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:heavy_check_mark: verify/ds/segtree/persistent_dynamic_segtree.test.cpp

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Code

#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"

#include "../../../ds/segtree/persistent_dynamic_segtree.hpp"

#include "../../../utilities/fast_io.hpp"
#include <algorithm>
#include <cassert>
#include <cstdint>
#include <limits>
#include <string>
#include <vector>

#include "../../../monoid/add.hpp"

namespace {

struct Concat {
    using value_type = std::string;

    static value_type id() { return ""; }

    static value_type op(const value_type& a, const value_type& b) { return a + b; }
};

void test_versions() {
    using Sum = m1une::monoid::Add<long long>;
    using Seg = m1une::ds::PersistentDynamicSegtree<Sum>;

    Seg base(-1'000'000'000'000'000'000LL, 1'000'000'000'000'000'000LL, 0);
    base.reserve(4096);
    Seg first = base.set(-3, 2);
    Seg second = first.set(2, 4);
    Seg branch = base.set(7, 10);

    assert(base.size() == 2'000'000'000'000'000'000ULL);
    assert(base.node_count() == branch.node_count());
    assert(base.all_prod() == 0);
    assert(first.all_prod() == 2);
    assert(second.all_prod() == 6);
    assert(branch.all_prod() == 10);
    assert(base.get(-3) == 0);
    assert(first.get(-3) == 2);
    assert(second.get(2) == 4);
    assert(branch.get(7) == 10);
    assert(second.prod(-4, 3) == 6);
    assert(second.max_right(-8, [](long long x) { return x <= 3; }) == 2);
    assert(second.min_left(8, [](long long x) { return x <= 4; }) == -2);

    std::size_t live = base.node_count();
    second.release();
    assert(base.node_count() < live);
    assert(second.all_prod() == 0);
    assert(first.get(-3) == 2);
    first.release();
    branch.release();
    assert(base.node_count() == 0);
}

void test_uniform_and_non_commutative() {
    using Sum = m1une::monoid::Add<long long>;
    using Seg = m1une::ds::PersistentDynamicSegtree<Sum, int>;

    Seg uniform(-5, 6, 3);
    assert(uniform.all_prod() == 33);
    assert(uniform.prod(-2, 4) == 18);
    Seg changed = uniform.set(0, 10);
    assert(uniform.all_prod() == 33);
    assert(changed.all_prod() == 40);

    m1une::ds::PersistentDynamicSegtree<Concat, int> text(-2, 3, "x");
    auto first = text.set(1, "a");
    auto second = first.set(-1, "b");
    assert(text.all_prod() == "xxxxx");
    assert(first.all_prod() == "xxxax");
    assert(second.all_prod() == "xbxax");
}

void test_extreme_coordinates() {
    using Sum = m1une::monoid::Add<long long>;
    using Seg = m1une::ds::PersistentDynamicSegtree<Sum>;

    constexpr long long left = std::numeric_limits<long long>::min();
    constexpr long long right = std::numeric_limits<long long>::max();
    Seg base(left, right);
    Seg first = base.set(left, 3);
    Seg second = first.set(right - 1, 4);

    assert(base.size() == std::numeric_limits<unsigned long long>::max());
    assert(base.all_prod() == 0);
    assert(first.get(left) == 3);
    assert(second.get(right - 1) == 4);
    assert(second.all_prod() == 7);
}

void test_randomized() {
    using Sum = m1une::monoid::Add<long long>;
    using Seg = m1une::ds::PersistentDynamicSegtree<Sum, int>;
    constexpr int left = -29;
    constexpr int right = 37;

    std::vector<Seg> versions;
    versions.emplace_back(left, right, 1);
    std::vector<std::vector<long long>> expected(1, std::vector<long long>(right - left, 1));

    std::uint64_t state = 13;
    auto random = [&state]() {
        state ^= state << 7;
        state ^= state >> 9;
        return state;
    };

    for (int step = 0; step < 1500; step++) {
        std::size_t from = std::size_t(random() % versions.size());
        int p = left + int(random() % expected[from].size());
        long long x = static_cast<long long>(random() % 20);

        versions.push_back(versions[from].set(p, x));
        expected.push_back(expected[from]);
        expected.back()[std::size_t(p - left)] = x;

        std::size_t check = std::size_t(random() % versions.size());
        int l = left + int(random() % (expected[check].size() + 1));
        int r = left + int(random() % (expected[check].size() + 1));
        if (r < l) std::swap(l, r);
        long long sum = 0;
        for (int i = l; i < r; i++) {
            sum += expected[check][std::size_t(i - left)];
        }
        assert(versions[check].prod(l, r) == sum);

        long long limit = static_cast<long long>(random() % 100);
        int start = left + int(random() % (expected[check].size() + 1));
        int max_right = start;
        sum = 0;
        while (max_right < right && sum + expected[check][std::size_t(max_right - left)] <= limit) {
            sum += expected[check][std::size_t(max_right - left)];
            max_right++;
        }
        assert(versions[check].max_right(start, [limit](long long value) { return value <= limit; }) == max_right);

        int finish = left + int(random() % (expected[check].size() + 1));
        int min_left = finish;
        sum = 0;
        while (left < min_left && expected[check][std::size_t(min_left - 1 - left)] + sum <= limit) {
            min_left--;
            sum += expected[check][std::size_t(min_left - left)];
        }
        assert(versions[check].min_left(finish, [limit](long long value) { return value <= limit; }) == min_left);
    }
}

}  // namespace

int main() {
    m1une::utilities::FastInput fast_input;
    m1une::utilities::FastOutput fast_output;

    test_versions();
    test_uniform_and_non_commutative();
    test_extreme_coordinates();
    test_randomized();

    long long a, b;
    fast_input >> a >> b;
    fast_output << a + b << '\n';
}
#line 1 "verify/ds/segtree/persistent_dynamic_segtree.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"

#line 1 "ds/segtree/persistent_dynamic_segtree.hpp"



#include <cassert>
#include <concepts>
#include <cstddef>
#include <limits>
#include <memory>
#include <numeric>
#include <type_traits>
#include <utility>
#include <vector>

#line 1 "monoid/concept.hpp"



#line 5 "monoid/concept.hpp"

namespace m1une {
namespace monoid {

// Concept to check if a type satisfies the requirements of a Monoid.
// A Monoid must have a `value_type`, an identity element `id()`, and an associative binary operation `op()`.
template <typename M>
concept IsMonoid = requires(typename M::value_type a, typename M::value_type b) {
    // 1. Must define `value_type`
    typename M::value_type;

    // 2. Must have a static method `id()` returning `value_type`
    { M::id() } -> std::same_as<typename M::value_type>;

    // 3. Must have a static method `op(a, b)` returning `value_type`
    { M::op(a, b) } -> std::same_as<typename M::value_type>;
};

// Concept for groups. A type satisfying this concept must also obey the group
// laws; concepts can check the interface but not the algebraic properties.
template <typename M>
concept IsGroup = IsMonoid<M> && requires(typename M::value_type a) {
    { M::inv(a) } -> std::same_as<typename M::value_type>;
};

// Concept for commutative groups. Commutativity is a semantic requirement and
// cannot be checked by a C++ concept.
template <typename M>
concept IsCommutativeGroup = IsGroup<M>;

}  // namespace monoid
}  // namespace m1une


#line 1 "ds/segtree/dynamic_segtree_common.hpp"



#line 11 "ds/segtree/dynamic_segtree_common.hpp"

namespace m1une {
namespace ds {
namespace detail {

template <std::integral Index>
using dynamic_size_type = std::make_unsigned_t<Index>;

template <std::integral Index>
constexpr dynamic_size_type<Index> dynamic_distance(Index left, Index right) {
    return static_cast<dynamic_size_type<Index>>(right) - static_cast<dynamic_size_type<Index>>(left);
}

template <class Monoid, class Size>
typename Monoid::value_type monoid_repeat(typename Monoid::value_type value, Size count) {
    typename Monoid::value_type result = Monoid::id();
    while (count != 0) {
        if (count & 1) result = Monoid::op(result, value);
        count >>= 1;
        if (count != 0) value = Monoid::op(value, value);
    }
    return result;
}

template <class ActedMonoid>
typename ActedMonoid::value_type dynamic_mapping(
    const typename ActedMonoid::operator_type& f,
    const typename ActedMonoid::value_type& value
) {
    using F = typename ActedMonoid::operator_type;
    using T = typename ActedMonoid::value_type;
    if constexpr (requires(F g, T x, long long ord) { ActedMonoid::mapping(g, x, ord); }) {
        return ActedMonoid::mapping(f, value, 0);
    } else {
        return ActedMonoid::mapping(f, value);
    }
}

template <class ActedMonoid, class Size>
typename ActedMonoid::operator_type dynamic_shift(
    const typename ActedMonoid::operator_type& f,
    Size offset
) {
    using F = typename ActedMonoid::operator_type;
    if constexpr (requires(F g, long long ord) { ActedMonoid::op_shift(g, ord); }) {
        assert(offset <= static_cast<Size>(std::numeric_limits<long long>::max()));
        return ActedMonoid::op_shift(f, static_cast<long long>(offset));
    } else {
        return f;
    }
}

template <class Monoid, std::integral Index>
class UniformMonoidDomain {
   public:
    using T = typename Monoid::value_type;
    using size_type = dynamic_size_type<Index>;

   private:
    struct Level {
        size_type small_length;
        T small_value;
        T large_value;
    };

    Index _left;
    Index _right;
    T _initial_value;
    std::vector<Level> _levels;

   public:
    UniformMonoidDomain(Index left, Index right, T initial_value)
        : _left(left), _right(right), _initial_value(std::move(initial_value)) {
        assert(left <= right);
        size_type n = size();
        constexpr int digits = std::numeric_limits<size_type>::digits;
        _levels.reserve(digits + 1);
        for (int depth = 0; depth <= digits; depth++) {
            size_type small = depth == digits ? 0 : n >> depth;
            size_type large = small;
            if (depth != 0) {
                bool has_remainder;
                if (depth == digits) {
                    has_remainder = n != 0;
                } else {
                    size_type mask = (size_type(1) << depth) - 1;
                    has_remainder = (n & mask) != 0;
                }
                if (has_remainder) large++;
            }
            _levels.push_back(Level{
                small,
                monoid_repeat<Monoid>(_initial_value, small),
                monoid_repeat<Monoid>(_initial_value, large),
            });
        }
    }

    Index left_bound() const {
        return _left;
    }

    Index right_bound() const {
        return _right;
    }

    size_type size() const {
        return dynamic_distance(_left, _right);
    }

    bool empty() const {
        return _left == _right;
    }

    const T& initial_value() const {
        return _initial_value;
    }

    const T& default_product(int depth, Index left, Index right) const {
        assert(0 <= depth && depth < int(_levels.size()));
        const Level& level = _levels[depth];
        size_type length = dynamic_distance(left, right);
        if (length == level.small_length) return level.small_value;
        assert(length == level.small_length + 1);
        return level.large_value;
    }
};

}  // namespace detail
}  // namespace ds
}  // namespace m1une


#line 1 "ds/segtree/persistent_node_pool.hpp"



#line 9 "ds/segtree/persistent_node_pool.hpp"

namespace m1une {
namespace ds {
namespace detail {

// Node must have integer `left`, `right`, and `references` members.
template <class Node>
struct PersistentNodePool {
    std::vector<Node> nodes;
    int first_free = 0;
    std::size_t live_nodes = 0;

   private:
    void release_zero(int node) {
        int left = nodes[node].left;
        int right = nodes[node].right;
        nodes[node] = Node();
        nodes[node].left = first_free;
        first_free = node;
        --live_nodes;
        if (left && --nodes[left].references == 0) release_zero(left);
        if (right && --nodes[right].references == 0) release_zero(right);
    }

   public:
    PersistentNodePool() { nodes.emplace_back(); }

    void reserve(std::size_t capacity) { nodes.reserve(capacity + 1); }

    Node& operator[](int node) { return nodes[node]; }

    const Node& operator[](int node) const { return nodes[node]; }

    void retain(int node) {
        if (node) ++nodes[node].references;
    }

    void release(int node) {
        if (!node) return;
        assert(nodes[node].references > 0);
        if (--nodes[node].references == 0) release_zero(node);
    }

    template <class... Args>
    int emplace(Args&&... args) {
        int result;
        if (!first_free) {
            assert(nodes.size() < std::size_t(std::numeric_limits<int>::max()));
            nodes.emplace_back(std::forward<Args>(args)...);
            result = int(nodes.size()) - 1;
        } else {
            result = first_free;
            first_free = nodes[result].left;
            nodes[result] = Node(std::forward<Args>(args)...);
        }
        Node& node = nodes[result];
        node.references = 0;
        retain(node.left);
        retain(node.right);
        ++live_nodes;
        return result;
    }

    int clone(int node) {
        assert(node);
        Node copy = nodes[node];
        return emplace(std::move(copy));
    }

    bool unique(int node) const {
        return !node || nodes[node].references == 1;
    }

    // Returns node itself when it has one owner, otherwise an unowned clone.
    // The caller must attach a returned clone with replace() before it can be
    // released or exposed as a root.
    int clone_if_shared(int node) {
        if (unique(node)) return node;
        return clone(node);
    }

    void replace(int& edge, int node) {
        if (edge == node) return;
        retain(node);
        int old = edge;
        edge = node;
        release(old);
    }

    std::size_t size() const { return live_nodes; }
};

}  // namespace detail
}  // namespace ds
}  // namespace m1une


#line 17 "ds/segtree/persistent_dynamic_segtree.hpp"

namespace m1une {
namespace ds {

// A persistent sparse segment tree over an integral half-open interval.
template <m1une::monoid::IsMonoid Monoid, std::integral Index = long long>
    requires(!std::same_as<std::remove_cv_t<Index>, bool>)
struct PersistentDynamicSegtree {
    using T = typename Monoid::value_type;
    using index_type = Index;
    using size_type = detail::dynamic_size_type<Index>;

   private:
    struct Node {
        T val;
        int left;
        int right;
        int references;

        Node() : val(Monoid::id()), left(0), right(0), references(0) {}
        explicit Node(T value) : val(std::move(value)), left(0), right(0), references(0) {}
    };

    struct Config {
        detail::UniformMonoidDomain<Monoid, Index> domain;

        Config(Index left, Index right, T initial_value) : domain(left, right, std::move(initial_value)) {}
    };

    std::shared_ptr<const Config> _config;
    using Pool = detail::PersistentNodePool<Node>;
    std::shared_ptr<Pool> _pool;
    int _root;

    PersistentDynamicSegtree(std::shared_ptr<const Config> config, std::shared_ptr<Pool> pool, int root)
        : _config(std::move(config)), _pool(std::move(pool)), _root(root) {
        _pool->retain(_root);
    }

    int new_node(T value) const { return _pool->emplace(std::move(value)); }

    const T& value(int t, Index left, Index right, int depth) const {
        if (t) return (*_pool)[t].val;
        return _config->domain.default_product(depth, left, right);
    }

    int set_node(int t, Index left, Index right, int depth, Index p, T x, bool copy_on_write = false) const {
        Index middle = std::midpoint(left, right);
        if (copy_on_write) {
            int result = t ? _pool->clone_if_shared(t) : new_node(value(0, left, right, depth));
            if (middle == left) {
                (*_pool)[result].val = std::move(x);
                return result;
            }
            int child;
            if (p < middle) {
                child = set_node((*_pool)[result].left, left, middle, depth + 1, p, std::move(x), true);
                _pool->replace((*_pool)[result].left, child);
            } else {
                child = set_node((*_pool)[result].right, middle, right, depth + 1, p, std::move(x), true);
                _pool->replace((*_pool)[result].right, child);
            }
            Node& node = (*_pool)[result];
            node.val = Monoid::op(value(node.left, left, middle, depth + 1),
                                  value(node.right, middle, right, depth + 1));
            return result;
        }
        if (middle == left) return new_node(std::move(x));

        int left_child = t ? (*_pool)[t].left : 0;
        int right_child = t ? (*_pool)[t].right : 0;
        if (p < middle) {
            left_child = set_node(left_child, left, middle, depth + 1, p, std::move(x));
        } else {
            right_child = set_node(right_child, middle, right, depth + 1, p, std::move(x));
        }

        int result = new_node(
            Monoid::op(value(left_child, left, middle, depth + 1), value(right_child, middle, right, depth + 1)));
        _pool->replace((*_pool)[result].left, left_child);
        _pool->replace((*_pool)[result].right, right_child);
        return result;
    }

    T prod_node(int t, Index left, Index right, int depth, Index query_left, Index query_right) const {
        if (query_right <= left || right <= query_left) return Monoid::id();
        if (query_left <= left && right <= query_right) {
            return value(t, left, right, depth);
        }
        Index middle = std::midpoint(left, right);
        return Monoid::op(prod_node(t ? (*_pool)[t].left : 0, left, middle, depth + 1, query_left, query_right),
                          prod_node(t ? (*_pool)[t].right : 0, middle, right, depth + 1, query_left, query_right));
    }

    template <class F>
    Index max_right_node(int t, Index left, Index right, int depth, Index query_left, T& product, F& predicate) const {
        if (right <= query_left) return right;
        if (query_left <= left) {
            T next = Monoid::op(product, value(t, left, right, depth));
            if (predicate(next)) {
                product = std::move(next);
                return right;
            }
            Index middle = std::midpoint(left, right);
            if (middle == left) return left;
        }

        Index middle = std::midpoint(left, right);
        Index result =
            max_right_node(t ? (*_pool)[t].left : 0, left, middle, depth + 1, query_left, product, predicate);
        if (result < middle) return result;
        return max_right_node(t ? (*_pool)[t].right : 0, middle, right, depth + 1, query_left, product, predicate);
    }

    template <class F>
    Index min_left_node(int t, Index left, Index right, int depth, Index query_right, T& product, F& predicate) const {
        if (query_right <= left) return left;
        if (right <= query_right) {
            T next = Monoid::op(value(t, left, right, depth), product);
            if (predicate(next)) {
                product = std::move(next);
                return left;
            }
            Index middle = std::midpoint(left, right);
            if (middle == left) return right;
        }

        Index middle = std::midpoint(left, right);
        Index result =
            min_left_node(t ? (*_pool)[t].right : 0, middle, right, depth + 1, query_right, product, predicate);
        if (middle < result) return result;
        return min_left_node(t ? (*_pool)[t].left : 0, left, middle, depth + 1, query_right, product, predicate);
    }

   public:
    PersistentDynamicSegtree() : PersistentDynamicSegtree(Index(0), Index(0), Monoid::id()) {}

    explicit PersistentDynamicSegtree(Index n) : PersistentDynamicSegtree(Index(0), n, Monoid::id()) {
        if constexpr (std::signed_integral<Index>) assert(Index(0) <= n);
    }

    PersistentDynamicSegtree(Index left, Index right) : PersistentDynamicSegtree(left, right, Monoid::id()) {}

    PersistentDynamicSegtree(Index left, Index right, T initial_value)
        : _config(std::make_shared<Config>(left, right, std::move(initial_value))),
          _pool(std::make_shared<Pool>()),
          _root(0) {}

    PersistentDynamicSegtree(const PersistentDynamicSegtree& other)
        : _config(other._config), _pool(other._pool), _root(other._root) {
        if (_pool) _pool->retain(_root);
    }
    PersistentDynamicSegtree(PersistentDynamicSegtree&& other) noexcept
        : _config(std::move(other._config)), _pool(std::move(other._pool)), _root(other._root) {
        other._root = 0;
    }
    PersistentDynamicSegtree& operator=(const PersistentDynamicSegtree& other) {
        if (this == &other) return *this;
        if (other._pool) other._pool->retain(other._root);
        if (_pool) _pool->release(_root);
        _config = other._config;
        _pool = other._pool;
        _root = other._root;
        return *this;
    }
    PersistentDynamicSegtree& operator=(PersistentDynamicSegtree&& other) noexcept {
        if (this == &other) return *this;
        if (_pool) _pool->release(_root);
        _config = std::move(other._config);
        _pool = std::move(other._pool);
        _root = other._root;
        other._root = 0;
        return *this;
    }
    ~PersistentDynamicSegtree() {
        if (_pool) _pool->release(_root);
    }

    size_type size() const { return _config->domain.size(); }

    bool empty() const { return _config->domain.empty(); }

    Index left_bound() const { return _config->domain.left_bound(); }

    Index right_bound() const { return _config->domain.right_bound(); }

    const T& initial_value() const { return _config->domain.initial_value(); }

    void reserve(std::size_t node_capacity) const {
        assert(node_capacity < std::numeric_limits<std::size_t>::max());
        _pool->reserve(node_capacity);
    }

    std::size_t node_count() const { return _pool->size(); }

    void release() {
        if (_pool) _pool->release(_root);
        _pool = std::make_shared<Pool>();
        _root = 0;
    }

    PersistentDynamicSegtree set(Index p, T x) const {
        assert(left_bound() <= p && p < right_bound());
        return PersistentDynamicSegtree(_config, _pool,
                                        set_node(_root, left_bound(), right_bound(), 0, p, std::move(x)));
    }

    void set_inplace(Index p, T x) {
        assert(left_bound() <= p && p < right_bound());
        int root = set_node(_root, left_bound(), right_bound(), 0, p, std::move(x), true);
        _pool->replace(_root, root);
    }

    T get(Index p) const {
        assert(left_bound() <= p && p < right_bound());
        int t = _root;
        Index left = left_bound();
        Index right = right_bound();

        while (t) {
            Index middle = std::midpoint(left, right);
            if (middle == left) return (*_pool)[t].val;
            if (p < middle) {
                t = (*_pool)[t].left;
                right = middle;
            } else {
                t = (*_pool)[t].right;
                left = middle;
            }
        }
        return initial_value();
    }

    T operator[](Index p) const { return get(p); }

    T prod(Index left, Index right) const {
        assert(left_bound() <= left && left <= right && right <= right_bound());
        if (left == right) return Monoid::id();
        return prod_node(_root, left_bound(), right_bound(), 0, left, right);
    }

    T all_prod() const { return value(_root, left_bound(), right_bound(), 0); }

    template <class F>
    Index max_right(Index left, F predicate) const {
        assert(left_bound() <= left && left <= right_bound());
        assert(predicate(Monoid::id()));
        if (left == right_bound()) return right_bound();
        T product = Monoid::id();
        return max_right_node(_root, left_bound(), right_bound(), 0, left, product, predicate);
    }

    template <class F>
    Index min_left(Index right, F predicate) const {
        assert(left_bound() <= right && right <= right_bound());
        assert(predicate(Monoid::id()));
        if (right == left_bound()) return left_bound();
        T product = Monoid::id();
        return min_left_node(_root, left_bound(), right_bound(), 0, right, product, predicate);
    }
};

}  // namespace ds
}  // namespace m1une


#line 4 "verify/ds/segtree/persistent_dynamic_segtree.test.cpp"

#line 1 "utilities/fast_io.hpp"



#include <algorithm>
#include <array>
#include <cerrno>
#include <charconv>
#line 9 "utilities/fast_io.hpp"
#include <cstdio>
#include <cstdlib>
#include <cstdint>
#include <cstring>
#include <iterator>
#include <string>
#include <sys/stat.h>
#line 18 "utilities/fast_io.hpp"
#include <unistd.h>
#line 20 "utilities/fast_io.hpp"

namespace m1une {
namespace utilities {

struct FastOutput;

namespace internal {

// Shared with the convenience helpers in template.hpp.
inline FastOutput* standard_output_instance = nullptr;

// Detect std::begin(x), std::end(x).
template <class T, class = void>
struct is_range : std::false_type {};

template <class T>
struct is_range<T, std::void_t<
    decltype(std::begin(std::declval<T&>())),
    decltype(std::end(std::declval<T&>()))
>> : std::true_type {};

template <class T>
inline constexpr bool is_range_v = is_range<T>::value;

template <class T>
using range_reference_t = decltype(*std::begin(std::declval<T&>()));

template <class T>
using range_value_t = std::remove_cv_t<std::remove_reference_t<range_reference_t<T>>>;

template <class T, class = void>
struct range_stored_value {
    using type = range_value_t<T>;
};

template <class T>
struct range_stored_value<T, std::void_t<typename std::remove_cv_t<std::remove_reference_t<T>>::value_type>> {
    using type = typename std::remove_cv_t<std::remove_reference_t<T>>::value_type;
};

template <class T>
using range_stored_value_t = typename range_stored_value<T>::type;

// Treat strings and C strings as scalar output objects, not as ranges.
template <class T>
struct is_char_array : std::false_type {};

template <class T, std::size_t N>
struct is_char_array<T[N]>
    : std::bool_constant<std::is_same_v<std::remove_cv_t<T>, char>> {};

template <class T>
struct is_string_like
    : std::bool_constant<
          std::is_same_v<std::decay_t<T>, std::string>
          || std::is_same_v<std::decay_t<T>, const char*>
          || std::is_same_v<std::decay_t<T>, char*>
          || is_char_array<std::remove_reference_t<T>>::value
      > {};

template <class T>
inline constexpr bool is_string_like_v = is_string_like<T>::value;

// ModInt-like type: x.val() is printable, and x can be assigned from long long.
template <class T, class = void>
struct has_val_method : std::false_type {};

template <class T>
struct has_val_method<T, std::void_t<decltype(std::declval<const T&>().val())>>
    : std::true_type {};

template <class T>
inline constexpr bool has_val_method_v = has_val_method<T>::value;

template <class T, class = void>
struct has_static_mod_raw : std::false_type {};

template <class T>
struct has_static_mod_raw<
    T, std::void_t<decltype(T::mod()), decltype(T::raw(std::declval<uint32_t>()))>>
    : std::true_type {};

template <class T>
inline constexpr bool has_static_mod_raw_v = has_static_mod_raw<T>::value;

// libstdc++ before GCC 16 does not classify __int128 as an integral type in
// strict ISO modes such as -std=c++23. Keep the fast-I/O interface independent
// of that implementation detail.
template <class T>
inline constexpr bool is_integral_v =
    std::is_integral_v<T>
    || std::is_same_v<std::remove_cv_t<T>, __int128_t>
    || std::is_same_v<std::remove_cv_t<T>, __uint128_t>;

template <class T>
inline constexpr bool is_signed_v =
    std::is_signed_v<T>
    || std::is_same_v<std::remove_cv_t<T>, __int128_t>;

template <class T>
struct make_unsigned {
    using type = std::make_unsigned_t<T>;
};

template <>
struct make_unsigned<__int128_t> {
    using type = __uint128_t;
};

template <>
struct make_unsigned<__uint128_t> {
    using type = __uint128_t;
};

template <class T>
using make_unsigned_t = typename make_unsigned<std::remove_cv_t<T>>::type;

}  // namespace internal

struct FastInput {
    static constexpr int buffer_size = 1 << 20;

   private:
    std::FILE* _stream;
    char _buffer[buffer_size];
    int _position;
    int _length;
    int _file_descriptor;
    bool _streaming;

    bool refill() {
        _position = 0;
        if (_streaming) {
            ssize_t length;
            do {
                length = ::read(_file_descriptor, _buffer, buffer_size);
            } while (length < 0 && errno == EINTR);
            if (length <= 0) {
                _length = 0;
                return false;
            }
            _length = int(length);
        } else {
            _length = int(std::fread(_buffer, 1, buffer_size, _stream));
        }
        return _length != 0;
    }

    template <class T>
    bool read_integer_from_stream(T& value) {
        if (!skip_spaces()) return false;
        int c = read_char_raw();

        bool negative = false;
        if (c == '-') {
            negative = true;
            c = read_char_raw();
        }

        if constexpr (internal::is_signed_v<T>) {
            T result = 0;
            while ('0' <= c && c <= '9') {
                result = negative ? result * 10 - (c - '0')
                                  : result * 10 + (c - '0');
                c = read_char_raw();
            }
            value = result;
        } else {
            T result = 0;
            while ('0' <= c && c <= '9') {
                result = result * 10 + T(c - '0');
                c = read_char_raw();
            }
            value = negative ? T(0) - result : result;
        }
        return true;
    }

    bool prepare_number() {
        if (_length - _position >= 64) return true;
        const int remaining = _length - _position;
        if (remaining > 0) std::memmove(_buffer, _buffer + _position, remaining);
        const int added = int(std::fread(_buffer + remaining, 1, buffer_size - remaining, _stream));
        _position = 0;
        _length = remaining + added;
        if (_length < buffer_size) _buffer[_length] = '\0';
        return _length != 0;
    }

   public:
    explicit FastInput(std::FILE* stream = stdin)
        : _stream(stream),
          _position(0),
          _length(0),
          _file_descriptor(::fileno(stream)),
          _streaming([&] {
              struct stat status;
              return _file_descriptor >= 0
                     && ::fstat(_file_descriptor, &status) == 0
                     && !S_ISREG(status.st_mode);
          }()) {}

    FastInput(const FastInput&) = delete;
    FastInput& operator=(const FastInput&) = delete;

    int read_char_raw() {
        if (_position == _length && !refill()) return EOF;
        return _buffer[_position++];
    }

    bool skip_spaces() {
        int c = read_char_raw();
        while (c != EOF && c <= ' ') c = read_char_raw();
        if (c == EOF) return false;
        --_position;
        return true;
    }

    bool read(char& value) {
        if (!skip_spaces()) return false;
        value = char(read_char_raw());
        return true;
    }

    bool read(std::string& value) {
        if (!skip_spaces()) return false;
        value.clear();
        while (true) {
            const int begin = _position;
            while (_position < _length &&
                   static_cast<unsigned char>(_buffer[_position]) > ' ') {
                ++_position;
            }
            value.append(_buffer + begin, _position - begin);
            if (_position < _length) {
                ++_position;
                return true;
            }
            if (!refill()) return true;
        }
    }

    bool read(bool& value) {
        int x;
        if (!read(x)) return false;
        value = x != 0;
        return true;
    }

    template <class T>
    std::enable_if_t<
        internal::is_integral_v<T>
            && !std::is_same_v<std::remove_cv_t<T>, bool>
            && !std::is_same_v<std::remove_cv_t<T>, char>,
        bool
    >
    read(T& value) {
        if (_streaming) return read_integer_from_stream(value);
        if (!prepare_number()) return false;
        int c = static_cast<unsigned char>(_buffer[_position++]);
        while (c <= ' ') c = static_cast<unsigned char>(_buffer[_position++]);

        bool negative = false;
        if (c == '-') {
            negative = true;
            c = static_cast<unsigned char>(_buffer[_position++]);
        }

        if constexpr (internal::is_signed_v<T>) {
            T result = 0;
            while ('0' <= c && c <= '9') {
                const int first = c - '0';
                const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
                if (0 <= second && second <= 9) {
                    result = negative ? result * 100 - (first * 10 + second)
                                      : result * 100 + (first * 10 + second);
                    ++_position;
                } else {
                    result = negative ? result * 10 - first : result * 10 + first;
                }
                c = static_cast<unsigned char>(_buffer[_position++]);
            }
            value = result;
        } else {
            T result = 0;
            while ('0' <= c && c <= '9') {
                const unsigned first = unsigned(c - '0');
                const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
                if (0 <= second && second <= 9) {
                    result = result * 100 + T(first * 10 + unsigned(second));
                    ++_position;
                } else {
                    result = result * 10 + T(first);
                }
                c = static_cast<unsigned char>(_buffer[_position++]);
            }
            value = negative ? T(0) - result : result;
        }
        if (_position > _length) _position = _length;
        return true;
    }

    template <class T>
    std::enable_if_t<std::is_floating_point_v<T>, bool>
    read(T& value) {
        if (!skip_spaces()) return false;
        int c = read_char_raw();
        bool negative = false;
        if (c == '-' || c == '+') {
            negative = c == '-';
            c = read_char_raw();
        }

        long double result = 0;
        while ('0' <= c && c <= '9') {
            result = result * 10 + (c - '0');
            c = read_char_raw();
        }
        if (c == '.') {
            long double place = 0.1L;
            c = read_char_raw();
            while ('0' <= c && c <= '9') {
                result += (c - '0') * place;
                place *= 0.1L;
                c = read_char_raw();
            }
        }
        if (c == 'e' || c == 'E') {
            c = read_char_raw();
            bool exponent_negative = false;
            if (c == '-' || c == '+') {
                exponent_negative = c == '-';
                c = read_char_raw();
            }
            int exponent = 0;
            while ('0' <= c && c <= '9') {
                exponent = exponent * 10 + (c - '0');
                c = read_char_raw();
            }
            long double scale = 1;
            long double power = 10;
            while (exponent > 0) {
                if (exponent & 1) scale *= power;
                power *= power;
                exponent >>= 1;
            }
            result = exponent_negative ? result / scale : result * scale;
        }
        value = static_cast<T>(negative ? -result : result);
        return true;
    }

    template <class T>
    std::enable_if_t<
        internal::has_val_method_v<T>
            && !internal::is_integral_v<T>
            && !internal::is_range_v<T>,
        bool
    >
    read(T& value) {
        long long x;
        if (!read(x)) return false;
        if constexpr (internal::has_static_mod_raw_v<T>) {
            if (x >= 0 && uint64_t(x) < uint64_t(T::mod())) {
                value = T::raw(uint32_t(x));
            } else {
                value = T(x);
            }
        } else {
            value = T(x);
        }
        return true;
    }

    template <class First, class Second>
    bool read(std::pair<First, Second>& value) {
        if (!read(value.first)) return false;
        return read(value.second);
    }

    template <class Range>
    std::enable_if_t<
        internal::is_range_v<Range>
            && !internal::is_string_like_v<Range>,
        bool
    >
    read(Range& range) {
        using StoredValue = internal::range_stored_value_t<Range>;
        constexpr bool nested = internal::is_range_v<StoredValue>
                                && !internal::is_string_like_v<StoredValue>;

        for (auto&& value : range) {
            if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
                bool x;
                if (!read(x)) return false;
                value = x;
            } else {
                if (!read(value)) return false;
            }
        }
        return true;
    }

    template <class First, class Second, class... Rest>
    bool read(First& first, Second& second, Rest&... rest) {
        if (!read(first)) return false;
        return read(second, rest...);
    }

    template <class T>
    FastInput& operator>>(T& value) {
        if (!read(value)) std::abort();
        return *this;
    }
};

struct FastOutput {
    static constexpr int buffer_size = 1 << 20;

   private:
    inline static const auto digit_quads = [] {
        std::array<char, 40000> result{};
        for (int i = 0; i < 10000; i++) {
            int value = i;
            for (int j = 3; j >= 0; j--) {
                result[4 * i + j] = char('0' + value % 10);
                value /= 10;
            }
        }
        return result;
    }();

    std::FILE* _stream;
    char _buffer[buffer_size];
    int _position;
    int _precision;
    std::chars_format _float_format;
    char _range_separator;
    std::string* _capture = nullptr;

    template <class T>
    std::string format_cell(const T& value) {
        std::string result;
        struct CaptureGuard {
            std::string*& target;
            std::string* previous;
            ~CaptureGuard() { target = previous; }
        } guard{_capture, _capture};
        _capture = &result;
        write(value);
        return result;
    }

    template <class Matrix>
    void write_aligned_matrix(const Matrix& matrix) {
        std::vector<std::vector<std::string>> rows;
        std::vector<std::size_t> widths;
        for (const auto& row : matrix) {
            auto& cells = rows.emplace_back();
            std::size_t column = 0;
            for (const auto& value : row) {
                cells.push_back(format_cell(value));
                if (column == widths.size()) widths.push_back(0);
                widths[column] = std::max(widths[column], cells.back().size());
                ++column;
            }
        }
        bool first = true;
        for (const auto& row : rows) {
            if (!first) write_char('\n');
            first = false;
            for (std::size_t column = 0; column < row.size(); ++column) {
                if (column != 0) write_char(_range_separator);
                for (std::size_t padding = row[column].size();
                     padding < widths[column]; ++padding) {
                    write_char(' ');
                }
                write(row[column]);
            }
        }
    }

   public:
    explicit FastOutput(std::FILE* stream = stdout)
        : _stream(stream),
          _position(0),
          _precision(6),
          _float_format(std::chars_format::general),
          _range_separator(' ') {
        if (_stream == stdout
            && internal::standard_output_instance == nullptr) {
            internal::standard_output_instance = this;
        }
    }

    FastOutput(const FastOutput&) = delete;
    FastOutput& operator=(const FastOutput&) = delete;

    ~FastOutput() {
        flush();
        if (internal::standard_output_instance == this) {
            internal::standard_output_instance = nullptr;
        }
    }

    void flush() {
        if (_position != 0) {
            std::fwrite(_buffer, 1, _position, _stream);
            _position = 0;
        }
        std::fflush(_stream);
    }

    void write_char(char c) {
        if (_capture != nullptr) {
            _capture->push_back(c);
            return;
        }
        if (_position == buffer_size) flush();
        _buffer[_position++] = c;
    }

    void write(const char* s) {
        while (*s != '\0') write_char(*s++);
    }

    void write(const std::string& s) {
        if (_capture != nullptr) {
            _capture->append(s);
            return;
        }
        std::size_t position = 0;
        while (position < s.size()) {
            if (_position == buffer_size) flush();
            const std::size_t copied =
                std::min<std::size_t>(buffer_size - _position, s.size() - position);
            std::memcpy(_buffer + _position, s.data() + position, copied);
            _position += int(copied);
            position += copied;
        }
    }

    void write(char c) {
        write_char(c);
    }

    void write(bool value) {
        write_char(value ? '1' : '0');
    }

    template <class T>
    std::enable_if_t<std::is_floating_point_v<T>>
    write(T value) {
        char digits[128];
        auto [end, error] = std::to_chars(
            digits,
            digits + sizeof(digits),
            value,
            _float_format,
            _precision
        );
        if (error != std::errc()) std::abort();
        for (const char* pointer = digits; pointer != end; pointer++) {
            write_char(*pointer);
        }
    }

    template <class T>
    std::enable_if_t<
        internal::is_integral_v<T>
            && !std::is_same_v<std::remove_cv_t<T>, bool>
            && !std::is_same_v<std::remove_cv_t<T>, char>
    >
    write(T value) {
        using Raw = std::remove_cv_t<T>;
        using Unsigned = internal::make_unsigned_t<Raw>;

        Unsigned magnitude;
        if constexpr (internal::is_signed_v<Raw>) {
            if (value < 0) {
                write_char('-');
                magnitude = Unsigned(0) - Unsigned(value);
            } else {
                magnitude = Unsigned(value);
            }
        } else {
            magnitude = value;
        }

        if (magnitude == 0) {
            write_char('0');
            return;
        }

        unsigned chunks[16];
        int count = 0;
        while (magnitude >= 10000) {
            const Unsigned quotient = magnitude / 10000;
            chunks[count++] = unsigned(magnitude - quotient * 10000);
            magnitude = quotient;
        }
        if (_capture == nullptr && _position > buffer_size - 64) flush();
        char captured[64];
        char* const begin = _capture != nullptr ? captured : _buffer + _position;
        char* destination = begin;
        const unsigned leading = unsigned(magnitude);
        const char* first = digit_quads.data() + 4 * leading;
        int skip = leading < 10 ? 3 : leading < 100 ? 2 : leading < 1000 ? 1 : 0;
        for (; skip < 4; skip++) *destination++ = first[skip];
        while (count--) {
            const char* digits = digit_quads.data() + 4 * chunks[count];
            std::memcpy(destination, digits, 4);
            destination += 4;
        }
        if (_capture != nullptr) {
            _capture->append(begin, destination - begin);
        } else {
            _position += int(destination - begin);
        }
    }

    template <class T>
    std::enable_if_t<
        internal::has_val_method_v<T>
            && !internal::is_integral_v<T>
            && !internal::is_range_v<T>
    >
    write(const T& value) {
        write(value.val());
    }

    template <class First, class Second>
    void write(const std::pair<First, Second>& value) {
        write(value.first);
        write_char(' ');
        write(value.second);
    }

    template <class Range>
    std::enable_if_t<
        internal::is_range_v<Range>
            && !internal::is_string_like_v<Range>
    >
    write(const Range& range) {
        using StoredValue = internal::range_stored_value_t<const Range>;
        constexpr bool nested = internal::is_range_v<StoredValue>
                                && !internal::is_string_like_v<StoredValue>;

        bool first = true;
        for (const auto& value : range) {
            if (!first) write_char(nested ? '\n' : _range_separator);
            first = false;
            if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
                write(static_cast<bool>(value));
            } else {
                write(value);
            }
        }
    }

    template <class First, class... Rest>
    void print(const First& first, const Rest&... rest) {
        write(first);
        ((write_char(' '), write(rest)), ...);
    }

    void println() {
        write_char('\n');
    }

    void set_precision(int precision) {
        _precision = precision;
    }

    void set_fixed(int precision = 6) {
        _float_format = std::chars_format::fixed;
        _precision = precision;
    }

    void set_general(int precision = 6) {
        _float_format = std::chars_format::general;
        _precision = precision;
    }

    void set_range_separator(char separator) {
        _range_separator = separator;
    }

    template <class Matrix>
    void write_aligned(const Matrix& matrix) {
        using Row = internal::range_stored_value_t<const Matrix>;
        using Cell = internal::range_stored_value_t<const Row>;
        static_assert(internal::is_range_v<Row> && !internal::is_string_like_v<Row>,
                      "write_aligned requires a two-dimensional range");
        static_assert(!internal::is_range_v<Cell> || internal::is_string_like_v<Cell>,
                      "write_aligned requires scalar cells");
        write_aligned_matrix(matrix);
    }

    template <class Matrix>
    void println_aligned(const Matrix& matrix) {
        write_aligned(matrix);
        write_char('\n');
    }

    template <class... Args>
    void println(const Args&... args) {
        print(args...);
        write_char('\n');
    }

    template <class T>
    FastOutput& operator<<(const T& value) {
        write(value);
        return *this;
    }
};

}  // namespace utilities
}  // namespace m1une


#line 12 "verify/ds/segtree/persistent_dynamic_segtree.test.cpp"

#line 1 "monoid/add.hpp"



namespace m1une {
namespace monoid {

// Monoid for addition (Range Sum).
template <typename T>
struct Add {
    using value_type = T;
    static constexpr bool commutative = true;

    // Returns the identity element for addition, which is 0.
    static constexpr T id() {
        return T(0);
    }

    // Returns the sum of a and b.
    static constexpr T op(const T& a, const T& b) {
        return a + b;
    }

    static constexpr T inv(const T& x) {
        return -x;
    }
};

}  // namespace monoid
}  // namespace m1une


#line 14 "verify/ds/segtree/persistent_dynamic_segtree.test.cpp"

namespace {

struct Concat {
    using value_type = std::string;

    static value_type id() { return ""; }

    static value_type op(const value_type& a, const value_type& b) { return a + b; }
};

void test_versions() {
    using Sum = m1une::monoid::Add<long long>;
    using Seg = m1une::ds::PersistentDynamicSegtree<Sum>;

    Seg base(-1'000'000'000'000'000'000LL, 1'000'000'000'000'000'000LL, 0);
    base.reserve(4096);
    Seg first = base.set(-3, 2);
    Seg second = first.set(2, 4);
    Seg branch = base.set(7, 10);

    assert(base.size() == 2'000'000'000'000'000'000ULL);
    assert(base.node_count() == branch.node_count());
    assert(base.all_prod() == 0);
    assert(first.all_prod() == 2);
    assert(second.all_prod() == 6);
    assert(branch.all_prod() == 10);
    assert(base.get(-3) == 0);
    assert(first.get(-3) == 2);
    assert(second.get(2) == 4);
    assert(branch.get(7) == 10);
    assert(second.prod(-4, 3) == 6);
    assert(second.max_right(-8, [](long long x) { return x <= 3; }) == 2);
    assert(second.min_left(8, [](long long x) { return x <= 4; }) == -2);

    std::size_t live = base.node_count();
    second.release();
    assert(base.node_count() < live);
    assert(second.all_prod() == 0);
    assert(first.get(-3) == 2);
    first.release();
    branch.release();
    assert(base.node_count() == 0);
}

void test_uniform_and_non_commutative() {
    using Sum = m1une::monoid::Add<long long>;
    using Seg = m1une::ds::PersistentDynamicSegtree<Sum, int>;

    Seg uniform(-5, 6, 3);
    assert(uniform.all_prod() == 33);
    assert(uniform.prod(-2, 4) == 18);
    Seg changed = uniform.set(0, 10);
    assert(uniform.all_prod() == 33);
    assert(changed.all_prod() == 40);

    m1une::ds::PersistentDynamicSegtree<Concat, int> text(-2, 3, "x");
    auto first = text.set(1, "a");
    auto second = first.set(-1, "b");
    assert(text.all_prod() == "xxxxx");
    assert(first.all_prod() == "xxxax");
    assert(second.all_prod() == "xbxax");
}

void test_extreme_coordinates() {
    using Sum = m1une::monoid::Add<long long>;
    using Seg = m1une::ds::PersistentDynamicSegtree<Sum>;

    constexpr long long left = std::numeric_limits<long long>::min();
    constexpr long long right = std::numeric_limits<long long>::max();
    Seg base(left, right);
    Seg first = base.set(left, 3);
    Seg second = first.set(right - 1, 4);

    assert(base.size() == std::numeric_limits<unsigned long long>::max());
    assert(base.all_prod() == 0);
    assert(first.get(left) == 3);
    assert(second.get(right - 1) == 4);
    assert(second.all_prod() == 7);
}

void test_randomized() {
    using Sum = m1une::monoid::Add<long long>;
    using Seg = m1une::ds::PersistentDynamicSegtree<Sum, int>;
    constexpr int left = -29;
    constexpr int right = 37;

    std::vector<Seg> versions;
    versions.emplace_back(left, right, 1);
    std::vector<std::vector<long long>> expected(1, std::vector<long long>(right - left, 1));

    std::uint64_t state = 13;
    auto random = [&state]() {
        state ^= state << 7;
        state ^= state >> 9;
        return state;
    };

    for (int step = 0; step < 1500; step++) {
        std::size_t from = std::size_t(random() % versions.size());
        int p = left + int(random() % expected[from].size());
        long long x = static_cast<long long>(random() % 20);

        versions.push_back(versions[from].set(p, x));
        expected.push_back(expected[from]);
        expected.back()[std::size_t(p - left)] = x;

        std::size_t check = std::size_t(random() % versions.size());
        int l = left + int(random() % (expected[check].size() + 1));
        int r = left + int(random() % (expected[check].size() + 1));
        if (r < l) std::swap(l, r);
        long long sum = 0;
        for (int i = l; i < r; i++) {
            sum += expected[check][std::size_t(i - left)];
        }
        assert(versions[check].prod(l, r) == sum);

        long long limit = static_cast<long long>(random() % 100);
        int start = left + int(random() % (expected[check].size() + 1));
        int max_right = start;
        sum = 0;
        while (max_right < right && sum + expected[check][std::size_t(max_right - left)] <= limit) {
            sum += expected[check][std::size_t(max_right - left)];
            max_right++;
        }
        assert(versions[check].max_right(start, [limit](long long value) { return value <= limit; }) == max_right);

        int finish = left + int(random() % (expected[check].size() + 1));
        int min_left = finish;
        sum = 0;
        while (left < min_left && expected[check][std::size_t(min_left - 1 - left)] + sum <= limit) {
            min_left--;
            sum += expected[check][std::size_t(min_left - left)];
        }
        assert(versions[check].min_left(finish, [limit](long long value) { return value <= limit; }) == min_left);
    }
}

}  // namespace

int main() {
    m1une::utilities::FastInput fast_input;
    m1une::utilities::FastOutput fast_output;

    test_versions();
    test_uniform_and_non_commutative();
    test_extreme_coordinates();
    test_randomized();

    long long a, b;
    fast_input >> a >> b;
    fast_output << a + b << '\n';
}
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