#line 2 "data-structure/slope-trick-weighted.hpp"
#include <cassert>
#include <iostream>
using namespace std;
#line 2 "segment-tree/rbst-segment-tree.hpp"
#include <functional>
#include <type_traits>
#line 2 "internal/internal-type-traits.hpp"
#line 4 "internal/internal-type-traits.hpp"
using namespace std;
namespace nyaan_internal {
template <typename T>
using is_broadly_integral =
typename conditional_t<is_integral_v<T> || is_same_v<T, __int128_t> ||
is_same_v<T, __uint128_t>,
true_type, false_type>::type;
template <typename T>
using is_broadly_signed =
typename conditional_t<is_signed_v<T> || is_same_v<T, __int128_t>,
true_type, false_type>::type;
template <typename T>
using is_broadly_unsigned =
typename conditional_t<is_unsigned_v<T> || is_same_v<T, __uint128_t>,
true_type, false_type>::type;
#define ENABLE_VALUE(x) \
template <typename T> \
constexpr bool x##_v = x<T>::value;
ENABLE_VALUE(is_broadly_integral);
ENABLE_VALUE(is_broadly_signed);
ENABLE_VALUE(is_broadly_unsigned);
#undef ENABLE_VALUE
#define ENABLE_HAS_TYPE(var) \
template <class, class = void> \
struct has_##var : false_type {}; \
template <class T> \
struct has_##var<T, void_t<typename T::var>> : true_type {}; \
template <class T> \
constexpr auto has_##var##_v = has_##var<T>::value;
#define ENABLE_HAS_VAR(var) \
template <class, class = void> \
struct has_##var : false_type {}; \
template <class T> \
struct has_##var<T, void_t<decltype(T::var)>> : true_type {}; \
template <class T> \
constexpr auto has_##var##_v = has_##var<T>::value;
} // namespace nyaan_internal
#line 7 "segment-tree/rbst-segment-tree.hpp"
ENABLE_HAS_VAR(lazy);
ENABLE_HAS_VAR(shift);
template <typename Node, typename I, typename T, typename E, T (*f)(T, T),
T (*g)(T, E), E (*h)(E, E), T (*ti)(), E (*ei)()>
struct RBSTSegmentTreeBase {
protected:
using Ptr = Node *;
template <typename... Args>
static Ptr _my_new(Args... args) {
return new Node(args...);
}
static void _my_del(Ptr t) { delete t; }
static int _count(const Ptr t) { return t ? t->cnt : 0; }
static T _sum(const Ptr &t) { return t ? t->sum : ti(); }
static uint64_t _rng() {
static uint64_t x_ = 88172645463325252ULL;
return x_ ^= x_ << 7, x_ ^= x_ >> 9, x_ & 0xFFFFFFFFull;
}
static Ptr _merge(Ptr l, Ptr r) {
if (!l || !r) return l ? l : r;
if (int((_rng() * (l->cnt + r->cnt)) >> 32) < l->cnt) {
_push(l);
l->r = _merge(l->r, r);
return _update(l);
} else {
_push(r);
r->l = _merge(l, r->l);
return _update(r);
}
}
static Ptr _build(int l, int r, const vector<pair<I, T>> &dat) {
if (l == r) return nullptr;
if (l + 1 == r) return _my_new(dat[l].first, dat[l].second);
int m = (l + r) / 2;
return _merge(_build(l, m, dat), _build(m, r, dat));
};
static void _push([[maybe_unused]] Ptr t) {
if constexpr (has_lazy_v<Node>) {
if (!t) return;
if (t->lazy != ei()) {
if (t->l) _propagate(t->l, t->lazy);
if (t->r) _propagate(t->r, t->lazy);
t->lazy = ei();
}
}
if constexpr (has_shift_v<Node>) {
if (!t) return;
if (t->shift != I{}) {
if (t->l) _shift(t->l, t->shift);
if (t->r) _shift(t->r, t->shift);
t->shift = I{};
}
}
}
static void _propagate([[maybe_unused]] Ptr t, [[maybe_unused]] const E &x) {
if constexpr (has_lazy_v<Node>) {
if (!t) return;
t->lazy = h(t->lazy, x);
t->val = g(t->val, x);
t->sum = g(t->sum, x);
}
}
static void _shift([[maybe_unused]] Ptr t, [[maybe_unused]] const I &sh) {
if constexpr (has_shift_v<Node>) {
if (!t) return;
t->key += sh, t->shift += sh;
}
}
static Ptr _update(Ptr t) {
if (!t) return t;
t->cnt = 1;
t->sum = t->val;
if (t->l) t->cnt += t->l->cnt, t->sum = f(t->l->sum, t->sum);
if (t->r) t->cnt += t->r->cnt, t->sum = f(t->sum, t->r->sum);
return t;
}
// key が k であるノードを探す, なければ nullptr
static Ptr _find(Ptr t, I k) {
while (t) {
_push(t);
if (k == t->key) return t;
t = k < t->key ? t->l : t->r;
}
return nullptr;
}
static void _erase(Ptr &t, I k) {
if (!t) return;
_push(t);
if (k == t->key) {
Ptr tl = t->l, tr = t->r;
_my_del(t);
t = _merge(tl, tr);
} else if (k < t->key) {
_erase(t->l, k);
_update(t);
} else {
_erase(t->r, k);
_update(t);
}
}
// [k 未満, k 以上]
static pair<Ptr, Ptr> _split_by_key(Ptr t, I k) {
if (!t) return {nullptr, nullptr};
_push(t);
if (k == t->key) {
Ptr tl = t->l;
t->l = nullptr;
return {tl, _update(t)};
} else if (k < t->key) {
auto s = _split_by_key(t->l, k);
t->l = s.second;
return {s.first, _update(t)};
} else {
auto s = _split_by_key(t->r, k);
t->r = s.first;
return {_update(t), s.second};
}
}
// [k 未満, k, k 超過]
static array<Ptr, 3> _split_by_key3(Ptr t, I k) {
if (!t) return {{nullptr, nullptr, nullptr}};
_push(t);
if (k == t->key) {
Ptr tl = t->l, tr = t->r;
t->l = t->r = nullptr;
return {{tl, _update(t), tr}};
} else if (k < t->key) {
auto s = _split_by_key3(t->l, k);
t->l = s[2];
return {{s[0], s[1], _update(t)}};
} else {
auto s = _split_by_key3(t->r, k);
t->r = s[0];
return {{_update(t), s[1], s[2]}};
}
}
// (-inf, i] の prod について check(prod) の (true / false) で切る
template <typename C>
static pair<Ptr, Ptr> _split_max_right(Ptr t, const C &check, T prod = ti()) {
assert(check(prod));
if (!t) return {nullptr, nullptr};
_push(t);
T p1 = f(prod, _sum(t->l));
if (check(p1)) {
prod = p1;
} else {
auto s = _split_max_right(t->l, check, prod);
t->l = s.second;
return {s.first, _update(t)};
}
prod = f(prod, t->val);
if (!check(prod)) {
Ptr tl = t->l;
t->l = nullptr;
return {tl, _update(t)};
}
p1 = f(prod, _sum(t->r));
if (check(p1)) {
return {t, nullptr};
} else {
auto s = _split_max_right(t->r, check, prod);
t->r = s.first;
return {_update(t), s.second};
}
}
// [i, inf) の prod について check(prod) の (false / true) で切る
template <typename C>
static pair<Ptr, Ptr> _split_min_left(Ptr t, const C &check, T prod = ti()) {
assert(check(prod));
if (!t) return {nullptr, nullptr};
_push(t);
T p1 = f(_sum(t->r), prod);
if (check(p1)) {
prod = p1;
} else {
auto s = _split_min_left(t->r, check, prod);
t->r = s.first;
return {_update(t), s.second};
}
prod = f(t->val, prod);
if (!check(prod)) {
Ptr tr = t->r;
t->r = nullptr;
return {_update(t), tr};
}
p1 = f(_sum(t->l), prod);
if (check(p1)) {
return {nullptr, t};
} else {
auto s = _split_min_left(t->l, check, prod);
t->l = s.second;
return {s.first, _update(t)};
}
}
// [l, inf) である地点に apply
static void _apply_left(Ptr t, I l, const E &e) {
if (!t) return;
_push(t);
if (t->key < l) {
_apply_left(t->r, l, e);
} else if (t->key == l) {
t->val = g(t->val, e);
_propagate(t->r, e);
} else {
_apply_left(t->l, l, e);
t->val = g(t->val, e);
_propagate(t->r, e);
}
_update(t);
}
// [-inf, r) である地点に apply
static void _apply_right(Ptr t, I r, const E &e) {
if (!t) return;
_push(t);
if (t->key < r) {
_propagate(t->l, e);
t->val = g(t->val, e);
_apply_right(t->r, r, e);
} else if (t->key == r) {
_propagate(t->l, e);
} else {
_apply_right(t->l, r, e);
}
_update(t);
}
// [l, r) に apply
static void _apply(Ptr t, I l, I r, const E &e) {
if (!t) return;
_push(t);
if (t->key < l) {
_apply(t->r, l, r, e);
} else if (t->key == l) {
t->val = g(t->val, e);
_apply_right(t->r, r, e);
} else if (t->key < r) {
_apply_left(t->l, l, e);
t->val = g(t->val, e);
_apply_right(t->r, r, e);
} else if (t->key == r) {
_apply_left(t->l, l, e);
} else {
_apply(t->l, l, r, e);
}
_update(t);
}
// l 以上
static T _fold_left(Ptr t, I l) {
if (!t) return ti();
_push(t);
if (t->key < l) {
return _fold_left(t->r, l);
} else if (t->key == l) {
return f(t->val, _fold_left(t->r, l));
} else {
T tl = _fold_left(t->l, l);
return f(f(tl, t->val), _sum(t->r));
}
}
// r 未満
static T _fold_right(Ptr t, I r) {
if (!t) return ti();
_push(t);
if (t->key < r) {
T tr = _fold_right(t->r, r);
return f(f(_sum(t->l), t->val), tr);
} else if (t->key == r) {
return _sum(t->l);
} else {
return _fold_right(t->l, r);
}
}
static T _fold(Ptr t, I l, I r) {
if (!t) return ti();
_push(t);
if (t->key < l) {
return _fold(t->r, l, r);
} else if (t->key == l) {
return f(t->val, _fold_right(t->r, r));
} else if (t->key < r) {
T tl = _fold_left(t->l, l);
T tr = _fold_right(t->r, r);
return f(f(tl, t->val), tr);
} else if (t->key == r) {
return _fold_left(t->l, l);
} else {
return _fold(t->l, l, r);
}
}
// t を根とする木の上で最小の key は？ (t が空の場合は failed)
static pair<I, T> _get_min_keyval(Ptr t, const I &failed) {
if (!t) return {failed, ti()};
while (t->l) _push(t), t = t->l;
return {t->key, t->val};
}
// t を根とする木の上で最小の key は？ (t が空の場合は failed)
static pair<I, T> _get_max_keyval(Ptr t, const I &failed) {
if (!t) return {failed, ti()};
while (t->r) _push(t), t = t->r;
return {t->key, t->val};
}
// t を根とする木のうち、[0, i の区間 fold が true になる最大の i は何か？
// exclusive かつ (空 または[0,右]が真の場合) の場合は failed(inf)
// inclusive かつ (空 または[0,0] が偽の場合) の場合は failed
template <typename C, bool exclusive>
static I _max_right(Ptr t, C check, const I &failed) {
if (!t) return failed;
_push(t);
Ptr now = t;
T prod_now = ti();
[[maybe_unused]] I prev = failed;
while (true) {
if (now->l != nullptr) {
_push(now->l);
auto pl = f(prod_now, now->l->sum);
if (check(pl)) {
prod_now = pl;
} else {
now = now->l;
continue;
}
}
auto pl = f(prod_now, now->val);
if (!check(pl)) {
if constexpr (exclusive) {
return now->key;
} else {
return now->l ? _get_max_keyval(now->l, failed).first : prev;
}
}
prod_now = pl;
if (now->r == nullptr) {
if constexpr (exclusive) {
return failed;
} else {
return now->key;
}
}
_push(now->r);
if constexpr (!exclusive) prev = now->key;
now = now->r;
}
}
// t を根とする木のうち、i, inf) の区間 fold が true になる最小の i は何か？
// inclusive かつ (空 または 存在しない) 場合は failed
// exlucisve かつ (空 または [左, inf) が真) の場合は failed
template <typename C, bool inclusive>
static I _min_left(Ptr t, C check, const I &failed) {
if (!t) return failed;
_push(t);
Ptr now = t;
T prod_now = ti();
[[maybe_unused]] I prev = failed;
while (true) {
if (now->r != nullptr) {
_push(now->r);
auto pr = f(now->r->sum, prod_now);
if (check(pr)) {
prod_now = pr;
} else {
now = now->r;
continue;
}
}
auto pr = f(now->val, prod_now);
if (!check(pr)) {
if constexpr (inclusive) {
return now->r ? _get_min_keyval(now->r, failed).first : prev;
} else {
return now->key;
}
}
prod_now = pr;
if (now->l == nullptr) {
if constexpr (inclusive) {
return now->key;
} else {
return failed;
}
}
_push(now->l);
if constexpr (inclusive) prev = now->key;
now = now->l;
}
}
static void _clear(Ptr t) {
if (!t) return;
if (t->l) _clear(t->l);
if (t->r) _clear(t->r);
_my_del(t);
}
static Ptr _deepcopy(Ptr t) {
if (!t) return nullptr;
Ptr u = _my_new(*t);
if (u->l) u->l = _deepcopy(u->l);
if (u->r) u->r = _deepcopy(u->r);
return u;
}
static void _dump(Ptr t) {
if (!t) return;
_push(t);
_dump(t->l);
cerr << "## key = " << t->key << ",";
cerr << "\tval = " << t->val << ", ";
cerr << "\tsum = " << t->sum << ", ";
cerr << "\tchild = ";
cerr << "( ";
if (t->l) cerr << t->l->key;
if (!t->l) cerr << "nil";
cerr << ", ";
if (t->r) cerr << t->r->key;
if (!t->r) cerr << "nil";
cerr << " )" << endl;
_dump(t->r);
}
static void _make_array(Ptr t, vector<pair<I, T>> &v) {
if (!t) return;
_push(t);
if (t->l) _make_array(t->l, v);
v.emplace_back(t->key, t->val);
if (t->r) _make_array(t->r, v);
}
public:
Ptr root;
RBSTSegmentTreeBase() : root(nullptr) {}
RBSTSegmentTreeBase(Ptr t) : root(t) {}
RBSTSegmentTreeBase(const vector<T> xs, const vector<I> &vals = {}) {
if (!vals.empty()) assert(xs.size() == vals.size());
int n = xs.size();
vector<pair<I, T>> dat(n);
for (int i = 0; i < n; i++) dat[i] = {vals.empty() ? i : vals[i], xs[i]};
root = _build(0, n, dat);
}
RBSTSegmentTreeBase(RBSTSegmentTreeBase &&rhs) noexcept { root = rhs.root; }
RBSTSegmentTreeBase(const RBSTSegmentTreeBase &rhs) { root = rhs.root; }
~RBSTSegmentTreeBase() = default;
using RBST = RBSTSegmentTreeBase;
RBST &operator=(RBST &&rhs) noexcept {
root = rhs.root;
return *this;
}
RBST &operator=(const RBST &rhs) {
root = rhs.root;
return *this;
}
RBST deepcopy() { return _deepcopy(root); }
friend void swap(RBST &lhs, RBST &rhs) { swap(lhs.root, rhs.root); }
void swap(RBST &rhs) { swap(root, rhs.root); }
// destructive ordered _merge (max(lhs) < min(rhs))
friend RBST ordered_merge(RBST &lhs, RBST &rhs) {
assert(lhs.get_max_key() < rhs.get_min_key());
return RBST{_merge(lhs.root, rhs.root)};
}
// 1 点 値の書き換え
void set_val(I i, T x) {
auto s = _split_by_key3(root, i);
if (s[1] == nullptr) {
s[1] = _my_new(i, x);
} else {
s[1]->val = x;
}
root = _merge(_merge(s[0], _update(s[1])), s[2]);
}
// すでに要素が存在するときに値を set する。おそらく少し早い
void set_val_fast(I i, T x) {
static vector<Ptr> ps;
ps.clear();
Ptr t = root;
while (t) {
_push(t);
ps.push_back(t);
if (i == t->key) break;
t = i < t->key ? t->l : t->r;
}
if (!t) {
set_val(i, x);
return;
}
t->val = x;
for (int j = ps.size() - 1; j >= 0; j--) _update(ps[j]);
}
// 1 点取得
T get_val(I i) {
Ptr p = _find(root, i);
return p ? p->val : ti();
}
bool exist(I i) {
Ptr p = _find(root, i);
return p != nullptr;
}
// 1 点 値の書き換え
// func の返り値は void !!!!!!(参照された値を直接更新する)
template <typename F>
auto apply_val(I i, F&& func)
-> enable_if_t<is_invocable_r_v<void, F&, T&>> {
auto s = _split_by_key3(root, i);
if (s[1] == nullptr) s[1] = _my_new(i);
std::invoke(func, s[1]->val);
root = _merge(_merge(s[0], _update(s[1])), s[2]);
}
// 1 点 値の書き換え 値が既に存在するときに早い
// func の返り値は void !!!!!!(参照された値を直接更新する)
template <typename F>
auto apply_val_fast(I i, F&& func)
-> enable_if_t<is_invocable_r_v<void, F&, T&>> {
static vector<Ptr> ps;
ps.clear();
Ptr t = root;
while (t) {
_push(t);
ps.push_back(t);
if (i == t->key) break;
t = i < t->key ? t->l : t->r;
}
if (!t) {
apply_val(i, func);
return;
}
std::invoke(func, t->val);
for (int j = ps.size() - 1; j >= 0; j--) _update(ps[j]);
}
// 頂点の削除
virtual void erase(I i) { _erase(root, i); }
// 範囲作用
void apply(I l, I r, const E &e) {
if (l >= r) return;
_apply(root, l, r, e);
}
void apply_all(const E &e) { _propagate(root, e); }
// 範囲取得
T fold(I l, I r) {
if (l >= r) return ti();
return _fold(root, l, r);
}
T fold_all() { return _sum(root); }
void shift(const I &sh) { _shift(root, sh); }
// key 最小を取得
I get_min_key(I failed = {}) { return _get_min_keyval(root, failed).first; }
// key 最大を取得
I get_max_key(I failed = {}) { return _get_max_keyval(root, failed).first; }
// (key, val) 最小を取得
pair<I, T> get_min_keyval(I failed = {}) {
return _get_min_keyval(root, failed);
}
// (key, val) 最大を取得
pair<I, T> get_max_keyval(I failed = {}) {
return _get_max_keyval(root, failed);
}
// (key, val) 最小を pop
pair<I, T> pop_min_keyval(I failed = {}) {
assert(root != nullptr);
auto kv = _get_min_keyval(root, failed);
erase(kv.first);
return kv;
}
// (key, val) 最大を取得
pair<I, T> pop_max_keyval(I failed = {}) {
assert(root != nullptr);
auto kv = _get_max_keyval(root, failed);
erase(kv.first);
return kv;
}
// n 未満の i のうち、[i, n) の区間 fold が true になる最小の i は何か？
// (存在しない場合は failed を返す)
template <typename C>
I min_left(I n, C check, I failed) {
assert(check(ti()) == true);
auto [x, y] = _split_by_key(root, n);
I res = _min_left<C, true>(x, check, failed);
root = _merge(x, y);
return res;
}
// n 未満の i のうち、(i, n) の区間 fold が true になる最小の i は何か？
// (空だったり (左端, n) が 真の場合は minus_infty を返す)
template <typename C>
I min_left_exclusive(I n, C check, I minus_infty) {
assert(check(ti()) == true);
auto [x, y] = _split_by_key(root, n);
I res = _min_left<C, false>(x, check, minus_infty);
root = _merge(x, y);
return res;
}
// n 以上の i のうち、[n, i) の区間 fold が true になる最大の i は何か？
// (空だったり [n, 右端] が true の場合は infty を返す)
template <typename C>
I max_right(I n, C check, I infty) {
assert(check(ti()) == true);
auto [x, y] = _split_by_key(root, n);
I res = _max_right<C, true>(y, check, infty);
root = _merge(x, y);
return res;
}
// n 以上の i のうち、[n, i] の区間 fold が true になる最大の i は何か？
// (存在しない場合は failed を返す)
template <typename C>
I max_right_inclusive(I n, C check, I failed) {
assert(check(ti()) == true);
auto [x, y] = _split_by_key(root, n);
I res = _max_right<C, false>(y, check, failed);
root = _merge(x, y);
return res;
}
// (key 未満, key 以上) で分割
// 呼び出し後のオブジェクトは空のセグ木になる
pair<RBST, RBST> split_by_key(const I &key) {
auto [x, y] = _split_by_key(root, key);
root = nullptr;
return make_pair(RBST{x}, RBST{y});
}
// [i, inf) の区間積が (false, true) になる境界で分割
// 呼び出し後のオブジェクトは空のセグ木になる
template <typename C>
pair<RBST, RBST> split_min_left(const C &check) {
assert(check(ti()) == true);
auto [x, y] = _split_min_left(root, check);
root = nullptr;
return make_pair(RBST{x}, RBST{y});
}
// (-inf, i] の区間積が (true, false) になる境界で分割
// 呼び出し後のオブジェクトは空のセグ木になる
template <typename C>
pair<RBST, RBST> split_max_right(const C &check) {
assert(check(ti()) == true);
auto [x, y] = _split_max_right(root, check);
root = nullptr;
return make_pair(RBST{x}, RBST{y});
}
void clear() { _clear(root), root = nullptr; }
int size() { return _count(root); }
bool empty() { return !root; }
void dump() {
cerr << "***** dump start *****" << endl;
_dump(root);
cerr << "****** dump end ******" << endl;
}
// 列を配列に変換して返す
vector<pair<I, T>> make_array() {
vector<pair<I, T>> res;
_make_array(root, res);
return res;
}
};
namespace RBSTSegmentTreeImpl {
bool _ei() { return false; }
template <typename I, typename T, typename E, T (*f)(T, T), T (*g)(T, E),
E (*h)(E, E), T (*ti)(), E (*ei)()>
struct ShiftableLazySegNode {
ShiftableLazySegNode *l, *r;
I key, shift;
T val, sum;
E lazy;
int cnt;
ShiftableLazySegNode(const I &i, const T &t = ti())
: l(), r(), key(i), shift(I{}), val(t), sum(t), lazy(ei()), cnt(1) {}
};
template <typename I, typename T, typename E, T (*f)(T, T), T (*g)(T, E),
E (*h)(E, E), T (*ti)(), E (*ei)()>
using RBSTShiftableLazySegmentTree =
RBSTSegmentTreeBase<ShiftableLazySegNode<I, T, E, f, g, h, ti, ei>, I, T, E,
f, g, h, ti, ei>;
template <typename I, typename T, typename E, T (*f)(T, T), T (*g)(T, E),
E (*h)(E, E), T (*ti)(), E (*ei)()>
struct LazySegNode {
LazySegNode *l, *r;
I key;
T val, sum;
E lazy;
int cnt;
LazySegNode(const I &i, const T &t = ti())
: l(), r(), key(i), val(t), sum(t), lazy(ei()), cnt(1) {}
};
template <typename I, typename T, typename E, T (*f)(T, T), T (*g)(T, E),
E (*h)(E, E), T (*ti)(), E (*ei)()>
using RBSTLazySegmentTree =
RBSTSegmentTreeBase<LazySegNode<I, T, E, f, g, h, ti, ei>, I, T, E, f, g, h,
ti, ei>;
template <typename I, typename T, T (*f)(T, T), T (*ti)()>
struct SegNode {
SegNode *l, *r;
I key;
T val, sum;
int cnt;
SegNode(const I &i, const T &t = ti())
: l(), r(), key(i), val(t), sum(t), cnt(1) {}
};
template <typename I, typename T, T (*f)(T, T), T (*ti)()>
using RBSTSegmentTree = RBSTSegmentTreeBase<SegNode<I, T, f, ti>, I, T, bool, f,
nullptr, nullptr, ti, _ei>;
} // namespace RBSTSegmentTreeImpl
using RBSTSegmentTreeImpl::RBSTLazySegmentTree;
using RBSTSegmentTreeImpl::RBSTSegmentTree;
using RBSTSegmentTreeImpl::RBSTShiftableLazySegmentTree;
/**
* @brief RBST-based Dynamic Lazy Segment Tree
*/
#line 8 "data-structure/slope-trick-weighted.hpp"
namespace SlopeTrickImpl {
template <typename Int>
using T = pair<Int, Int>;
template <typename Int>
using E = Int;
template <typename Int>
T<Int> f(T<Int> a, T<Int> b) {
return {a.first + b.first, a.second + b.second};
}
template <typename Int>
T<Int> g(T<Int> a, E<Int> b) {
return {a.first, a.second + a.first * b};
}
template <typename Int>
E<Int> h(E<Int> a, E<Int> b) {
return a + b;
}
template <typename Int>
T<Int> ti() {
return {};
}
template <typename Int>
E<Int> ei() {
return {};
}
template <typename Int>
using SegTree = RBSTShiftableLazySegmentTree<Int, T<Int>, E<Int>, f<Int>,
g<Int>, h<Int>, ti<Int>, ei<Int>>;
} // namespace SlopeTrickImpl
template <typename I>
struct WeightedSlopeTrick {
static constexpr I inf = (I{1} << (sizeof(I) * 8 - 2)) - 1;
using Seg = typename SlopeTrickImpl::SegTree<I>;
using T = SlopeTrickImpl::T<I>;
using E = SlopeTrickImpl::E<I>;
// x : 座標, c : 傾きの変化量
struct P {
I x, c;
P(I _x, I _c) : x(_x), c(_c) {}
};
private:
Seg L, R;
I min_y;
// seg[x] += c
void _apply(Seg& seg, I x, I c) {
if (c == 0) return;
seg.apply_val_fast(x, [&](T& t) {
t.first += c;
t.second += x * c;
});
}
void _push_L(I x, I c = 1) { _apply(L, x, c); }
void _push_R(I x, I c = 1) { _apply(R, x, c); }
P _get_L() {
assert(!L.empty());
auto kv = L.get_max_keyval();
return P{kv.first, kv.second.first};
}
P _get_R() {
assert(!R.empty());
auto kv = R.get_min_keyval();
return P{kv.first, kv.second.first};
}
P _getpop_L() {
assert(!L.empty());
auto kv = L.pop_max_keyval();
return P{kv.first, kv.second.first};
}
P _getpop_R() {
assert(!R.empty());
auto kv = R.pop_min_keyval();
return P{kv.first, kv.second.first};
}
pair<Seg, Seg> _split_L(I c) {
assert(L.fold_all().first >= c);
pair<Seg, Seg> res =
L.split_min_left([&](const T& t) { return t.first <= c; });
c -= res.second.fold_all().first;
if (c != 0) {
I k = res.first.get_max_key();
_apply(res.first, k, -c);
_apply(res.second, k, c);
}
return res;
}
pair<Seg, Seg> _split_R(I c) {
assert(R.fold_all().first >= c);
pair<Seg, Seg> res =
R.split_max_right([&](const T& t) { return t.first <= c; });
c -= res.first.fold_all().first;
if (c != 0) {
I k = res.second.get_min_key();
_apply(res.first, k, c);
_apply(res.second, k, -c);
}
return res;
}
// destructive merge
Seg _unite(Seg& lhs, Seg& rhs) {
if (lhs.empty()) return rhs;
if (rhs.empty()) return lhs;
assert(lhs.get_max_key() <= rhs.get_min_key() && "WeightSlopTrick::_unite");
if (lhs.get_max_key() == rhs.get_min_key()) {
auto [x, p] = lhs.pop_max_keyval();
_apply(rhs, x, p.first);
}
if (lhs.empty()) return rhs;
if (rhs.empty()) return lhs;
return ordered_merge(lhs, rhs);
}
public:
WeightedSlopeTrick() : min_y(0) {}
void debug() {
auto LL = L.make_array(), RR = R.make_array();
cerr << "L : ";
for (auto& [k, v] : LL) cerr << "( " << k << ", " << v.first << " ), ";
cerr << endl << "R : ";
for (auto& [k, v] : RR) cerr << "( " << k << ", " << v.first << " ), ";
cerr << endl << "min : ( ";
cerr << get_min().first << ", " << get_min().second << " )" << endl;
}
// return {x, y} s.t. {argmin, min}
pair<I, I> get_min() {
I x = L.empty() ? R.empty() ? 0 : _get_R().x : _get_L().x;
return {x, min_y};
}
void shift_L(I a) { L.shift(a), L.apply_all(a); }
void shift_R(I a) { R.shift(a), R.apply_all(a); }
// f(x) <- f(x - a)
void shift_x(I a) { shift_L(a), shift_R(a); }
// f(x) <- f(x) + a
void shift_y(I a) { min_y += a; }
// add (x-a)_+ _____/
void add_xma(I a, I c = 1) {
_apply(L, a, c);
auto [L1, L2] = _split_L(c);
auto [c_sum, ac_sum] = L2.fold(a, inf);
min_y += ac_sum - c_sum * a;
auto c2 = L2.get_val(a);
L2.erase(a);
R = _unite(L2, R);
_apply(R, a, c2.first);
L = L1;
}
// add (a-x)_+ \_____
void add_amx(I a, I c = 1) {
_apply(R, a, c);
auto [R1, R2] = _split_R(c);
auto [c_sum, ac_sum] = R1.fold(-inf, a);
min_y += c_sum * a - ac_sum;
auto c2 = R1.get_val(a);
R1.erase(a);
L = _unite(L, R1);
_apply(L, a, c2.first);
R = R2;
}
// add |x-a| \____/
void add_abs_xma(I a, I c = 1) {
add_xma(a, c);
add_amx(a, c);
}
// chmin right side \_/ -> \__
void chmin_right() { R.clear(); }
// chmin left side \_/ -> __/
void chmin_left() { L.clear(); }
// destructive merge
void merge(WeightedSlopeTrick& r) {
if (L.size() + R.size() < r.L.size() + r.R.size()) swap(*this, r);
for (auto& [x, t] : r.L.make_array()) add_amx(x, t.first);
for (auto& [x, t] : r.R.make_array()) add_xma(x, t.first);
shift_y(r.min_y);
}
I eval(I x) {
I res = min_y;
if (!L.empty() && _get_L().x > x) {
auto [L1, L2] = L.split_by_key(x);
auto [c_sum, xc_sum] = L2.fold_all();
res += xc_sum - c_sum * x;
L = _unite(L1, L2);
}
if (!R.empty() && _get_R().x < x) {
auto [R1, R2] = R.split_by_key(x);
auto [c_sum, xc_sum] = R1.fold_all();
res += c_sum * x - xc_sum;
R = _unite(R1, R2);
}
return res;
}
void clear() { L.clear(), R.clear(), min_y = 0; }
};
using SlopeTrick = WeightedSlopeTrick<__int128_t>;
/**
* @brief Weighted Slope Trick
*/
Weighted Slope Trick