AtCoder Regular Contest 085: F - NRE
solution
DP。単純には$l_j$の小さい順に$i$番目の操作まで使って位置$l_i$以降は位置$j$まで全て$1$に書き変わっているときの位置$j$までの範囲のHamming距離の最小値を$\mathrm{dp}(i, j)$とする。 このままでは$O(NQ)$で間に合わないので、数列$a$中における位置$j$までの$0$の数を$z(j)$としてこれを引いた$\mathrm{dp}(i, j) - z(j)$を区間加算と区間最小値の遅延伝播segment木を用いて管理し上手くやる。 つまり実家。$O(Q \log N)$。
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 = int(n) - 1; (i) >= 0; -- (i))
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;
const Monoid mon;
const 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)));
}
}
};
constexpr int inf = 1e9 + 7;
struct min_monoid {
typedef int underlying_type;
int unit() const { return inf; }
int append(int a, int b) const { return min(a, b); }
};
struct plus_operator_monoid {
typedef int underlying_type;
typedef int target_type;
int identity() const { return 0; }
int apply(underlying_type a, target_type b) const { return min(inf, a + b); }
int compose(underlying_type a, underlying_type b) const { return a + b; }
};
int main() {
// input
int n; scanf("%d", &n);
vector<bool> b(n);
REP (i, n) {
int b_i; scanf("%d", &b_i);
b[i] = b_i;
}
int q; scanf("%d", &q);
vector<int> l(q), r(q);
REP (i, q) {
scanf("%d%d", &l[i], &r[i]);
-- l[i];
}
// solve
vector<int> zero(n + 1);
REP (i, n) {
zero[i + 1] = zero[i] + not b[i];
}
vector<vector<int> > from_l(n);
REP (i, q) {
from_l[l[i]].push_back(i);
}
lazy_propagation_segment_tree<min_monoid, plus_operator_monoid> segtree(n + 1);
segtree.point_set(0, 0);
REP (l_i, n) {
for (int i : from_l[l_i]) {
int value = segtree.range_concat(l_i, r[i] + 1) + zero[r[i]];
segtree.point_set(r[i], value - zero[r[i]]);
}
int x = segtree.range_concat(l_i, l_i + 1) + zero[l_i ] + b[l_i];
int y = segtree.range_concat(l_i + 1, l_i + 2) + zero[l_i + 1];
segtree.point_set(l_i + 1, min(x, y) - zero[l_i + 1]);
}
// output
int result = segtree.range_concat(n, n + 1) + zero[n];
printf("%d\n", result);
return 0;
}