program_contest_library
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test/yuki42_1.test.cpp
- category: test
-
View this file on GitHub
- Last commit date: 2020-04-08 17:47:46+09:00
- see: https://yukicoder.me/problems/no/42
Depends on
Code
#define PROBLEM "https://yukicoder.me/problems/no/42"
#include "../memo/macro.hpp"
#include "../math/modint.cpp"
using mint = modint<1000000009>;
#include "../math/lagrange_interpolation.cpp"
signed main(void) {
vector<ll> c({1, 5, 10, 50, 100, 500});
vector<mint> dp(3010, 1);
FOR(i, 1, 6) FOR(j, c[i], 3010) dp[j] += dp[j-c[i]];
ll t;
cin >> t;
while(t--) {
ll M;
cin >> M;
if(M < 3010) {
cout << dp[M] << endl;
continue;
}
mint m = M/500;
ll q = M % 500;
vector<mint> x(6), y(6);
REP(i, 6) {
x[i] = i;
y[i] = dp[i*500+q];
}
cout << lagrange_interpolation(x, y, m) << endl;
}
return 0;
}
#line 1 "test/yuki42_1.test.cpp"
#define PROBLEM "https://yukicoder.me/problems/no/42"
#line 1 "memo/macro.hpp"
#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using PII = pair<ll, ll>;
#define FOR(i, a, n) for (ll i = (ll)a; i < (ll)n; ++i)
#define REP(i, n) FOR(i, 0, n)
#define ALL(x) x.begin(), x.end()
template<typename T> void chmin(T &a, const T &b) { a = min(a, b); }
template<typename T> void chmax(T &a, const T &b) { a = max(a, b); }
struct FastIO {FastIO() { cin.tie(0); ios::sync_with_stdio(0); }}fastiofastio;
const ll INF = 1LL<<60;
#line 1 "math/modint.cpp"
// BEGIN CUT
template<ll MOD>
struct modint {
ll x;
modint(): x(0) {}
modint(ll y) : x(y>=0 ? y%MOD : y%MOD+MOD) {}
static constexpr ll mod() { return MOD; }
// e乗
modint pow(ll e) {
ll a = 1, p = x;
while(e > 0) {
if(e%2 == 0) {p = (p*p) % MOD; e /= 2;}
else {a = (a*p) % MOD; e--;}
}
return modint(a);
}
modint inv() const {
ll a=x, b=MOD, u=1, y=1, v=0, z=0;
while(a) {
ll q = b/a;
swap(z -= q*u, u);
swap(y -= q*v, v);
swap(b -= q*a, a);
}
return z;
}
// Comparators
bool operator <(modint b) { return x < b.x; }
bool operator >(modint b) { return x > b.x; }
bool operator<=(modint b) { return x <= b.x; }
bool operator>=(modint b) { return x >= b.x; }
bool operator!=(modint b) { return x != b.x; }
bool operator==(modint b) { return x == b.x; }
// Basic Operations
modint operator+(modint r) const { return modint(*this) += r; }
modint operator-(modint r) const { return modint(*this) -= r; }
modint operator*(modint r) const { return modint(*this) *= r; }
modint operator/(modint r) const { return modint(*this) /= r; }
modint &operator+=(modint r) {
if((x += r.x) >= MOD) x -= MOD;
return *this;
}
modint &operator-=(modint r) {
if((x -= r.x) < 0) x += MOD;
return *this;
}
modint &operator*=(modint r) {
#if !defined(_WIN32) || defined(_WIN64)
x = (ll)x * r.x % MOD; return *this;
#endif
unsigned long long y = (unsigned long long)x * r.x;
unsigned xh = (unsigned) (y >> 32), xl = (unsigned) y, d, m;
asm(
"divl %4; \n\t"
: "=a" (d), "=d" (m)
: "d" (xh), "a" (xl), "r" (MOD)
);
x = m;
return *this;
}
modint &operator/=(modint r) { return *this *= r.inv(); }
// increment, decrement
modint operator++() { x++; return *this; }
modint operator++(signed) { modint t = *this; x++; return t; }
modint operator--() { x--; return *this; }
modint operator--(signed) { modint t = *this; x--; return t; }
// 平方剰余のうち一つを返す なければ-1
friend modint sqrt(modint a) {
if(a == 0) return 0;
ll q = MOD-1, s = 0;
while((q&1)==0) q>>=1, s++;
modint z=2;
while(1) {
if(z.pow((MOD-1)/2) == MOD-1) break;
z++;
}
modint c = z.pow(q), r = a.pow((q+1)/2), t = a.pow(q);
ll m = s;
while(t.x>1) {
modint tp=t;
ll k=-1;
FOR(i, 1, m) {
tp *= tp;
if(tp == 1) { k=i; break; }
}
if(k==-1) return -1;
modint cp=c;
REP(i, m-k-1) cp *= cp;
c = cp*cp, t = c*t, r = cp*r, m = k;
}
return r.x;
}
template<class T>
friend modint operator*(T l, modint r) { return modint(l) *= r; }
template<class T>
friend modint operator+(T l, modint r) { return modint(l) += r; }
template<class T>
friend modint operator-(T l, modint r) { return modint(l) -= r; }
template<class T>
friend modint operator/(T l, modint r) { return modint(l) /= r; }
template<class T>
friend bool operator==(T l, modint r) { return modint(l) == r; }
template<class T>
friend bool operator!=(T l, modint r) { return modint(l) != r; }
// Input/Output
friend ostream &operator<<(ostream& os, modint a) { return os << a.x; }
friend istream &operator>>(istream& is, modint &a) {
is >> a.x;
a.x = ((a.x%MOD)+MOD)%MOD;
return is;
}
friend string to_frac(modint v) {
static map<ll, PII> mp;
if(mp.empty()) {
mp[0] = mp[MOD] = {0, 1};
FOR(i, 2, 1001) FOR(j, 1, i) if(__gcd(i, j) == 1) {
mp[(modint(i) / j).x] = {i, j};
}
}
auto itr = mp.lower_bound(v.x);
if(itr != mp.begin() && v.x - prev(itr)->first < itr->first - v.x) --itr;
string ret = to_string(itr->second.first + itr->second.second * ((int)v.x - itr->first));
if(itr->second.second > 1) {
ret += '/';
ret += to_string(itr->second.second);
}
return ret;
}
};
// END CUT
#line 4 "test/yuki42_1.test.cpp"
using mint = modint<1000000009>;
#line 1 "math/lagrange_interpolation.cpp"
// BEGIN CUT
// verify: https://atcoder.jp/contests/arc033/submissions/6839946
// x座標が相異なるn+1点(x_i,y_i)を通るn次以下の多項式f(T)の値を返す
// x_i = a + i*d 0<=i<=n (等差数列)
// 0割りを起こさないようにTが小さいときに注意
// O(nlog(MOD))
mint lagrange_interpolation_arithmetic(mint a, mint d, vector<mint> y, mint T) {
const ll n = y.size() - 1;
mint ret = 0, ft = 1;
REP(i, n+1) ft *= T-(a+d*i);
// f_0(x_0)
mint f = 1;
FOR(i, 1, n+1) f *= -1*i*d;
ret += y[0] / f * ft / (T-a);
// f_i(x_i) → f_{i+1}(x_{i+1})
REP(i, n) {
f *= d*(i+1) / (d*(i-n));
ret += y[i+1] / f * ft / (T-a-d*(i+1));
}
return ret;
}
// verify: https://atcoder.jp/contests/arc033/submissions/6839930
// x座標が相異なるn+1点(x_i,y_i)を通るn次以下の多項式f(T)の値を返す
// O(n^2)
mint lagrange_interpolation(vector<mint> x, vector<mint> y, mint T) {
const ll n = x.size() - 1;
mint ret = 0;
REP(i, n+1) {
mint t = 1;
REP(j, n+1) {
if(i == j) continue;
t *= T-x[j];
t /= x[i]-x[j];
}
ret += t * y[i];
}
return ret;
}
// verify: https://atcoder.jp/contests/abc137/submissions/6839902
// x座標が相異なるn+1点(x_i,y_i)を通るn次以下の多項式f(x)を返す
// O(n^2) 定数倍がかなり重い
vector<mint> lagrange_interpolation(vector<mint> x, vector<mint> y) {
const ll n = x.size() - 1;
REP(i, n+1) {
mint t = 1;
REP(j, n+1) if(i != j) t *= x[i]-x[j];
y[i] /= t;
}
ll cur = 0, nxt = 1;
vector<vector<mint>> dp(2, vector<mint>(n+2));
dp[0][0] = -1 * x[0], dp[0][1] = 1;
FOR(i, 1, n+1) {
REP(j, n+2) {
dp[nxt][j] = dp[cur][j] * -1 * x[i];
if(j >= 1) dp[nxt][j] += dp[cur][j-1];
}
swap(nxt, cur);
}
REP(i, n+1) x[i] = x[i].inv();
vector<mint> ret(n+1); // f(x)
REP(i, n+1) {
if(y[i]==0) continue;
// 0割り対策の場合分け
if(x[i] == 0) {
REP(j, n+1) ret[j] += dp[cur][j+1] * y[i];
} else {
ret[0] -= dp[cur][0] * x[i] * y[i];
mint pre = -1 * dp[cur][0] * x[i];
FOR(j, 1, n+1) {
ret[j] -= (dp[cur][j] - pre) * x[i] * y[i];
pre = -1 * (dp[cur][j] - pre) * x[i];
}
}
}
return ret;
}
// END CUT
#line 6 "test/yuki42_1.test.cpp"
signed main(void) {
vector<ll> c({1, 5, 10, 50, 100, 500});
vector<mint> dp(3010, 1);
FOR(i, 1, 6) FOR(j, c[i], 3010) dp[j] += dp[j-c[i]];
ll t;
cin >> t;
while(t--) {
ll M;
cin >> M;
if(M < 3010) {
cout << dp[M] << endl;
continue;
}
mint m = M/500;
ll q = M % 500;
vector<mint> x(6), y(6);
REP(i, 6) {
x[i] = i;
y[i] = dp[i*500+q];
}
cout << lagrange_interpolation(x, y, m) << endl;
}
return 0;
}