HackerRank World CodeSprint 12: Factorial Array
$10^9+7$と誤読すると解けない。気を付けよう。 ついでに$1$減らすクエリがあっても難しくなる (無理矢理やれば解けそうだが)。
problem
数列$A$が与えられる。次のようなクエリがたくさん与えられるので処理せよ。
- 区間$[l, r]$が与えられる。$i \in [l, r]$のそれぞれに対し$A_i$を$1$増やせ。
- 区間$[l, r]$が与えられる。$i \in [l, r]$のそれぞれに対し階乗$A_i!$を考え、その総和を$\bmod 10^9$で答えよ。
- 添字$i$と値$v$が与えられる。$A_i$に$v$を代入せよ。
solution
法が$10^9$なのですぐに$0$になる。 $40! \equiv 0 \bmod 10^9$なので区間中の$40$以下の数を数えておくようなsegment木を書く。 $O((N + Q) \log N)$。
implementation
#include <bits/stdc++.h>
#define REP(i, n) for (int i = 0; (i) < int(n); ++ (i))
#define REP_R(i, n) for (int i = (n) - 1; (i) >= 0; -- (i))
using ll = long long;
using namespace std;
template <class Monoid, class OperatorMonoid>
struct lazy_propagation_segment_tree { // on monoids
static_assert (is_same<typename Monoid::underlying_type, typename OperatorMonoid::target_type>::value, "");
typedef typename Monoid::underlying_type underlying_type;
typedef typename OperatorMonoid::underlying_type operator_type;
Monoid mon;
OperatorMonoid op;
int n;
vector<underlying_type> a;
vector<operator_type> f;
lazy_propagation_segment_tree() = default;
lazy_propagation_segment_tree(int a_n, underlying_type initial_value = Monoid().unit(), Monoid const & a_mon = Monoid(), OperatorMonoid const & a_op = OperatorMonoid())
: mon(a_mon), op(a_op) {
n = 1; while (n <= a_n) n *= 2;
a.resize(2 * n - 1, mon.unit());
fill(a.begin() + (n - 1), a.begin() + ((n - 1) + a_n), initial_value); // set initial values
REP_R (i, n - 1) a[i] = mon.append(a[2 * i + 1], a[2 * i + 2]); // propagate initial values
f.resize(max(0, (2 * n - 1) - n), op.identity());
}
void point_set(int i, underlying_type z) {
assert (0 <= i and i < n);
point_set(0, 0, n, i, z);
}
void point_set(int i, int il, int ir, int j, underlying_type z) {
if (i == n + j - 1) { // 0-based
a[i] = z;
} else if (ir <= j or j+1 <= il) {
// nop
} else {
range_apply(2 * i + 1, il, (il + ir) / 2, 0, n, f[i]);
range_apply(2 * i + 2, (il + ir) / 2, ir, 0, n, f[i]);
f[i] = op.identity();
point_set(2 * i + 1, il, (il + ir) / 2, j, z);
point_set(2 * i + 2, (il + ir) / 2, ir, j, z);
a[i] = mon.append(a[2 * i + 1], a[2 * i + 2]);
}
}
void range_apply(int l, int r, operator_type z) {
assert (0 <= l and l <= r and r <= n);
range_apply(0, 0, n, l, r, z);
}
void range_apply(int i, int il, int ir, int l, int r, operator_type z) {
if (l <= il and ir <= r) { // 0-based
a[i] = op.apply(z, a[i]);
if (i < f.size()) f[i] = op.compose(z, f[i]);
} else if (ir <= l or r <= il) {
// nop
} else {
range_apply(2 * i + 1, il, (il + ir) / 2, 0, n, f[i]);
range_apply(2 * i + 2, (il + ir) / 2, ir, 0, n, f[i]);
f[i] = op.identity();
range_apply(2 * i + 1, il, (il + ir) / 2, l, r, z);
range_apply(2 * i + 2, (il + ir) / 2, ir, l, r, z);
a[i] = mon.append(a[2 * i + 1], a[2 * i + 2]);
}
}
underlying_type range_concat(int l, int r) {
assert (0 <= l and l <= r and r <= n);
return range_concat(0, 0, n, l, r);
}
underlying_type range_concat(int i, int il, int ir, int l, int r) {
if (l <= il and ir <= r) { // 0-based
return a[i];
} else if (ir <= l or r <= il) {
return mon.unit();
} else {
return op.apply(f[i], mon.append(
range_concat(2 * i + 1, il, (il + ir) / 2, l, r),
range_concat(2 * i + 2, (il + ir) / 2, ir, l, r)));
}
}
};
template <int N>
struct count_monoid {
typedef array<int, N> underlying_type;
underlying_type unit() const { return underlying_type(); }
underlying_type append(underlying_type a, underlying_type b) const {
underlying_type c = {};
REP (i, N) c[i] = a[i] + b[i];
return c;
}
};
template <int N>
struct increment_operator_monoid {
typedef int underlying_type;
typedef array<int, N> target_type;
underlying_type identity() const { return 0; }
target_type apply(underlying_type a, target_type b) const {
if (a == 0) return b;
target_type c = {};
REP (i, N - a) c[i + a] = b[i];
return c;
}
underlying_type compose(underlying_type a, underlying_type b) const { return a + b; }
};
constexpr int mod = 1e9; // not 1e9+7
constexpr int width = 41;
int main() {
// prepare
int fact[width] = {};
fact[0] = 1;
REP (i, width - 1) {
fact[i + 1] = fact[i] *(ll) (i + 1) % mod;
}
// input initial values
int n, m; scanf("%d%d", &n, &m);
lazy_propagation_segment_tree<count_monoid<width>, increment_operator_monoid<width> > a(n);
auto point_set = [&](int i, int a_i) {
auto cnt = a.mon.unit();
if (a_i < width) {
cnt[a_i] = 1;
}
a.point_set(i, cnt);
};
REP (i, n) {
int a_i; scanf("%d", &a_i);
point_set(i, a_i);
}
// operate
while (m --) {
int type; scanf("%d", &type);
if (type == 1) {
int l, r; scanf("%d%d", &l, &r); -- l;
a.range_apply(l, r, 1);
} else if (type == 2) {
int l, r; scanf("%d%d", &l, &r); -- l;
auto cnt = a.range_concat(l, r);
ll acc = 0;
REP (i, width) {
acc += fact[i] *(ll) cnt[i] % mod;
}
printf("%lld\n", acc % mod);
} else if (type == 3) {
int i, a_i; scanf("%d%d", &i, &a_i); -- i;
point_set(i, a_i);
}
}
return 0;
}