>(value) * static_cast<int64_t>(rhs.value);
uint32_t xh = static_cast<uint32_t>(x >> 32), xl = static_cast<uint32_t>(x), d, m;
asm(
"divl %4; \n\t"
: "=a" (d), "=d" (m)
: "d" (xh), "a" (xl), "r" (mod())
);
value = m;
#else
value = normalize(static_cast<int64_t>(value) * static_cast<int64_t>(rhs.value));
#endif
return *this;
}
template <typename U = T>
typename enable_if<is_same<typename Modular<U>::Type, int64_t>::value, Modular>::type& operator*=(const Modular& rhs) {
int64_t q = static_cast<int64_t>(static_cast<long double>(value) * rhs.value / mod());
value = normalize(value * rhs.value - q * mod());
return *this;
}
template <typename U = T>
typename enable_if<!is_integral<typename Modular<U>::Type>::value, Modular>::type& operator*=(const Modular& rhs) {
value = normalize(value * rhs.value);
return *this;
}
Modular& operator/=(const Modular& other) { return *this *= Modular(inverse(other.value, mod())); }
template <typename U>
friend bool operator==(const Modular<U>& lhs, const Modular<U>& rhs);
template <typename U>
friend bool operator<(const Modular<U>& lhs, const Modular<U>& rhs);
template <typename U>
friend std::istream& operator>>(std::istream& stream, Modular<U>& number);
private:
Type value;
};
template <typename T> bool operator==(const Modular<T>& lhs, const Modular<T>& rhs) { return lhs.value == rhs.value; }
template <typename T, typename U> bool operator==(const Modular<T>& lhs, U rhs) { return lhs == Modular<T>(rhs); }
template <typename T, typename U> bool operator==(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) == rhs; }
template <typename T> bool operator!=(const Modular<T>& lhs, const Modular<T>& rhs) { return !(lhs == rhs); }
template <typename T, typename U> bool operator!=(const Modular<T>& lhs, U rhs) { return !(lhs == rhs); }
template <typename T, typename U> bool operator!=(U lhs, const Modular<T>& rhs) { return !(lhs == rhs); }
template <typename T> bool operator<(const Modular<T>& lhs, const Modular<T>& rhs) { return lhs.value < rhs.value; }
template <typename T> Modular<T> operator+(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) += rhs; }
template <typename T, typename U> Modular<T> operator+(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) += rhs; }
template <typename T, typename U> Modular<T> operator+(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) += rhs; }
template <typename T> Modular<T> operator-(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) -= rhs; }
template <typename T, typename U> Modular<T> operator-(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) -= rhs; }
template <typename T, typename U> Modular<T> operator-(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) -= rhs; }
template <typename T> Modular<T> operator*(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) *= rhs; }
template <typename T, typename U> Modular<T> operator*(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) *= rhs; }
template <typename T, typename U> Modular<T> operator*(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) *= rhs; }
template <typename T> Modular<T> operator/(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) /= rhs; }
template <typename T, typename U> Modular<T> operator/(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) /= rhs; }
template <typename T, typename U> Modular<T> operator/(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) /= rhs; }
template<typename T, typename U>
Modular<T> power(const Modular<T>& a, const U& b) {
assert(b >= 0);
Modular<T> x = a, res = 1;
U p = b;
while (p > 0) {
if (p & 1) res *= x;
x *= x;
p >>= 1;
}
return res;
}
template <typename T>
string to_string(const Modular<T>& number) {
for (int i = 1; i < 100; i++) {
if ((int) (number * i) < 1000) return to_string((int) (number * i)) + "/" + to_string(i);
}
return to_string(number());
}
template <typename T>
std::ostream& operator<<(std::ostream& stream, const Modular<T>& number) {
return stream << number();
}
template <typename T>
std::istream& operator>>(std::istream& stream, Modular<T>& number) {
typename common_type<typename Modular<T>::Type, int64_t>::type x;
stream >> x;
number.value = Modular<T>::normalize(x);
return stream;
}
/*
using ModType = int;
struct VarMod { static ModType value; };
ModType VarMod::value;
ModType& md = VarMod::value;
using Mint = Modular<VarMod>;
*/
constexpr int md = 998244353;
using Mint = Modular<std::integral_constant<decay<decltype(md)>::type, md>>;
Mint solve(int n, vector<int> const &a, vector<int> const &b) {
int m = (int) a.size();
vector<vector<Mint>> f(1 << m, vector<Mint>(n + 1));
f[0][0] = 1;
vector<vector<Mint>> ff(1 << m, vector<Mint>(n + 1));
ff[0][0] = 1;
vector<vector<Mint>> c(n + 1, vector<Mint>(n + 1));
for (int i = 0; i <= n; i++) {
c[i][0] = 1;
for (int j = 1; j <= i; j++) {
c[i][j] = c[i - 1][j - 1] + c[i - 1][j];
}
}
for (int mask = 1; mask < (1 << m); mask++) {
int any = 0;
while (((mask >> any) & 1) == 0) ++any;
// f[mask & ~(1 << any)], any
int without = mask & ~(1 << any);
for (int was = 0; was <= n; was++) {
Mint numerator = 1;
for (int get = 0; get + was <= n && get <= b[any]; get++) {
numerator *= a[any] - get;
ff[mask][get + was] += ff[without][was] * c[get + was][get] * numerator;
}
Mint value = f[without][was];
if (value == 0) continue;
numerator = 1;
for (int get = 0; get + was <= n && get < b[any]; get++) {
numerator *= a[any] - get;
f[mask][get + was] += value * c[get + was][get] * numerator;
}
}
}
for (int mask = 0; mask < 1 << m; mask++) {
for (int i = 0; i < m; i++) {
if (((mask >> i) & 1) == 0) {
Mint numerator = 1;
for (int j = 0; j < b[i]; j++) {
numerator *= a[i] - j;
}
for (int was = n - b[i]; was >= 0; was--) {
f[mask][was + b[i]] += f[mask][was] * c[was + b[i]][b[i]] * numerator;
}
}
}
}
int targetSum = 0;
for (int x: b) targetSum += x;
vector<int> sumB(1 << m);
vector<int> sumA(1 << m);
for (int mask = 0; mask < 1 << m; mask++) {
for (int i = 0; i < m; i++) {
if ((mask >> i) & 1) {
sumB[mask] += b[i];
sumA[mask] += a[i];
}
}
}
// vector<vector<Mint>> g(1 << m, vector<Mint>(targetSum + 1));
// g[(1 << m) - 1][targetSum] = 0;
// for (int mask = (1 << m) - 2; mask >= 0; mask--) {
// for (int k = targetSum - 1; k >= sumB[mask]; k--) {
// Mint sumAddBit = 0;
// int left = k - sumB[mask];
// Mint good = Mint(n) - Mint(sumA[mask]) - Mint(left);
// Mint bad = sumA[mask] - sumB[mask];
// // debug(good);
// Mint expected = (good + bad) / good;
// Mint allWays = f[((1 << m) - 1) ^ mask][left];
// if (allWays == 0) continue;
// for (int i = 0; i < m; i++) {
// if (((mask >> i) & 1) == 1) continue;
// if (left < b[i] - 1) continue;
// Mint goodWays = f[((1 << m) - 1) ^ mask ^ (1 << i)][left - (b[i] - 1)] * c[left][b[i] - 1];
// // debug(good);
// Mint prob = goodWays / allWays * (a[i] - (b[i] - 1)) / good;
// sumAddBit += prob;
// g[mask][k] += (g[mask | (1 << i)][k + 1] + expected) * prob;
// }
// Mint probStay = 1 - sumAddBit;
// g[mask][k] += (g[mask][k + 1] + expected) * probStay;
// debug(mask, k, g[mask][k]);
// }
// }
Mint answer = 0;
for (int mask = (1 << m) - 2; mask >= 0; mask--) {
for (int k = sumB[mask]; k < targetSum; k++) {
int left = k - sumB[mask];
int good = n - sumA[mask] - left;
Mint denominator = 1;
for (int i = 0; i < k; i++) {
denominator *= good - i;
}
int bad = sumA[mask] - sumB[mask];
// debug(good);
Mint expected = Mint(good + bad) / good;
Mint goodWays = f[((1 << m) - 1) ^ mask][left];
for (int i = 0, count = left; i < m; i++) {
if (((mask >> i) & 1) == 1) {
goodWays *= c[count + b[i]][b[i]];
count += b[i];
}
}
Mint allWays = ff[((1 << m) - 1)][k];
debug(mask, k, goodWays, allWays, expected);
answer += goodWays / allWays * expected;
}
}
// return g[0][0];
return answer;
}
Mint solve2(int n, vector<int> const &a, vector<int> const &b) {
int m = (int) a.size();
vector<vector<Mint>> f(1 << m, vector<Mint>(n + 1));
f[0][0] = 1;
vector<vector<Mint>> c(n + 1, vector<Mint>(n + 1));
for (int i = 0; i <= n; i++) {
c[i][0] = 1;
for (int j = 1; j <= i; j++) {
c[i][j] = c[i - 1][j - 1] + c[i - 1][j];
}
}
for (int mask = 1; mask < (1 << m); mask++) {
int any = 0;
while (((mask >> any) & 1) == 0) ++any;
// f[mask & ~(1 << any)], any
int without = mask & ~(1 << any);
for (int was = 0; was <= n; was++) {
Mint value = f[without][was];
if (value == 0) continue;
Mint numerator = 1;
for (int get = 0; get + was <= n && get < b[any]; get++) {
f[mask][get + was] += value * numerator * c[get + was][get];
numerator *= a[any] - get;
}
}
}
int targetSum = 0;
for (int x: b) targetSum += x;
vector<int> sumB(1 << m);
vector<int> sumA(1 << m);
for (int mask = 0; mask < 1 << m; mask++) {
for (int i = 0; i < m; i++) {
if ((mask >> i) & 1) {
sumB[mask] += b[i];
sumA[mask] += a[i];
}
}
}
vector<vector<Mint>> fDen(1 << m, vector<Mint>(targetSum + 1));
vector<Mint> fullNumerator(m);
for (int i = 0; i < m; i++) {
fullNumerator[i] = 1;
for (int j = 0; j < b[i]; j++) {
fullNumerator[i] *= a[i] - j;
}
}
fDen[0][0] = 1;
for (int mask = 0; mask < 1 << m; mask++) {
for (int count = sumB[mask]; count < targetSum; count++) {
int ways = (n - sumA[mask]) - (count - sumB[mask]);
if (ways == 0) continue;
Mint denominator = Mint(1) / ways;
for (int i = 0; i < m; i++) {
if (((mask >> i) & 1) == 0 && count + 1 >= sumB[mask] + b[i]) {
fDen[mask | (1 << i)][count + 1] += fDen[mask][count] * c[count - sumB[mask]][b[i] - 1] * fullNumerator[i] * denominator;
}
}
fDen[mask][count + 1] += fDen[mask][count] * denominator;
}
}
Mint answer = 0;
for (int mask = 0; mask + 1 < 1 << m; mask++) {
for (int k = sumB[mask]; k < targetSum; k++) {
int left = k - sumB[mask];
int good = n - sumA[mask] - left;
int bad = sumA[mask] - sumB[mask];
Mint expected = Mint(good + bad) / good;
Mint prob = f[((1 << m) - 1) ^ mask][left] * fDen[mask][k];
answer += prob * expected;
// debug(mask, k, prob, expected, f[((1 << m) - 1) ^ mask][left], fDen[mask][k]);
}
}
// return g[0][0];
return answer;
}
struct state {
vector<int> taken;
bool operator==(state const &s) const {
return s.taken == taken;
}
};
namespace std {
template<>
struct hash<state> {
size_t operator()(state const &s) const {
size_t h = 0;
for (int x : s.taken) {
h = h * 239017 + x + 1;
}
return h;
}
};
};
string to_string(state const &s) {
return to_string(s.taken);
}
Mint solveStupid(int n, vector<int> const &a, vector<int> const &b) {
unordered_map<state, Mint> expected;
expected[{b}] = 0;
int m = (int) a.size();
Mint ans2 = 0;
function<Mint(state const &, Mint)> calc = [&](state const &s, Mint prob) {
// if (expected.find(s) != expected.end()) {
// return expected[s];
// }
if (s.taken == b) {
return Mint(0);
}
int ways = 0;
int allWays = 0;
for (int i = 0; i < m; i++) {
if (s.taken[i] < b[i]) {
ways += a[i] - s.taken[i];
}
allWays += a[i] - s.taken[i];
}
Mint res = Mint(allWays) / Mint(ways);
ans2 += prob * Mint(allWays) / Mint(ways);
debug(s, prob, Mint(allWays) / Mint(ways));
for (int i = 0; i < m; i++) {
if (s.taken[i] < b[i]) {
state ns = s;
ns.taken[i]++;
Mint got = calc(ns, prob * Mint(a[i] - s.taken[i]) / Mint(ways));
res += got * Mint(a[i] - s.taken[i]) / ways;
}
}
return expected[s] = res;
};
Mint res = calc({vector<int>(m, 0)}, 1);
debug(expected);
debug(res, ans2);
return res;
}
void stress() {
mt19937 rng(239);
for (int it = 0; it < 10000; it++) {
int n = 5;
vector<int> a(n), b(n);
for (int i = 0; i < n; i++) {
b[i] = rng() % 2 + 1;
a[i] = b[i] + rng() % 3;
}
int sum = accumulate(a.begin(), a.end(), 0);
Mint ans1 = solve2(sum, a, b);
Mint ans2 = solveStupid(sum, a, b);
if (ans1 != ans2) {
debug(a, b);
debug(ans1, ans2);
assert(ans1 == ans2);
}
}
}
int main() {
std::cin.tie(NULL); std::ios::sync_with_stdio(false);
// stress();
int n, m;
cin >> n >> m;
vector<int> a(m);
for (int &x: a) cin >> x;
vector<int> b(m);
for (int &x: b) cin >> x;
Mint ans = solve2(n, a, b);
// Mint ans2 = solveStupid(n, a, b);
// for (int i = 1; i < 100; i++) {
// if ((int) (ans * i) < 1000) {
// cout << ans * i << '/' << i << endl;
// break;
// }
// }
debug(ans);
cout << ans << '\n';
}
Details
Tip: Click on the bar to expand more detailed information
Test #1:
score: 100
Accepted
time: 0ms
memory: 3484kb
input:
2 2 1 1 1 1
output:
2
result:
ok answer is '2'
Test #2:
score: 0
Accepted
time: 1ms
memory: 3468kb
input:
4 2 2 2 2 1
output:
582309210
result:
ok answer is '582309210'
Test #3:
score: -100
Wrong Answer
time: 1ms
memory: 3468kb
input:
5 5 1 1 1 1 1 0 0 0 0 1
output:
0
result:
wrong answer expected '5', found '0'