library
This documentation is automatically generated by online-judge-tools/verification-helper
LazySegmentTree(遅延セグメント木)
(data-struct/segment/LazySegmentTree.hpp)
- View this file on GitHub
- Last update: 2024-05-12 17:35:55+09:00
- Include:
#include "data-struct/segment/LazySegmentTree.hpp"
概要
準同型な作用付きモノイドを扱える。
分配則は成り立たないが、任意の $f \in U$ と $k \in \mathbb N$ に対して、ある $g_{f, k} \in U$ が存在して、どのような $x_1, x_2, \ldots, x_k \in T$ に対しても $g_{f, k}(x_1 \cdot x_2 \cdot \cdots \cdot x_k) = f(x_1) \cdot f(x_2) \cdot \cdots \cdot f(x_k)$ となり、かつ $f, k$ から $g_{f, k}$ を効率よく計算できるときは、 mul_op を定義することで使える。
- コンストラクタ
-
LazySegmentTree(): 長さ $0$ に初期化する。 $\Theta(N)$ 。 -
LazySegmentTree(int n): 長さnで初期化する。初期値はe。 $\Theta(N)$ 。 -
LazySegmentTree(vector<T> v): 列vで初期化する。 $\Theta(N)$ 。 -
void init(vector<T> v): 列vで初期化する。 $\Theta(N)$ 。
-
- 取得クエリ
-
T prod(int l, int r):op(a[l], a[l+1], ..., a[r-1])を返す。 $\Theta(\log N)$ 。 -
T get(int k):a[k]を返す。 $\Theta(\log N)$ 。 -
T all_prod():op(a[0], a[1], ..., a[N-1])を返す。 $\Theta(1)$ 。
-
- 更新クエリ
-
void set(int k, T x):a[k]にxを代入する。 $\Theta(\log N)$ 。 -
void apply(int k, U f):a[k]にmp(f, a[k])を代入する。 $\Theta(\log N)$ 。 -
void update(int k, T upd(T)):a[k]にupd(a[k])を代入する。 $\Theta(\log N)$ 。 -
void apply(int l, int r, U f):a[l], a[l+1], ..., a[r-1]にmp(f, a[l]), mp(f, a[l+1]), ..., mp(f, a[r-1])を代入する。 $\Theta(\log N)$ 。
-
- セグメント木上の二分探索
-
int max_right(int l, bool f(T)):
[l, r)に対してfがtrueを返すような最大のrを返す。f(e) = trueである必要がある。 $\Theta(\log N)$ 。
厳密には、以下の条件を共に満たすr(のうち1つ)を返す。-
r = lまたはf(op(a[l], a[l+1], ..., a[r-1])) = true -
r = nまたはf(op(a[l], a[l+1], ..., a[r])) = false
-
-
int min_left(int r, bool f(T)):
[l, r)に対してfがtrueを返すような最小のlを返す。f(e) = trueである必要がある。 $\Theta(\log N)$ 。
厳密には、以下の条件を共に満たすl(のうち1つ)を返す。-
l = rまたはf(op(a[l], a[l+1], ..., a[r-1])) = true -
l = 0またはf(op(a[l-1], a[l], ..., a[r-1])) = false
-
-
また、以下のクエリに対する遅延セグメント木が LazySegmentTree のエイリアスとして作られている。
-
RangeUpdateQueryRangeSumQuery: 区間代入クエリと区間和クエリを扱える。 -
RangeUpdateQueryRangeMinimumQuery: 区間代入クエリと区間最小値クエリを扱える。 -
RangeUpdateQueryRangeMaximumQuery: 区間代入クエリと区間最大値クエリを扱える。 -
RangeAddQueryRangeSumQuery: 区間加算クエリと区間和クエリを扱える。 -
RangeAddQueryRangeMinimumQuery: 区間加算クエリと区間最小値クエリを扱える。 -
RangeAddQueryRangeMaximumQuery: 区間加算クエリと区間最大値クエリを扱える。 -
RangeChminQueryRangeMinimumQuery: 区間chminクエリと区間最小値クエリを扱える。 -
RangeChminQueryRangeMaximumQuery: 区間chminクエリと区間最大値クエリを扱える。 -
RangeChmaxQueryRangeMinimumQuery: 区間chmaxクエリと区間最小値クエリを扱える。 -
RangeChmaxQueryRangeMaximumQuery: 区間chmaxクエリと区間最大値クエリを扱える。
Depends on
- other/monoid.hpp
- other/template.hpp
- template/alias.hpp
- template/bitop.hpp
- template/func.hpp
- template/in.hpp
- template/macros.hpp
- template/out.hpp
- template/type_traits.hpp
- template/util.hpp
Required by
- data-struct/2D/AreaofUnionofRectangles.hpp
- RangeAssignSegmentTree
(data-struct/segment/RangeAssignSegmentTree.hpp)
Verified with
- test/aoj/DSL/DSL_2_F-RUQRMQ.test.cpp
- test/aoj/DSL/DSL_2_G-RAQRSQ.test.cpp
- test/aoj/DSL/DSL_2_H-RAQRMQ.test.cpp
- test/aoj/DSL/DSL_2_I-RUQRSQ.test.cpp
- test/aoj/GRL/GRL_5_E-HLD.test.cpp
- test/yosupo/data_structure/area_of_union_of_rectangles.test.cpp
- test/yosupo/data_structure/range_affine_range_sum.test.cpp
- test/yosupo/data_structure/range_set_range_composite.test.cpp
Code
#pragma once
#include "../../other/template.hpp"
#include "../../other/monoid.hpp"
template<class A> class LazySegmentTree {
static_assert(Monoid::is_action<A>::value, "A must be action");
private:
using M = typename A::M;
using E = typename A::E;
using T = typename M::value_type;
using U = typename E::value_type;
int h, n, ori;
std::vector<T> data;
std::vector<U> lazy;
std::vector<bool> lazyflag;
template<bool AlwaysTrue = true,
typename std::enable_if<!Monoid::has_mul_op<A>::value &&
AlwaysTrue>::type* = nullptr>
static inline T Aop(const U& a, const T& b, int) {
return A::op(a, b);
}
template<bool AlwaysTrue = true,
typename std::enable_if<Monoid::has_mul_op<A>::value &&
AlwaysTrue>::type* = nullptr>
static inline T Aop(const U& a, const T& b, int c) {
return A::mul_op(a, c, b);
}
void all_apply(int k, const U& x, int d) {
data[k] = Aop(x, data[k], d);
if (k < n) {
if (lazyflag[k]) {
lazy[k] = E::op(lazy[k], x);
}
else {
lazy[k] = x;
lazyflag[k] = true;
}
}
}
void eval(int k, int d) {
if (lazyflag[k]) {
all_apply(k << 1, lazy[k], d >> 1);
all_apply(k << 1 ^ 1, lazy[k], d >> 1);
lazyflag[k] = false;
}
}
void calc(int k) { data[k] = M::op(data[k << 1], data[k << 1 ^ 1]); }
public:
LazySegmentTree() : LazySegmentTree(0) {}
LazySegmentTree(int n) : LazySegmentTree(std::vector<T>(n, M::id())) {}
LazySegmentTree(int n, const T& v)
: LazySegmentTree(std::vector<T>(n, v)) {}
LazySegmentTree(const std::vector<T>& v) { init(v); }
void init(const std::vector<T>& v) {
ori = v.size();
h = bitop::ceil_log2(ori);
n = 1 << h;
data.assign(n << 1, M::id());
rep (i, ori) data[n + i] = v[i];
rrep (i, n, 1) calc(i);
lazy.resize(n);
lazyflag.assign(n, false);
}
void eval_all() {
rep (i, h) {
rep (j, 1 << i) eval((1 << i) + j, 1 << (h - i));
}
}
T prod(int l, int r) {
assert(0 <= l && l <= r && r <= ori);
if (l == r) return M::id();
l += n, r += n;
rreps (i, h) {
bool seen = false;
if (((l >> i) << i) != l) eval(l >> i, 1 << i), seen = true;
if (((r >> i) << i) != r) eval((r - 1) >> i, 1 << i), seen = true;
if (!seen) break;
}
T lsm = M::id(), rsm = M::id();
while (l != r) {
if (l & 1) lsm = M::op(lsm, data[l++]);
if (r & 1) rsm = M::op(data[--r], rsm);
l >>= 1, r >>= 1;
}
return M::op(lsm, rsm);
}
T get(int k) {
assert(0 <= k && k < ori);
k += n;
rreps (i, h) eval(k >> i, 1 << i);
return data[k];
}
T all_prod() const { return data[1]; }
template<class Upd> void update(int k, const Upd& upd) {
assert(0 <= k && k < ori);
k += n;
rreps (i, h) eval(k >> i, 1 << i);
data[k] = upd(data[k]);
reps (i, h) calc(k >> i);
}
void set(int k, const T& x) {
update(k, [&](const T&) -> T { return x; });
}
void apply(int k, const U& x) {
update(k, [&](const T& a) -> T { return A::op(x, a); });
}
void apply(int l, int r, const U& x) {
assert(0 <= l && l <= r && r <= ori);
if (l == r) return;
l += n, r += n;
int lst = h + 1;
rreps (i, h) {
if (((l >> i) << i) != l) eval(l >> i, 1 << i), lst = i;
if (((r >> i) << i) != r) eval((r - 1) >> i, 1 << i), lst = i;
if (lst != i) break;
}
for (int l2 = l, r2 = r, d = 1; l2 != r2; l2 >>= 1, r2 >>= 1, d <<= 1) {
if (l2 & 1) all_apply(l2++, x, d);
if (r2 & 1) all_apply(--r2, x, d);
}
rep (i, lst, h + 1) {
if (((l >> i) << i) != l) calc(l >> i);
if (((r >> i) << i) != r) calc((r - 1) >> i);
}
}
template<class C> int max_right(int l, const C& cond) {
assert(0 <= l && l <= ori);
assert(cond(M::id()));
if (l == ori) return ori;
l += n;
rreps (i, h) {
if (((l >> i) << i) != l) eval(l >> i, 1 << i);
else break;
}
T sm = M::id();
int d = 1;
do {
while ((l & 1) == 0) l >>= 1, d <<= 1;
if (!cond(M::op(sm, data[l]))) {
while (l < n) {
eval(l, d);
l <<= 1;
d >>= 1;
if (cond(M::op(sm, data[l]))) sm = M::op(sm, data[l++]);
}
return l - n;
}
sm = M::op(sm, data[l++]);
} while ((l & -l) != l);
return ori;
}
template<class C> int min_left(int r, const C& cond) {
assert(0 <= r && r <= ori);
assert(cond(M::id()));
if (r == 0) return 0;
r += n;
rreps (i, h) {
if (((r >> i) << i) != r) eval((r - 1) >> i, 1 << i);
else break;
}
T sm = M::id();
int d = 1;
do {
--r;
while ((r & 1) && r > 1) r >>= 1, d <<= 1;
if (!cond(M::op(data[r], sm))) {
while (r < n) {
eval(r, d);
r = r << 1 ^ 1;
d >>= 1;
if (cond(M::op(data[r], sm))) sm = M::op(data[r--], sm);
}
return r + 1 - n;
}
sm = M::op(data[r], sm);
} while ((r & -r) != r);
return 0;
}
};
template<class T, int id = -1>
using RangeUpdateQueryRangeMinimumQuery =
LazySegmentTree<Monoid::AssignMin<T, id>>;
template<class T, int id = -1>
using RangeUpdateQueryRangeMaximumQuery =
LazySegmentTree<Monoid::AssignMax<T, id>>;
template<class T>
using RangeUpdateQueryRangeSumQuery = LazySegmentTree<Monoid::AssignSum<T>>;
template<class T, int id = -1>
using RangeAddQueryRangeMinimumQuery = LazySegmentTree<Monoid::AddMin<T, id>>;
template<class T, int id = -1>
using RangeAddQueryRangeMaximumQuery = LazySegmentTree<Monoid::AddMax<T, id>>;
template<class T>
using RangeAddQueryRangeSumQuery = LazySegmentTree<Monoid::AddSum<T>>;
template<class T, int id = -1>
using RangeChminQueryRangeMinimumQuery =
LazySegmentTree<Monoid::ChminMin<T, id>>;
template<class T, int id = -1>
using RangeChminQueryRangeMaximumQuery =
LazySegmentTree<Monoid::ChminMax<T, id>>;
template<class T, int id = -1>
using RangeChmaxQueryRangeMinimumQuery =
LazySegmentTree<Monoid::ChmaxMin<T, id>>;
template<class T, int id = -1>
using RangeChmaxQueryRangeMaximumQuery =
LazySegmentTree<Monoid::ChmaxMax<T, id>>;
/**
* @brief LazySegmentTree(遅延セグメント木)
* @docs docs/data-struct/segment/LazySegmentTree.md
*/#line 2 "data-struct/segment/LazySegmentTree.hpp"
#line 2 "other/template.hpp"
#include <bits/stdc++.h>
#line 2 "template/macros.hpp"
#line 4 "template/macros.hpp"
#ifndef __COUNTER__
#define __COUNTER__ __LINE__
#endif
#define OVERLOAD5(a, b, c, d, e, ...) e
#define REP1_0(b, c) REP1_1(b, c)
#define REP1_1(b, c) \
for (ll REP_COUNTER_##c = 0; REP_COUNTER_##c < (ll)(b); ++REP_COUNTER_##c)
#define REP1(b) REP1_0(b, __COUNTER__)
#define REP2(i, b) for (ll i = 0; i < (ll)(b); ++i)
#define REP3(i, a, b) for (ll i = (ll)(a); i < (ll)(b); ++i)
#define REP4(i, a, b, c) for (ll i = (ll)(a); i < (ll)(b); i += (ll)(c))
#define rep(...) OVERLOAD5(__VA_ARGS__, REP4, REP3, REP2, REP1)(__VA_ARGS__)
#define RREP2(i, a) for (ll i = (ll)(a)-1; i >= 0; --i)
#define RREP3(i, a, b) for (ll i = (ll)(a)-1; i >= (ll)(b); --i)
#define RREP4(i, a, b, c) for (ll i = (ll)(a)-1; i >= (ll)(b); i -= (ll)(c))
#define rrep(...) OVERLOAD5(__VA_ARGS__, RREP4, RREP3, RREP2)(__VA_ARGS__)
#define REPS2(i, b) for (ll i = 1; i <= (ll)(b); ++i)
#define REPS3(i, a, b) for (ll i = (ll)(a) + 1; i <= (ll)(b); ++i)
#define REPS4(i, a, b, c) for (ll i = (ll)(a) + 1; i <= (ll)(b); i += (ll)(c))
#define reps(...) OVERLOAD5(__VA_ARGS__, REPS4, REPS3, REPS2)(__VA_ARGS__)
#define RREPS2(i, a) for (ll i = (ll)(a); i > 0; --i)
#define RREPS3(i, a, b) for (ll i = (ll)(a); i > (ll)(b); --i)
#define RREPS4(i, a, b, c) for (ll i = (ll)(a); i > (ll)(b); i -= (ll)(c))
#define rreps(...) OVERLOAD5(__VA_ARGS__, RREPS4, RREPS3, RREPS2)(__VA_ARGS__)
#define each_for(...) for (auto&& __VA_ARGS__)
#define each_const(...) for (const auto& __VA_ARGS__)
#define all(v) std::begin(v), std::end(v)
#define rall(v) std::rbegin(v), std::rend(v)
#if __cpp_if_constexpr >= 201606L
#define IF_CONSTEXPR constexpr
#else
#define IF_CONSTEXPR
#endif
#define IO_BUFFER_SIZE (1 << 17)
#line 2 "template/alias.hpp"
#line 4 "template/alias.hpp"
using ll = long long;
using uint = unsigned int;
using ull = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;
using ld = long double;
using PLL = std::pair<ll, ll>;
template<class T>
using prique = std::priority_queue<T, std::vector<T>, std::greater<T>>;
template<class T> struct infinity {
static constexpr T value = std::numeric_limits<T>::max() / 2;
static constexpr T mvalue = std::numeric_limits<T>::lowest() / 2;
static constexpr T max = std::numeric_limits<T>::max();
static constexpr T min = std::numeric_limits<T>::lowest();
};
#if __cplusplus <= 201402L
template<class T> constexpr T infinity<T>::value;
template<class T> constexpr T infinity<T>::mvalue;
template<class T> constexpr T infinity<T>::max;
template<class T> constexpr T infinity<T>::min;
#endif
#if __cpp_variable_templates >= 201304L
template<class T> constexpr T INF = infinity<T>::value;
#endif
constexpr ll inf = infinity<ll>::value;
constexpr ld EPS = 1e-8;
constexpr ld PI = 3.1415926535897932384626;
#line 2 "template/type_traits.hpp"
#line 5 "template/type_traits.hpp"
template<class T, class... Args> struct function_traits_impl {
using result_type = T;
template<std::size_t idx>
using argument_type =
typename std::tuple_element<idx, std::tuple<Args...>>::type;
using argument_tuple = std::tuple<Args...>;
static constexpr std::size_t arg_size() { return sizeof...(Args); }
};
template<class> struct function_traits_helper;
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...)> {
using type = function_traits_impl<Res, Args...>;
};
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...)&> {
using type = function_traits_impl<Res, Args...>;
};
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...) const> {
using type = function_traits_impl<Res, Args...>;
};
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...) const&> {
using type = function_traits_impl<Res, Args...>;
};
#if __cpp_noexcept_function_type >= 201510L
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...) noexcept> {
using type = function_traits_impl<Res, Args...>;
};
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...)& noexcept> {
using type = function_traits_impl<Res, Args...>;
};
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...) const noexcept> {
using type = function_traits_impl<Res, Args...>;
};
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...) const& noexcept> {
using type = function_traits_impl<Res, Args...>;
};
#endif
template<class F>
using function_traits = typename function_traits_helper<
decltype(&std::remove_reference<F>::type::operator())>::type;
template<class F>
using function_result_type = typename function_traits<F>::result_type;
template<class F, std::size_t idx>
using function_argument_type =
typename function_traits<F>::template argument_type<idx>;
template<class F>
using function_argument_tuple = typename function_traits<F>::argument_tuple;
template<class T>
using is_signed_int =
std::integral_constant<bool, (std::is_integral<T>::value &&
std::is_signed<T>::value) ||
std::is_same<T, i128>::value>;
template<class T>
using is_unsigned_int =
std::integral_constant<bool, (std::is_integral<T>::value &&
std::is_unsigned<T>::value) ||
std::is_same<T, u128>::value>;
template<class T>
using is_int = std::integral_constant<bool, is_signed_int<T>::value ||
is_unsigned_int<T>::value>;
template<class T>
using make_signed_int = typename std::conditional<
std::is_same<T, i128>::value || std::is_same<T, u128>::value,
std::common_type<i128>, std::make_signed<T>>::type;
template<class T>
using make_unsigned_int = typename std::conditional<
std::is_same<T, i128>::value || std::is_same<T, u128>::value,
std::common_type<u128>, std::make_unsigned<T>>::type;
template<class T, class = void> struct is_range : std::false_type {};
template<class T>
struct is_range<
T,
decltype(all(std::declval<typename std::add_lvalue_reference<T>::type>()),
(void)0)> : std::true_type {};
template<class T, bool = is_range<T>::value>
struct range_rank : std::integral_constant<std::size_t, 0> {};
template<class T>
struct range_rank<T, true>
: std::integral_constant<std::size_t,
range_rank<typename T::value_type>::value + 1> {};
template<std::size_t size> struct int_least {
static_assert(size <= 128, "size must be less than or equal to 128");
using type = typename std::conditional<
size <= 8, std::int_least8_t,
typename std::conditional<
size <= 16, std::int_least16_t,
typename std::conditional<
size <= 32, std::int_least32_t,
typename std::conditional<size <= 64, std::int_least64_t,
i128>::type>::type>::type>::type;
};
template<std::size_t size> using int_least_t = typename int_least<size>::type;
template<std::size_t size> struct uint_least {
static_assert(size <= 128, "size must be less than or equal to 128");
using type = typename std::conditional<
size <= 8, std::uint_least8_t,
typename std::conditional<
size <= 16, std::uint_least16_t,
typename std::conditional<
size <= 32, std::uint_least32_t,
typename std::conditional<size <= 64, std::uint_least64_t,
u128>::type>::type>::type>::type;
};
template<std::size_t size> using uint_least_t = typename uint_least<size>::type;
template<class T>
using double_size_int = int_least<std::numeric_limits<T>::digits * 2 + 1>;
template<class T> using double_size_int_t = typename double_size_int<T>::type;
template<class T>
using double_size_uint = uint_least<std::numeric_limits<T>::digits * 2>;
template<class T> using double_size_uint_t = typename double_size_uint<T>::type;
template<class T>
using double_size =
typename std::conditional<is_signed_int<T>::value, double_size_int<T>,
double_size_uint<T>>::type;
template<class T> using double_size_t = typename double_size<T>::type;
#line 2 "template/in.hpp"
#line 4 "template/in.hpp"
#include <unistd.h>
#line 8 "template/in.hpp"
template<std::size_t buf_size = IO_BUFFER_SIZE,
std::size_t decimal_precision = 16>
class Scanner {
private:
template<class, class = void> struct has_scan : std::false_type {};
template<class T>
struct has_scan<
T, decltype(std::declval<T>().scan(std::declval<Scanner&>()), (void)0)>
: std::true_type {};
int fd;
int idx, sz;
bool state;
std::array<char, IO_BUFFER_SIZE + 1> buffer;
inline char cur() {
if (idx == sz) load();
if (idx == sz) {
state = false;
return '\0';
}
return buffer[idx];
}
inline void next() {
if (idx == sz) load();
if (idx == sz) return;
++idx;
}
public:
inline void load() {
int len = sz - idx;
if (idx < len) return;
std::memcpy(buffer.begin(), buffer.begin() + idx, len);
sz = len + read(fd, buffer.data() + len, buf_size - len);
buffer[sz] = 0;
idx = 0;
}
Scanner(int fd) : fd(fd), idx(0), sz(0), state(true) {}
Scanner(FILE* fp) : fd(fileno(fp)), idx(0), sz(0), state(true) {}
inline char scan_char() {
if (idx == sz) load();
return idx == sz ? '\0' : buffer[idx++];
}
Scanner ignore(int n = 1) {
if (idx + n > sz) load();
idx += n;
return *this;
}
inline void discard_space() {
if (idx == sz) load();
while (('\t' <= buffer[idx] && buffer[idx] <= '\r') ||
buffer[idx] == ' ') {
if (++idx == sz) load();
}
}
void scan(char& a) {
discard_space();
a = scan_char();
}
void scan(bool& a) {
discard_space();
a = scan_char() != '0';
}
void scan(std::string& a) {
discard_space();
a.clear();
while (cur() != '\0' && (buffer[idx] < '\t' || '\r' < buffer[idx]) &&
buffer[idx] != ' ') {
a += scan_char();
}
}
template<std::size_t len> void scan(std::bitset<len>& a) {
discard_space();
if (idx + len > sz) load();
rrep (i, len) a[i] = buffer[idx++] != '0';
}
template<class T,
typename std::enable_if<is_signed_int<T>::value &&
!has_scan<T>::value>::type* = nullptr>
void scan(T& a) {
discard_space();
if (buffer[idx] == '-') {
++idx;
if (idx + 40 > sz &&
(idx == sz || ('0' <= buffer[sz - 1] && buffer[sz - 1] <= '9')))
load();
a = 0;
while ('0' <= buffer[idx] && buffer[idx] <= '9') {
a = a * 10 - (buffer[idx++] - '0');
}
}
else {
if (idx + 40 > sz && '0' <= buffer[sz - 1] && buffer[sz - 1] <= '9')
load();
a = 0;
while ('0' <= buffer[idx] && buffer[idx] <= '9') {
a = a * 10 + (buffer[idx++] - '0');
}
}
}
template<class T,
typename std::enable_if<is_unsigned_int<T>::value &&
!has_scan<T>::value>::type* = nullptr>
void scan(T& a) {
discard_space();
if (idx + 40 > sz && '0' <= buffer[sz - 1] && buffer[sz - 1] <= '9')
load();
a = 0;
while ('0' <= buffer[idx] && buffer[idx] <= '9') {
a = a * 10 + (buffer[idx++] - '0');
}
}
template<class T,
typename std::enable_if<std::is_floating_point<T>::value &&
!has_scan<T>::value>::type* = nullptr>
void scan(T& a) {
discard_space();
bool sgn = false;
if (cur() == '-') {
sgn = true;
next();
}
a = 0;
while ('0' <= cur() && cur() <= '9') {
a = a * 10 + cur() - '0';
next();
}
if (cur() == '.') {
next();
T n = 0, d = 1;
for (int i = 0;
'0' <= cur() && cur() <= '9' && i < (int)decimal_precision;
++i) {
n = n * 10 + cur() - '0';
d *= 10;
next();
}
while ('0' <= cur() && cur() <= '9') next();
a += n / d;
}
if (sgn) a = -a;
}
private:
template<std::size_t i, class... Args> void scan(std::tuple<Args...>& a) {
if IF_CONSTEXPR (i < sizeof...(Args)) {
scan(std::get<i>(a));
scan<i + 1, Args...>(a);
}
}
public:
template<class... Args> void scan(std::tuple<Args...>& a) {
scan<0, Args...>(a);
}
template<class T, class U> void scan(std::pair<T, U>& a) {
scan(a.first);
scan(a.second);
}
template<class T,
typename std::enable_if<is_range<T>::value &&
!has_scan<T>::value>::type* = nullptr>
void scan(T& a) {
for (auto&& i : a) scan(i);
}
template<class T,
typename std::enable_if<has_scan<T>::value>::type* = nullptr>
void scan(T& a) {
a.scan(*this);
}
void operator()() {}
template<class Head, class... Args>
void operator()(Head& head, Args&... args) {
scan(head);
operator()(args...);
}
template<class T> Scanner& operator>>(T& a) {
scan(a);
return *this;
}
explicit operator bool() const { return state; }
friend Scanner& getline(Scanner& scan, std::string& a) {
a.erase();
char c;
if ((c = scan.scan_char()) == '\n' || c == '\0') return scan;
a += c;
while ((c = scan.scan_char()) != '\n' && c != '\0') a += c;
scan.state = true;
return scan;
}
};
Scanner<> scan(0);
#line 2 "template/out.hpp"
#line 8 "template/out.hpp"
struct NumberToString {
char buf[10000][4];
constexpr NumberToString() : buf{} {
rep (i, 10000) {
int n = i;
rrep (j, 4) {
buf[i][j] = (char)('0' + n % 10);
n /= 10;
}
}
}
} constexpr precalc_number_to_string;
template<std::size_t buf_size = IO_BUFFER_SIZE, bool debug = false>
class Printer {
private:
template<class, bool = debug, class = void>
struct has_print : std::false_type {};
template<class T>
struct has_print<T, false,
decltype(std::declval<T>().print(std::declval<Printer&>()),
(void)0)> : std::true_type {};
template<class T>
struct has_print<T, true,
decltype(std::declval<T>().debug(std::declval<Printer&>()),
(void)0)> : std::true_type {};
int fd;
std::size_t idx;
std::array<char, buf_size> buffer;
std::size_t decimal_precision;
public:
inline void print_char(char c) {
#if SHIO_LOCAL
buffer[idx++] = c;
if (idx == buf_size) flush();
#else
if IF_CONSTEXPR (!debug) {
buffer[idx++] = c;
if (idx == buf_size) flush();
}
#endif
}
inline void flush() {
idx = write(fd, buffer.begin(), idx);
idx = 0;
}
Printer(int fd) : fd(fd), idx(0), decimal_precision(16) {}
Printer(FILE* fp) : fd(fileno(fp)), idx(0), decimal_precision(16) {}
~Printer() { flush(); }
void set_decimal_precision(std::size_t decimal_precision) {
this->decimal_precision = decimal_precision;
}
void print(char c) {
if IF_CONSTEXPR (debug) print_char('\'');
print_char(c);
if IF_CONSTEXPR (debug) print_char('\'');
}
void print(bool b) { print_char((char)(b + '0')); }
void print(const char* a) {
if IF_CONSTEXPR (debug) print_char('"');
for (; *a != '\0'; ++a) print_char(*a);
if IF_CONSTEXPR (debug) print_char('"');
}
template<std::size_t len> void print(const char (&a)[len]) {
if IF_CONSTEXPR (debug) print_char('"');
for (auto i : a) print_char(i);
if IF_CONSTEXPR (debug) print_char('"');
}
void print(const std::string& a) {
if IF_CONSTEXPR (debug) print_char('"');
for (auto i : a) print_char(i);
if IF_CONSTEXPR (debug) print_char('"');
}
template<std::size_t len> void print(const std::bitset<len>& a) {
rrep (i, len) print_char((char)(a[i] + '0'));
}
template<class T,
typename std::enable_if<is_int<T>::value &&
!has_print<T>::value>::type* = nullptr>
void print(T a) {
if (!a) {
print_char('0');
return;
}
if IF_CONSTEXPR (is_signed_int<T>::value) {
if (a < 0) {
print_char('-');
using U = typename make_unsigned_int<T>::type;
print(static_cast<U>(-static_cast<U>(a)));
return;
}
}
if (idx + 40 >= buf_size) flush();
static char s[40];
int t = 40;
while (a >= 10000) {
int i = a % 10000;
a /= 10000;
t -= 4;
std::memcpy(s + t, precalc_number_to_string.buf[i], 4);
}
if (a >= 1000) {
std::memcpy(buffer.begin() + idx, precalc_number_to_string.buf[a],
4);
idx += 4;
}
else if (a >= 100) {
std::memcpy(buffer.begin() + idx,
precalc_number_to_string.buf[a] + 1, 3);
idx += 3;
}
else if (a >= 10) {
std::memcpy(buffer.begin() + idx,
precalc_number_to_string.buf[a] + 2, 2);
idx += 2;
}
else {
buffer[idx++] = '0' | a;
}
std::memcpy(buffer.begin() + idx, s + t, 40 - t);
idx += 40 - t;
}
template<class T,
typename std::enable_if<std::is_floating_point<T>::value &&
!has_print<T>::value>::type* = nullptr>
void print(T a) {
if (a == std::numeric_limits<T>::infinity()) {
print("inf");
return;
}
if (a == -std::numeric_limits<T>::infinity()) {
print("-inf");
return;
}
if (std::isnan(a)) {
print("nan");
return;
}
if (a < 0) {
print_char('-');
a = -a;
}
T b = a;
if (b < 1) {
print_char('0');
}
else {
std::string s;
while (b >= 1) {
s += (char)('0' + (int)std::fmod(b, 10.0));
b /= 10;
}
for (auto i = s.rbegin(); i != s.rend(); ++i) print_char(*i);
}
print_char('.');
rep (decimal_precision) {
a *= 10;
print_char((char)('0' + (int)std::fmod(a, 10.0)));
}
}
private:
template<std::size_t i, class... Args>
void print(const std::tuple<Args...>& a) {
if IF_CONSTEXPR (i < sizeof...(Args)) {
if IF_CONSTEXPR (debug) print_char(',');
print_char(' ');
print(std::get<i>(a));
print<i + 1, Args...>(a);
}
}
public:
template<class... Args> void print(const std::tuple<Args...>& a) {
if IF_CONSTEXPR (debug) print_char('(');
if IF_CONSTEXPR (sizeof...(Args) != 0) print(std::get<0>(a));
print<1, Args...>(a);
if IF_CONSTEXPR (debug) print_char(')');
}
template<class T, class U> void print(const std::pair<T, U>& a) {
if IF_CONSTEXPR (debug) print_char('(');
print(a.first);
if IF_CONSTEXPR (debug) print_char(',');
print_char(' ');
print(a.second);
if IF_CONSTEXPR (debug) print_char(')');
}
template<class T,
typename std::enable_if<is_range<T>::value &&
!has_print<T>::value>::type* = nullptr>
void print(const T& a) {
if IF_CONSTEXPR (debug) print_char('{');
for (auto i = std::begin(a); i != std::end(a); ++i) {
if (i != std::begin(a)) {
if IF_CONSTEXPR (debug) print_char(',');
print_char(' ');
}
print(*i);
}
if IF_CONSTEXPR (debug) print_char('}');
}
template<class T, typename std::enable_if<has_print<T>::value &&
!debug>::type* = nullptr>
void print(const T& a) {
a.print(*this);
}
template<class T, typename std::enable_if<has_print<T>::value &&
debug>::type* = nullptr>
void print(const T& a) {
a.debug(*this);
}
void operator()() {}
template<class Head, class... Args>
void operator()(const Head& head, const Args&... args) {
print(head);
operator()(args...);
}
template<class T> Printer& operator<<(const T& a) {
print(a);
return *this;
}
Printer& operator<<(Printer& (*pf)(Printer&)) { return pf(*this); }
};
template<std::size_t buf_size, bool debug>
Printer<buf_size, debug>& endl(Printer<buf_size, debug>& pr) {
pr.print_char('\n');
pr.flush();
return pr;
}
template<std::size_t buf_size, bool debug>
Printer<buf_size, debug>& flush(Printer<buf_size, debug>& pr) {
pr.flush();
return pr;
}
struct SetPrec {
int n;
template<class Pr> void print(Pr& pr) const { pr.set_decimal_precision(n); }
template<class Pr> void debug(Pr& pr) const { pr.set_decimal_precision(n); }
};
SetPrec setprec(int n) { return SetPrec{n}; };
Printer<> print(1), eprint(2);
void prints() { print.print_char('\n'); }
template<class T> auto prints(const T& v) -> decltype(print << v, (void)0) {
print << v;
print.print_char('\n');
}
template<class Head, class... Tail>
auto prints(const Head& head, const Tail&... tail)
-> decltype(print << head, (void)0) {
print << head;
print.print_char(' ');
prints(tail...);
}
Printer<IO_BUFFER_SIZE, true> debug(1), edebug(2);
void debugs() { debug.print_char('\n'); }
template<class T> auto debugs(const T& v) -> decltype(debug << v, (void)0) {
debug << v;
debug.print_char('\n');
}
template<class Head, class... Tail>
auto debugs(const Head& head, const Tail&... tail)
-> decltype(debug << head, (void)0) {
debug << head;
debug.print_char(' ');
debugs(tail...);
}
#line 2 "template/bitop.hpp"
#line 6 "template/bitop.hpp"
namespace bitop {
#define KTH_BIT(b, k) (((b) >> (k)) & 1)
#define POW2(k) (1ull << (k))
inline ull next_combination(int n, ull x) {
if (n == 0) return 1;
ull a = x & -x;
ull b = x + a;
return (x & ~b) / a >> 1 | b;
}
#define rep_comb(i, n, k) \
for (ull i = (1ull << (k)) - 1; i < (1ull << (n)); \
i = bitop::next_combination((n), i))
inline constexpr int msb(ull x) {
int res = x ? 0 : -1;
if (x & 0xFFFFFFFF00000000) x &= 0xFFFFFFFF00000000, res += 32;
if (x & 0xFFFF0000FFFF0000) x &= 0xFFFF0000FFFF0000, res += 16;
if (x & 0xFF00FF00FF00FF00) x &= 0xFF00FF00FF00FF00, res += 8;
if (x & 0xF0F0F0F0F0F0F0F0) x &= 0xF0F0F0F0F0F0F0F0, res += 4;
if (x & 0xCCCCCCCCCCCCCCCC) x &= 0xCCCCCCCCCCCCCCCC, res += 2;
return res + ((x & 0xAAAAAAAAAAAAAAAA) ? 1 : 0);
}
inline constexpr int ceil_log2(ull x) { return x ? msb(x - 1) + 1 : 0; }
inline constexpr ull reverse(ull x) {
x = ((x & 0xAAAAAAAAAAAAAAAA) >> 1) | ((x & 0x5555555555555555) << 1);
x = ((x & 0xCCCCCCCCCCCCCCCC) >> 2) | ((x & 0x3333333333333333) << 2);
x = ((x & 0xF0F0F0F0F0F0F0F0) >> 4) | ((x & 0x0F0F0F0F0F0F0F0F) << 4);
x = ((x & 0xFF00FF00FF00FF00) >> 8) | ((x & 0x00FF00FF00FF00FF) << 8);
x = ((x & 0xFFFF0000FFFF0000) >> 16) | ((x & 0x0000FFFF0000FFFF) << 16);
return (x >> 32) | (x << 32);
}
inline constexpr ull reverse(ull x, int n) { return reverse(x) >> (64 - n); }
} // namespace bitop
inline constexpr int popcnt(ull x) noexcept {
#if __cplusplus >= 202002L
return std::popcount(x);
#endif
x = (x & 0x5555555555555555) + ((x >> 1) & 0x5555555555555555);
x = (x & 0x3333333333333333) + ((x >> 2) & 0x3333333333333333);
x = (x & 0x0f0f0f0f0f0f0f0f) + ((x >> 4) & 0x0f0f0f0f0f0f0f0f);
x = (x & 0x00ff00ff00ff00ff) + ((x >> 8) & 0x00ff00ff00ff00ff);
x = (x & 0x0000ffff0000ffff) + ((x >> 16) & 0x0000ffff0000ffff);
return (x & 0x00000000ffffffff) + ((x >> 32) & 0x00000000ffffffff);
}
#line 2 "template/func.hpp"
#line 6 "template/func.hpp"
template<class T, class U, class Comp = std::less<>>
inline constexpr bool chmin(T& a, const U& b,
Comp cmp = Comp()) noexcept(noexcept(cmp(b, a))) {
return cmp(b, a) ? a = b, true : false;
}
template<class T, class U, class Comp = std::less<>>
inline constexpr bool chmax(T& a, const U& b,
Comp cmp = Comp()) noexcept(noexcept(cmp(a, b))) {
return cmp(a, b) ? a = b, true : false;
}
inline constexpr ll gcd(ll a, ll b) {
if (a < 0) a = -a;
if (b < 0) b = -b;
while (b) {
const ll c = a;
a = b;
b = c % b;
}
return a;
}
inline constexpr ll lcm(ll a, ll b) { return a / gcd(a, b) * b; }
inline constexpr bool is_prime(ll N) {
if (N <= 1) return false;
for (ll i = 2; i * i <= N; ++i) {
if (N % i == 0) return false;
}
return true;
}
inline std::vector<ll> prime_factor(ll N) {
std::vector<ll> res;
for (ll i = 2; i * i <= N; ++i) {
while (N % i == 0) {
res.push_back(i);
N /= i;
}
}
if (N != 1) res.push_back(N);
return res;
}
inline constexpr ll my_pow(ll a, ll b) {
ll res = 1;
while (b) {
if (b & 1) res *= a;
b >>= 1;
a *= a;
}
return res;
}
inline constexpr ll mod_pow(ll a, ll b, ll mod) {
assert(mod > 0);
if (mod == 1) return 0;
a %= mod;
ll res = 1;
while (b) {
if (b & 1) (res *= a) %= mod;
b >>= 1;
(a *= a) %= mod;
}
return res;
}
inline PLL extGCD(ll a, ll b) {
const ll n = a, m = b;
ll x = 1, y = 0, u = 0, v = 1;
ll t;
while (b) {
t = a / b;
std::swap(a -= t * b, b);
std::swap(x -= t * u, u);
std::swap(y -= t * v, v);
}
if (x < 0) {
x += m;
y -= n;
}
return {x, y};
}
inline ll mod_inv(ll a, ll mod) {
ll b = mod;
ll x = 1, u = 0;
ll t;
while (b) {
t = a / b;
std::swap(a -= t * b, b);
std::swap(x -= t * u, u);
}
if (x < 0) x += mod;
assert(a == 1);
return x;
}
#line 2 "template/util.hpp"
#line 6 "template/util.hpp"
template<class F> class RecLambda {
private:
F f;
public:
explicit constexpr RecLambda(F&& f_) : f(std::forward<F>(f_)) {}
template<class... Args>
constexpr auto operator()(Args&&... args)
-> decltype(f(*this, std::forward<Args>(args)...)) {
return f(*this, std::forward<Args>(args)...);
}
};
template<class F> inline constexpr RecLambda<F> rec_lambda(F&& f) {
return RecLambda<F>(std::forward<F>(f));
}
template<class Head, class... Tail> struct multi_dim_vector {
using type = std::vector<typename multi_dim_vector<Tail...>::type>;
};
template<class T> struct multi_dim_vector<T> { using type = T; };
template<class T, class Arg>
constexpr std::vector<T> make_vec(int n, Arg&& arg) {
return std::vector<T>(n, std::forward<Arg>(arg));
}
template<class T, class... Args>
constexpr typename multi_dim_vector<Args..., T>::type make_vec(int n,
Args&&... args) {
return typename multi_dim_vector<Args..., T>::type(
n, make_vec<T>(std::forward<Args>(args)...));
}
template<class T, class Comp = std::less<T>> class compressor {
private:
std::vector<T> dat;
Comp cmp;
bool sorted = false;
public:
compressor() : compressor(Comp()) {}
compressor(const Comp& cmp) : cmp(cmp) {}
compressor(const std::vector<T>& vec, bool f = false,
const Comp& cmp = Comp())
: dat(vec), cmp(cmp) {
if (f) build();
}
compressor(std::vector<T>&& vec, bool f = false, const Comp& cmp = Comp())
: dat(std::move(vec)), cmp(cmp) {
if (f) build();
}
compressor(std::initializer_list<T> il, bool f = false,
const Comp& cmp = Comp())
: dat(all(il)), cmp(cmp) {
if (f) build();
}
void reserve(int n) {
assert(!sorted);
dat.reserve(n);
}
void push_back(const T& v) {
assert(!sorted);
dat.push_back(v);
}
void push_back(T&& v) {
assert(!sorted);
dat.push_back(std::move(v));
}
template<class... Args> void emplace_back(Args&&... args) {
assert(!sorted);
dat.emplace_back(std::forward<Args>(args)...);
}
void push(const std::vector<T>& vec) {
assert(!sorted);
const int n = dat.size();
dat.resize(n + vec.size());
rep (i, vec.size()) dat[n + i] = vec[i];
}
int build() {
assert(!sorted);
sorted = true;
std::sort(all(dat), cmp);
dat.erase(std::unique(all(dat),
[&](const T& a, const T& b) -> bool {
return !cmp(a, b) && !cmp(b, a);
}),
dat.end());
return dat.size();
}
const T& operator[](int k) const& {
assert(sorted);
assert(0 <= k && k < (int)dat.size());
return dat[k];
}
int get(const T& val) const {
assert(sorted);
auto itr = std::lower_bound(all(dat), val, cmp);
assert(itr != dat.end() && !cmp(val, *itr));
return itr - dat.begin();
}
int lower_bound(const T& val) const {
assert(sorted);
auto itr = std::lower_bound(all(dat), val, cmp);
return itr - dat.begin();
}
int upper_bound(const T& val) const {
assert(sorted);
auto itr = std::upper_bound(all(dat), val, cmp);
return itr - dat.begin();
}
bool contains(const T& val) const {
assert(sorted);
return std::binary_search(all(dat), val, cmp);
}
std::vector<int> pressed(const std::vector<T>& vec) const {
assert(sorted);
std::vector<int> res(vec.size());
rep (i, vec.size()) res[i] = get(vec[i]);
return res;
}
void press(std::vector<T>& vec) const {
assert(sorted);
for (auto&& i : vec) i = get(i);
}
int size() const {
assert(sorted);
return dat.size();
}
};
#line 2 "other/monoid.hpp"
#line 4 "other/monoid.hpp"
namespace Monoid {
template<class M, class = void>
class has_value_type : public std::false_type {};
template<class M>
class has_value_type<M, decltype((void)std::declval<typename M::value_type>())>
: public std::true_type {};
template<class M, class = void> class has_op : public std::false_type {};
template<class M>
class has_op<M, decltype((void)M::op)> : public std::true_type {};
template<class M, class = void> class has_id : public std::false_type {};
template<class M>
class has_id<M, decltype((void)M::id)> : public std::true_type {};
template<class M, class = void> class has_inv : public std::false_type {};
template<class M>
class has_inv<M, decltype((void)M::inv)> : public std::true_type {};
template<class M, class = void> class has_get_inv : public std::false_type {};
template<class M>
class has_get_inv<M, decltype((void)M::get_inv)> : public std::true_type {};
template<class M, class = void> class has_init : public std::false_type {};
template<class M>
class has_init<M, decltype((void)M::init(0, 0))> : public std::true_type {};
template<class A, class = void> class has_mul_op : public std::false_type {};
template<class A>
class has_mul_op<A, decltype((void)A::mul_op)> : public std::true_type {};
template<class T, class = void> class is_semigroup : public std::false_type {};
template<class T>
class is_semigroup<T, decltype(std::declval<typename T::value_type>(),
(void)T::op)> : public std::true_type {};
template<class T, class = void> class is_monoid : public std::false_type {};
template<class T>
class is_monoid<T, decltype(std::declval<typename T::value_type>(), (void)T::op,
(void)T::id)> : public std::true_type {};
template<class T, class = void> class is_group : public std::false_type {};
template<class T>
class is_group<T, decltype(std::declval<typename T::value_type>(), (void)T::op,
(void)T::id, (void)T::get_inv)>
: public std::true_type {};
template<class T, class = void> class is_action : public std::false_type {};
template<class T>
class is_action<T, typename std::enable_if<is_monoid<typename T::M>::value &&
is_semigroup<typename T::E>::value &&
(has_op<T>::value ||
has_mul_op<T>::value)>::type>
: public std::true_type {};
template<class T, class = void>
class is_distributable_action : public std::false_type {};
template<class T>
class is_distributable_action<
T,
typename std::enable_if<is_action<T>::value && !has_mul_op<T>::value>::type>
: public std::true_type {};
template<class T> struct Sum {
using value_type = T;
static constexpr T op(const T& a, const T& b) { return a + b; }
static constexpr T id() { return T{0}; }
static constexpr T inv(const T& a, const T& b) { return a - b; }
static constexpr T get_inv(const T& a) { return -a; }
};
template<class T, int i = -1> struct Min {
using value_type = T;
static T max_value;
static T op(const T& a, const T& b) { return a < b ? a : b; }
static T id() { return max_value; }
};
template<class T> struct Min<T, -1> {
using value_type = T;
static constexpr T op(const T& a, const T& b) { return a < b ? a : b; }
static constexpr T id() { return infinity<T>::value; }
};
template<class T> struct Min<T, -2> {
using value_type = T;
static constexpr T op(const T& a, const T& b) { return a < b ? a : b; }
static constexpr T id() { return infinity<T>::max; }
};
template<class T, int id> T Min<T, id>::max_value;
template<class T, int i = -1> struct Max {
using value_type = T;
static T min_value;
static T op(const T& a, const T& b) { return a > b ? a : b; }
static T id() { return min_value; }
};
template<class T> struct Max<T, -1> {
using value_type = T;
static constexpr T op(const T& a, const T& b) { return a > b ? a : b; }
static constexpr T id() { return infinity<T>::mvalue; }
};
template<class T> struct Max<T, -2> {
using value_type = T;
static constexpr T op(const T& a, const T& b) { return a > b ? a : b; }
static constexpr T id() { return infinity<T>::min; }
};
template<class T> struct Assign {
using value_type = T;
static constexpr T op(const T&, const T& b) { return b; }
};
template<class T, int id = -1> struct AssignMin {
using M = Min<T, id>;
using E = Assign<T>;
static constexpr T op(const T& a, const T&) { return a; }
};
template<class T, int id = -1> struct AssignMax {
using M = Max<T, id>;
using E = Assign<T>;
static constexpr T op(const T& a, const T&) { return a; }
};
template<class T> struct AssignSum {
using M = Sum<T>;
using E = Assign<T>;
static constexpr T mul_op(const T& a, int b, const T&) { return a * b; }
};
template<class T, int id = -1> struct AddMin {
using M = Min<T, id>;
using E = Sum<T>;
static constexpr T op(const T& a, const T& b) { return b + a; }
};
template<class T, int id = -1> struct AddMax {
using M = Max<T, id>;
using E = Sum<T>;
static constexpr T op(const T& a, const T& b) { return b + a; }
};
template<class T> struct AddSum {
using M = Sum<T>;
using E = Sum<T>;
static constexpr T mul_op(const T& a, int b, const T& c) {
return c + a * b;
}
};
template<class T, int id = -1> struct ChminMin {
using M = Min<T, id>;
using E = Min<T>;
static constexpr T op(const T& a, const T& b) { return std::min(b, a); }
};
template<class T, int id = -1> struct ChminMax {
using M = Max<T, id>;
using E = Min<T>;
static constexpr T op(const T& a, const T& b) { return std::min(b, a); }
};
template<class T, int id = -1> struct ChmaxMin {
using M = Min<T, id>;
using E = Max<T>;
static constexpr T op(const T& a, const T& b) { return std::max(b, a); }
};
template<class T, int id = -1> struct ChmaxMax {
using M = Max<T, id>;
using E = Max<T>;
static constexpr T op(const T& a, const T& b) { return std::max(b, a); }
};
template<class M> struct ReverseMonoid {
using value_type = typename M::value_type;
static value_type op(const value_type& a, const value_type& b) {
return M::op(b, a);
}
static value_type id() {
static_assert(has_id<M>::value, "id is not defined");
return M::id();
}
static value_type inv(const value_type& a, const value_type& b) {
static_assert(has_inv<M>::value, "inv is not defined");
return M::inv(b, a);
}
static value_type get_inv(const value_type& a) {
static_assert(has_get_inv<M>::value, "get_inv is not defined");
return M::get_inv(a);
}
};
template<class E_> struct MakeAction {
using M = E_;
using E = E_;
using T = typename E_::value_type;
static T op(const T& a, const T& b) { return E_::op(b, a); }
};
} // namespace Monoid
#line 5 "data-struct/segment/LazySegmentTree.hpp"
template<class A> class LazySegmentTree {
static_assert(Monoid::is_action<A>::value, "A must be action");
private:
using M = typename A::M;
using E = typename A::E;
using T = typename M::value_type;
using U = typename E::value_type;
int h, n, ori;
std::vector<T> data;
std::vector<U> lazy;
std::vector<bool> lazyflag;
template<bool AlwaysTrue = true,
typename std::enable_if<!Monoid::has_mul_op<A>::value &&
AlwaysTrue>::type* = nullptr>
static inline T Aop(const U& a, const T& b, int) {
return A::op(a, b);
}
template<bool AlwaysTrue = true,
typename std::enable_if<Monoid::has_mul_op<A>::value &&
AlwaysTrue>::type* = nullptr>
static inline T Aop(const U& a, const T& b, int c) {
return A::mul_op(a, c, b);
}
void all_apply(int k, const U& x, int d) {
data[k] = Aop(x, data[k], d);
if (k < n) {
if (lazyflag[k]) {
lazy[k] = E::op(lazy[k], x);
}
else {
lazy[k] = x;
lazyflag[k] = true;
}
}
}
void eval(int k, int d) {
if (lazyflag[k]) {
all_apply(k << 1, lazy[k], d >> 1);
all_apply(k << 1 ^ 1, lazy[k], d >> 1);
lazyflag[k] = false;
}
}
void calc(int k) { data[k] = M::op(data[k << 1], data[k << 1 ^ 1]); }
public:
LazySegmentTree() : LazySegmentTree(0) {}
LazySegmentTree(int n) : LazySegmentTree(std::vector<T>(n, M::id())) {}
LazySegmentTree(int n, const T& v)
: LazySegmentTree(std::vector<T>(n, v)) {}
LazySegmentTree(const std::vector<T>& v) { init(v); }
void init(const std::vector<T>& v) {
ori = v.size();
h = bitop::ceil_log2(ori);
n = 1 << h;
data.assign(n << 1, M::id());
rep (i, ori) data[n + i] = v[i];
rrep (i, n, 1) calc(i);
lazy.resize(n);
lazyflag.assign(n, false);
}
void eval_all() {
rep (i, h) {
rep (j, 1 << i) eval((1 << i) + j, 1 << (h - i));
}
}
T prod(int l, int r) {
assert(0 <= l && l <= r && r <= ori);
if (l == r) return M::id();
l += n, r += n;
rreps (i, h) {
bool seen = false;
if (((l >> i) << i) != l) eval(l >> i, 1 << i), seen = true;
if (((r >> i) << i) != r) eval((r - 1) >> i, 1 << i), seen = true;
if (!seen) break;
}
T lsm = M::id(), rsm = M::id();
while (l != r) {
if (l & 1) lsm = M::op(lsm, data[l++]);
if (r & 1) rsm = M::op(data[--r], rsm);
l >>= 1, r >>= 1;
}
return M::op(lsm, rsm);
}
T get(int k) {
assert(0 <= k && k < ori);
k += n;
rreps (i, h) eval(k >> i, 1 << i);
return data[k];
}
T all_prod() const { return data[1]; }
template<class Upd> void update(int k, const Upd& upd) {
assert(0 <= k && k < ori);
k += n;
rreps (i, h) eval(k >> i, 1 << i);
data[k] = upd(data[k]);
reps (i, h) calc(k >> i);
}
void set(int k, const T& x) {
update(k, [&](const T&) -> T { return x; });
}
void apply(int k, const U& x) {
update(k, [&](const T& a) -> T { return A::op(x, a); });
}
void apply(int l, int r, const U& x) {
assert(0 <= l && l <= r && r <= ori);
if (l == r) return;
l += n, r += n;
int lst = h + 1;
rreps (i, h) {
if (((l >> i) << i) != l) eval(l >> i, 1 << i), lst = i;
if (((r >> i) << i) != r) eval((r - 1) >> i, 1 << i), lst = i;
if (lst != i) break;
}
for (int l2 = l, r2 = r, d = 1; l2 != r2; l2 >>= 1, r2 >>= 1, d <<= 1) {
if (l2 & 1) all_apply(l2++, x, d);
if (r2 & 1) all_apply(--r2, x, d);
}
rep (i, lst, h + 1) {
if (((l >> i) << i) != l) calc(l >> i);
if (((r >> i) << i) != r) calc((r - 1) >> i);
}
}
template<class C> int max_right(int l, const C& cond) {
assert(0 <= l && l <= ori);
assert(cond(M::id()));
if (l == ori) return ori;
l += n;
rreps (i, h) {
if (((l >> i) << i) != l) eval(l >> i, 1 << i);
else break;
}
T sm = M::id();
int d = 1;
do {
while ((l & 1) == 0) l >>= 1, d <<= 1;
if (!cond(M::op(sm, data[l]))) {
while (l < n) {
eval(l, d);
l <<= 1;
d >>= 1;
if (cond(M::op(sm, data[l]))) sm = M::op(sm, data[l++]);
}
return l - n;
}
sm = M::op(sm, data[l++]);
} while ((l & -l) != l);
return ori;
}
template<class C> int min_left(int r, const C& cond) {
assert(0 <= r && r <= ori);
assert(cond(M::id()));
if (r == 0) return 0;
r += n;
rreps (i, h) {
if (((r >> i) << i) != r) eval((r - 1) >> i, 1 << i);
else break;
}
T sm = M::id();
int d = 1;
do {
--r;
while ((r & 1) && r > 1) r >>= 1, d <<= 1;
if (!cond(M::op(data[r], sm))) {
while (r < n) {
eval(r, d);
r = r << 1 ^ 1;
d >>= 1;
if (cond(M::op(data[r], sm))) sm = M::op(data[r--], sm);
}
return r + 1 - n;
}
sm = M::op(data[r], sm);
} while ((r & -r) != r);
return 0;
}
};
template<class T, int id = -1>
using RangeUpdateQueryRangeMinimumQuery =
LazySegmentTree<Monoid::AssignMin<T, id>>;
template<class T, int id = -1>
using RangeUpdateQueryRangeMaximumQuery =
LazySegmentTree<Monoid::AssignMax<T, id>>;
template<class T>
using RangeUpdateQueryRangeSumQuery = LazySegmentTree<Monoid::AssignSum<T>>;
template<class T, int id = -1>
using RangeAddQueryRangeMinimumQuery = LazySegmentTree<Monoid::AddMin<T, id>>;
template<class T, int id = -1>
using RangeAddQueryRangeMaximumQuery = LazySegmentTree<Monoid::AddMax<T, id>>;
template<class T>
using RangeAddQueryRangeSumQuery = LazySegmentTree<Monoid::AddSum<T>>;
template<class T, int id = -1>
using RangeChminQueryRangeMinimumQuery =
LazySegmentTree<Monoid::ChminMin<T, id>>;
template<class T, int id = -1>
using RangeChminQueryRangeMaximumQuery =
LazySegmentTree<Monoid::ChminMax<T, id>>;
template<class T, int id = -1>
using RangeChmaxQueryRangeMinimumQuery =
LazySegmentTree<Monoid::ChmaxMin<T, id>>;
template<class T, int id = -1>
using RangeChmaxQueryRangeMaximumQuery =
LazySegmentTree<Monoid::ChmaxMax<T, id>>;
/**
* @brief LazySegmentTree(遅延セグメント木)
* @docs docs/data-struct/segment/LazySegmentTree.md
*/