cp-library
競技プログラミング用 C++ ライブラリ
Dynamic Segtree
data_structure/dynamic_segtree.hpp · GitHub で見る
共通ページ
#include "lib/data_structure/dynamic_segtree.hpp"
long long op(long long a, long long b) { return a + b; }
long long e() { return 0; }
DynamicSegtree<long long, op, e> seg(1LL << 60);
seg.reserve(200000);
seg.set(123, 10);
seg.multiply(123, 5);
auto x = seg.get(123);
auto sum = seg.prod(0, 1000);
auto r = seg.max_right(0, [](long long s) { return s <= 10; });
| 操作 | 内容・返り値 | 計算量 |
|---|---|---|
DynamicSegtree(n = 0) / DynamicSegtree(l,r)
|
\([0,n)\)／\([l,r)\) を単位元で初期化 | \(O(1)\) |
set(p,x) |
座標 p に代入 |
償却 \(O(\min(B,k))\) |
multiply(p,x) |
op(現在値,x) に更新 |
償却 \(O(\min(B,k))\) |
get(p) |
1 点の値。未更新なら e()
|
\(O(\min(B,k))\) |
prod(l,r) / all_prod()
|
\([l,r)\)／全体の集約 | prod: \(O(\min(B,k))\)、all_prod: \(O(1)\) |
max_right(l,f) / min_left(r,f)
|
区間積が f を満たす最大の右端／最小の左端 |
\(O(\min(B,k))\) |
size() / left_bound() / right_bound()
|
座標範囲の長さ／左端／右端 | \(O(1)\) |
node_count() |
更新済みの異なる座標数 | \(O(1)\) |
reserve(k) / clear()
|
領域予約／容量を残して全座標を単位元に戻す | \(O(k)\) |
memory_usage() |
確保済みノード領域のバイト数 | \(O(1)\) |
swap(other) |
内容と座標範囲を交換 | \(O(1)\) |
op は結合的な演算、e() は単位元です。非可換な演算にも対応します。未更新の座標の値は e() です。
構築範囲は DynamicSegtree(n) が \([0,n)\)、DynamicSegtree(l,r) が \([l,r)\) です。座標型は第 4 テンプレート引数で指定でき、既定は std::int64_t です。
set(p,x) は代入、multiply(p,x) は現在値を op(現在値,x) に更新します。prod(l,r) は \([l,r)\) の積、all_prod() は全体の積です。
max_right(l,f)・min_left(r,f) は区間積が f を満たす境界を返します。f(e()) が真で、区間を伸ばすと真から偽へ単調に変化する必要があります。
reserve(k) は更新する異なる座標数を目安に領域を予約します。node_count() はその個数、memory_usage() は確保済みノード領域のバイト数です。更新する異なる座標数の上限は \(2^{26}-1\) です。clear() は確保領域を残して初期状態に戻します。
座標型の bit 数を \(B\)、更新済み座標数を \(k\) として、更新は償却 \(O(\min(B,k))\)、get・prod・境界探索は \(O(\min(B,k))\) です。構築・all_prod・サイズ取得は \(O(1)\)、reserve・clear は \(O(k)\) です。空間は \(O(k)\)（予約容量が大きい場合はその容量分）です。
コード
#ifndef DYNAMIC_SEGTREE_HPP
#define DYNAMIC_SEGTREE_HPP
#include <algorithm>
#include <array>
#include <bit>
#include <cassert>
#include <cstddef>
#include <cstdint>
#include <limits>
#include <type_traits>
#include <utility>
#include <vector>
template <class S, auto op, auto e, class Coord = std::int64_t>
class DynamicSegtree {
static_assert(std::is_integral_v<Coord> && !std::is_same_v<Coord, bool> && sizeof(Coord) <= 8);
static constexpr unsigned B = 8, limit = (1u << 26) - 1;
using U = std::make_unsigned_t<Coord>;
static constexpr unsigned tag = 1u << 31;
static constexpr U sign = std::is_signed_v<Coord> ? U(1) << (std::numeric_limits<U>::digits - 1) : U(0);
template <std::size_t... I>
static auto units(std::index_sequence<I...>) { return std::array<S, sizeof...(I)>{(void(I), e())...}; }
struct Leaf {
std::array<U, B> keys{};
std::array<S, B> values = units(std::make_index_sequence<B>{});
S product = e();
unsigned count = 0;
};
struct Node {
U middle = 0;
unsigned left = 0, right = 0;
S product = e();
};
std::vector<Leaf> leaves;
std::vector<Node> nodes{Node{}};
unsigned root = 0, points = 0;
Coord first, last;
static U encode(Coord x) { return U(x) ^ sign; }
static Coord decode(U x) { return Coord(x ^ sign); }
Leaf &leaf(unsigned id) { return leaves[id & ~tag]; }
const Leaf &leaf(unsigned id) const { return leaves[id & ~tag]; }
const S &product(unsigned id) const { return id & tag ? leaf(id).product : nodes[id].product; }
unsigned make_leaf() {
leaves.emplace_back();
return unsigned(leaves.size() - 1) | tag;
}
unsigned make_node(U middle, unsigned left, unsigned right) {
S value = op(product(left), product(right));
nodes.push_back({middle, left, right, std::move(value)});
return unsigned(nodes.size() - 1);
}
template <unsigned N>
[[gnu::always_inline]] static inline S fold(const S *values, unsigned count) {
if constexpr (N == 1) return values[0];
else {
if (count <= N / 2) return fold<N / 2>(values, count);
return op(fold<N / 2>(values, N / 2), fold<N / 2>(values + N / 2, count - N / 2));
}
}
static void pull(Leaf &node) { node.product = fold<B>(node.values.data(), node.count); }
template <bool Append>
static void write(Leaf &node, unsigned p, U key, S value) {
if (p < node.count && node.keys[p] == key) {
if constexpr (Append) node.values[p] = op(node.values[p], value);
else node.values[p] = std::move(value);
} else {
std::move_backward(node.keys.begin() + p, node.keys.begin() + node.count, node.keys.begin() + node.count + 1);
std::move_backward(node.values.begin() + p, node.values.begin() + node.count, node.values.begin() + node.count + 1);
node.keys[p] = key;
node.values[p] = std::move(value);
++node.count;
}
pull(node);
}
template <bool Append>
void update(U key, S value) {
if (!root) {
root = make_leaf();
write<Append>(leaf(root), 0, key, std::move(value));
++points;
return;
}
std::array<unsigned, 64> path;
unsigned depth = 0, id = root;
bool outside = false;
while (!(id & tag)) {
unsigned h = std::countr_zero(nodes[id].middle);
if (((key ^ nodes[id].middle) >> h) > 1) { outside = true; break; }
assert(depth < path.size());
path[depth++] = id;
id = key < nodes[id].middle ? nodes[id].left : nodes[id].right;
}
unsigned replacement = id;
if (outside) {
assert(points < limit);
unsigned added = make_leaf();
write<Append>(leaf(added), 0, key, std::move(value));
unsigned h = std::bit_width(U(key ^ nodes[id].middle)) - 1;
U middle = ((key | nodes[id].middle) >> h) << h;
replacement = key < middle ? make_node(middle, added, id) : make_node(middle, id, added);
++points;
} else {
auto &node = leaf(id);
unsigned p = std::lower_bound(node.keys.begin(), node.keys.begin() + node.count, key) - node.keys.begin();
bool exists = p < node.count && node.keys[p] == key;
assert(exists || points < limit);
if (exists || node.count < B) write<Append>(node, p, key, std::move(value));
else {
U low = std::min(key, node.keys[0]), high = std::max(key, node.keys[B - 1]);
unsigned h = std::bit_width(U(low ^ high)) - 1;
U middle = ((low | high) >> h) << h;
unsigned cut = std::lower_bound(node.keys.begin(), node.keys.end(), middle) - node.keys.begin();
unsigned added = make_leaf();
auto &a = leaf(id), &b = leaf(added);
b.count = B - cut;
for (unsigned i = 0; i < b.count; ++i) b.keys[i] = a.keys[cut + i], b.values[i] = std::move(a.values[cut + i]);
a.count = cut;
if (key < middle) write<Append>(a, p, key, std::move(value));
else write<Append>(b, p - cut, key, std::move(value));
if (key < middle) pull(b);
else pull(a);
replacement = make_node(middle, id, added);
}
points += !exists;
}
if (depth) {
auto parent = path[depth - 1];
if (key < nodes[parent].middle) nodes[parent].left = replacement;
else nodes[parent].right = replacement;
} else root = replacement;
while (depth) {
unsigned k = path[--depth];
nodes[k].product = op(product(nodes[k].left), product(nodes[k].right));
}
}
S prefix(unsigned id, U r) const {
S result = e();
while (!(id & tag)) {
if (r <= nodes[id].middle) id = nodes[id].left;
else { result = op(result, product(nodes[id].left)); id = nodes[id].right; }
}
const auto &node = leaf(id);
for (unsigned i = 0; i < node.count && node.keys[i] < r; ++i) result = op(result, node.values[i]);
return result;
}
S suffix(unsigned id, U l) const {
S result = e();
while (!(id & tag)) {
if (nodes[id].middle <= l) id = nodes[id].right;
else { result = op(product(nodes[id].right), result); id = nodes[id].left; }
}
const auto &node = leaf(id);
for (unsigned i = node.count; i && l <= node.keys[i - 1];) result = op(node.values[--i], result);
return result;
}
template <class Pred>
bool search_right(unsigned id, U lower, U l, S &value, Pred &f, Coord &answer) const {
if (l <= lower) {
S next = op(value, product(id));
if (f(next)) { value = std::move(next); return false; }
}
if (id & tag) {
const auto &node = leaf(id);
unsigned p = std::lower_bound(node.keys.begin(), node.keys.begin() + node.count, l) - node.keys.begin();
for (; p < node.count; ++p) {
S next = op(value, node.values[p]);
if (!f(next)) { answer = decode(node.keys[p]); return true; }
value = std::move(next);
}
return false;
}
U middle = nodes[id].middle;
if (l < middle && search_right(nodes[id].left, lower, l, value, f, answer)) return true;
return search_right(nodes[id].right, middle, l, value, f, answer);
}
template <class Pred>
bool search_left(unsigned id, U upper, U r, S &value, Pred &f, Coord &answer) const {
if (upper <= r) {
S next = op(product(id), value);
if (f(next)) { value = std::move(next); return false; }
}
if (id & tag) {
const auto &node = leaf(id);
unsigned p = std::lower_bound(node.keys.begin(), node.keys.begin() + node.count, r) - node.keys.begin();
while (p) {
S next = op(node.values[--p], value);
if (!f(next)) { answer = decode(node.keys[p]) + 1; return true; }
value = std::move(next);
}
return false;
}
U middle = nodes[id].middle;
if (middle < r && search_left(nodes[id].right, upper, r, value, f, answer)) return true;
return search_left(nodes[id].left, middle, r, value, f, answer);
}
public:
using value_type = S;
using coordinate_type = Coord;
explicit DynamicSegtree(Coord n = 0) : DynamicSegtree(0, n) {}
DynamicSegtree(Coord l, Coord r) : first(l), last(r) { assert(l <= r); }
DynamicSegtree(const DynamicSegtree &) = default;
DynamicSegtree &operator=(const DynamicSegtree &) = default;
DynamicSegtree(DynamicSegtree &&other) : DynamicSegtree(other.first, other.last) { swap(other); }
DynamicSegtree &operator=(DynamicSegtree &&other) {
if (this != &other) { DynamicSegtree moved(std::move(other)); swap(moved); }
return *this;
}
void swap(DynamicSegtree &other) noexcept {
leaves.swap(other.leaves);
nodes.swap(other.nodes);
std::swap(root, other.root);
std::swap(points, other.points);
std::swap(first, other.first);
std::swap(last, other.last);
}
void reserve(std::size_t n) {
assert(n <= limit);
std::size_t capacity = (3 * n + 2 * B - 1) / (2 * B);
leaves.reserve(capacity);
nodes.reserve(capacity + 1);
}
std::size_t node_count() const { return points; }
std::size_t memory_usage() const { return leaves.capacity() * sizeof(Leaf) + nodes.capacity() * sizeof(Node); }
U size() const { return U(last) - U(first); }
Coord left_bound() const { return first; }
Coord right_bound() const { return last; }
void clear() { leaves.clear(); nodes.erase(nodes.begin() + 1, nodes.end()); root = points = 0; }
void set(Coord p, S value) { assert(first <= p && p < last); update<false>(encode(p), std::move(value)); }
void multiply(Coord p, S value) { assert(first <= p && p < last); update<true>(encode(p), std::move(value)); }
S get(Coord p) const {
assert(first <= p && p < last);
if (!root) return e();
U key = encode(p);
unsigned id = root;
while (!(id & tag)) id = key < nodes[id].middle ? nodes[id].left : nodes[id].right;
const auto &node = leaf(id);
unsigned i = std::lower_bound(node.keys.begin(), node.keys.begin() + node.count, key) - node.keys.begin();
return i < node.count && node.keys[i] == key ? node.values[i] : e();
}
S prod(Coord l, Coord r) const {
assert(first <= l && l <= r && r <= last);
if (!root || l == r) return e();
if (l == first && r == last) return product(root);
U a = encode(l), b = encode(r);
unsigned id = root;
while (!(id & tag)) {
U middle = nodes[id].middle;
if (b <= middle) id = nodes[id].left;
else if (middle <= a) id = nodes[id].right;
else return op(suffix(nodes[id].left, a), prefix(nodes[id].right, b));
}
const auto &node = leaf(id);
unsigned i = std::lower_bound(node.keys.begin(), node.keys.begin() + node.count, a) - node.keys.begin();
S result = e();
for (; i < node.count && node.keys[i] < b; ++i) result = op(result, node.values[i]);
return result;
}
S all_prod() const { return root ? product(root) : e(); }
template <class Pred>
Coord max_right(Coord l, Pred f) const {
assert(first <= l && l <= last && f(e()));
Coord answer = last;
if (root && l != last) { S value = e(); search_right(root, encode(first), encode(l), value, f, answer); }
return answer;
}
template <class Pred>
Coord min_left(Coord r, Pred f) const {
assert(first <= r && r <= last && f(e()));
Coord answer = first;
if (root && r != first) { S value = e(); search_left(root, encode(last), encode(r), value, f, answer); }
return answer;
}
};
#endif
#line 1 "data_structure/dynamic_segtree.hpp"
#include <algorithm>
#include <array>
#include <bit>
#include <cassert>
#include <cstddef>
#include <cstdint>
#include <limits>
#include <type_traits>
#include <utility>
#include <vector>
template <class S, auto op, auto e, class Coord = std::int64_t>
class DynamicSegtree {
static_assert(std::is_integral_v<Coord> && !std::is_same_v<Coord, bool> && sizeof(Coord) <= 8);
static constexpr unsigned B = 8, limit = (1u << 26) - 1;
using U = std::make_unsigned_t<Coord>;
static constexpr unsigned tag = 1u << 31;
static constexpr U sign = std::is_signed_v<Coord> ? U(1) << (std::numeric_limits<U>::digits - 1) : U(0);
template <std::size_t... I>
static auto units(std::index_sequence<I...>) { return std::array<S, sizeof...(I)>{(void(I), e())...}; }
struct Leaf {
std::array<U, B> keys{};
std::array<S, B> values = units(std::make_index_sequence<B>{});
S product = e();
unsigned count = 0;
};
struct Node {
U middle = 0;
unsigned left = 0, right = 0;
S product = e();
};
std::vector<Leaf> leaves;
std::vector<Node> nodes{Node{}};
unsigned root = 0, points = 0;
Coord first, last;
static U encode(Coord x) { return U(x) ^ sign; }
static Coord decode(U x) { return Coord(x ^ sign); }
Leaf &leaf(unsigned id) { return leaves[id & ~tag]; }
const Leaf &leaf(unsigned id) const { return leaves[id & ~tag]; }
const S &product(unsigned id) const { return id & tag ? leaf(id).product : nodes[id].product; }
unsigned make_leaf() {
leaves.emplace_back();
return unsigned(leaves.size() - 1) | tag;
}
unsigned make_node(U middle, unsigned left, unsigned right) {
S value = op(product(left), product(right));
nodes.push_back({middle, left, right, std::move(value)});
return unsigned(nodes.size() - 1);
}
template <unsigned N>
[[gnu::always_inline]] static inline S fold(const S *values, unsigned count) {
if constexpr (N == 1) return values[0];
else {
if (count <= N / 2) return fold<N / 2>(values, count);
return op(fold<N / 2>(values, N / 2), fold<N / 2>(values + N / 2, count - N / 2));
}
}
static void pull(Leaf &node) { node.product = fold<B>(node.values.data(), node.count); }
template <bool Append>
static void write(Leaf &node, unsigned p, U key, S value) {
if (p < node.count && node.keys[p] == key) {
if constexpr (Append) node.values[p] = op(node.values[p], value);
else node.values[p] = std::move(value);
} else {
std::move_backward(node.keys.begin() + p, node.keys.begin() + node.count, node.keys.begin() + node.count + 1);
std::move_backward(node.values.begin() + p, node.values.begin() + node.count, node.values.begin() + node.count + 1);
node.keys[p] = key;
node.values[p] = std::move(value);
++node.count;
}
pull(node);
}
template <bool Append>
void update(U key, S value) {
if (!root) {
root = make_leaf();
write<Append>(leaf(root), 0, key, std::move(value));
++points;
return;
}
std::array<unsigned, 64> path;
unsigned depth = 0, id = root;
bool outside = false;
while (!(id & tag)) {
unsigned h = std::countr_zero(nodes[id].middle);
if (((key ^ nodes[id].middle) >> h) > 1) { outside = true; break; }
assert(depth < path.size());
path[depth++] = id;
id = key < nodes[id].middle ? nodes[id].left : nodes[id].right;
}
unsigned replacement = id;
if (outside) {
assert(points < limit);
unsigned added = make_leaf();
write<Append>(leaf(added), 0, key, std::move(value));
unsigned h = std::bit_width(U(key ^ nodes[id].middle)) - 1;
U middle = ((key | nodes[id].middle) >> h) << h;
replacement = key < middle ? make_node(middle, added, id) : make_node(middle, id, added);
++points;
} else {
auto &node = leaf(id);
unsigned p = std::lower_bound(node.keys.begin(), node.keys.begin() + node.count, key) - node.keys.begin();
bool exists = p < node.count && node.keys[p] == key;
assert(exists || points < limit);
if (exists || node.count < B) write<Append>(node, p, key, std::move(value));
else {
U low = std::min(key, node.keys[0]), high = std::max(key, node.keys[B - 1]);
unsigned h = std::bit_width(U(low ^ high)) - 1;
U middle = ((low | high) >> h) << h;
unsigned cut = std::lower_bound(node.keys.begin(), node.keys.end(), middle) - node.keys.begin();
unsigned added = make_leaf();
auto &a = leaf(id), &b = leaf(added);
b.count = B - cut;
for (unsigned i = 0; i < b.count; ++i) b.keys[i] = a.keys[cut + i], b.values[i] = std::move(a.values[cut + i]);
a.count = cut;
if (key < middle) write<Append>(a, p, key, std::move(value));
else write<Append>(b, p - cut, key, std::move(value));
if (key < middle) pull(b);
else pull(a);
replacement = make_node(middle, id, added);
}
points += !exists;
}
if (depth) {
auto parent = path[depth - 1];
if (key < nodes[parent].middle) nodes[parent].left = replacement;
else nodes[parent].right = replacement;
} else root = replacement;
while (depth) {
unsigned k = path[--depth];
nodes[k].product = op(product(nodes[k].left), product(nodes[k].right));
}
}
S prefix(unsigned id, U r) const {
S result = e();
while (!(id & tag)) {
if (r <= nodes[id].middle) id = nodes[id].left;
else { result = op(result, product(nodes[id].left)); id = nodes[id].right; }
}
const auto &node = leaf(id);
for (unsigned i = 0; i < node.count && node.keys[i] < r; ++i) result = op(result, node.values[i]);
return result;
}
S suffix(unsigned id, U l) const {
S result = e();
while (!(id & tag)) {
if (nodes[id].middle <= l) id = nodes[id].right;
else { result = op(product(nodes[id].right), result); id = nodes[id].left; }
}
const auto &node = leaf(id);
for (unsigned i = node.count; i && l <= node.keys[i - 1];) result = op(node.values[--i], result);
return result;
}
template <class Pred>
bool search_right(unsigned id, U lower, U l, S &value, Pred &f, Coord &answer) const {
if (l <= lower) {
S next = op(value, product(id));
if (f(next)) { value = std::move(next); return false; }
}
if (id & tag) {
const auto &node = leaf(id);
unsigned p = std::lower_bound(node.keys.begin(), node.keys.begin() + node.count, l) - node.keys.begin();
for (; p < node.count; ++p) {
S next = op(value, node.values[p]);
if (!f(next)) { answer = decode(node.keys[p]); return true; }
value = std::move(next);
}
return false;
}
U middle = nodes[id].middle;
if (l < middle && search_right(nodes[id].left, lower, l, value, f, answer)) return true;
return search_right(nodes[id].right, middle, l, value, f, answer);
}
template <class Pred>
bool search_left(unsigned id, U upper, U r, S &value, Pred &f, Coord &answer) const {
if (upper <= r) {
S next = op(product(id), value);
if (f(next)) { value = std::move(next); return false; }
}
if (id & tag) {
const auto &node = leaf(id);
unsigned p = std::lower_bound(node.keys.begin(), node.keys.begin() + node.count, r) - node.keys.begin();
while (p) {
S next = op(node.values[--p], value);
if (!f(next)) { answer = decode(node.keys[p]) + 1; return true; }
value = std::move(next);
}
return false;
}
U middle = nodes[id].middle;
if (middle < r && search_left(nodes[id].right, upper, r, value, f, answer)) return true;
return search_left(nodes[id].left, middle, r, value, f, answer);
}
public:
using value_type = S;
using coordinate_type = Coord;
explicit DynamicSegtree(Coord n = 0) : DynamicSegtree(0, n) {}
DynamicSegtree(Coord l, Coord r) : first(l), last(r) { assert(l <= r); }
DynamicSegtree(const DynamicSegtree &) = default;
DynamicSegtree &operator=(const DynamicSegtree &) = default;
DynamicSegtree(DynamicSegtree &&other) : DynamicSegtree(other.first, other.last) { swap(other); }
DynamicSegtree &operator=(DynamicSegtree &&other) {
if (this != &other) { DynamicSegtree moved(std::move(other)); swap(moved); }
return *this;
}
void swap(DynamicSegtree &other) noexcept {
leaves.swap(other.leaves);
nodes.swap(other.nodes);
std::swap(root, other.root);
std::swap(points, other.points);
std::swap(first, other.first);
std::swap(last, other.last);
}
void reserve(std::size_t n) {
assert(n <= limit);
std::size_t capacity = (3 * n + 2 * B - 1) / (2 * B);
leaves.reserve(capacity);
nodes.reserve(capacity + 1);
}
std::size_t node_count() const { return points; }
std::size_t memory_usage() const { return leaves.capacity() * sizeof(Leaf) + nodes.capacity() * sizeof(Node); }
U size() const { return U(last) - U(first); }
Coord left_bound() const { return first; }
Coord right_bound() const { return last; }
void clear() { leaves.clear(); nodes.erase(nodes.begin() + 1, nodes.end()); root = points = 0; }
void set(Coord p, S value) { assert(first <= p && p < last); update<false>(encode(p), std::move(value)); }
void multiply(Coord p, S value) { assert(first <= p && p < last); update<true>(encode(p), std::move(value)); }
S get(Coord p) const {
assert(first <= p && p < last);
if (!root) return e();
U key = encode(p);
unsigned id = root;
while (!(id & tag)) id = key < nodes[id].middle ? nodes[id].left : nodes[id].right;
const auto &node = leaf(id);
unsigned i = std::lower_bound(node.keys.begin(), node.keys.begin() + node.count, key) - node.keys.begin();
return i < node.count && node.keys[i] == key ? node.values[i] : e();
}
S prod(Coord l, Coord r) const {
assert(first <= l && l <= r && r <= last);
if (!root || l == r) return e();
if (l == first && r == last) return product(root);
U a = encode(l), b = encode(r);
unsigned id = root;
while (!(id & tag)) {
U middle = nodes[id].middle;
if (b <= middle) id = nodes[id].left;
else if (middle <= a) id = nodes[id].right;
else return op(suffix(nodes[id].left, a), prefix(nodes[id].right, b));
}
const auto &node = leaf(id);
unsigned i = std::lower_bound(node.keys.begin(), node.keys.begin() + node.count, a) - node.keys.begin();
S result = e();
for (; i < node.count && node.keys[i] < b; ++i) result = op(result, node.values[i]);
return result;
}
S all_prod() const { return root ? product(root) : e(); }
template <class Pred>
Coord max_right(Coord l, Pred f) const {
assert(first <= l && l <= last && f(e()));
Coord answer = last;
if (root && l != last) { S value = e(); search_right(root, encode(first), encode(l), value, f, answer); }
return answer;
}
template <class Pred>
Coord min_left(Coord r, Pred f) const {
assert(first <= r && r <= last && f(e()));
Coord answer = first;
if (root && r != first) { S value = e(); search_left(root, encode(last), encode(r), value, f, answer); }
return answer;
}
};