Persistent dual segment tree (tools/persistent_dual_segtree.hpp)

• View this file on GitHub
• View document part on GitHub
• Last update: 2025-02-21 00:01:45+09:00
• Include: #include "tools/persistent_dual_segtree.hpp"

It is the data structure for monoids, a set $F$ of $S \to S$ mappings that satisfies the following properties.

• $F$ contains the identity map $\mathrm{id}$, where the identity map is the map that satisfies $\mathrm{id}(x) = x$ for all $x \in S$.
• $F$ is closed under composition, i.e., $f \circ g \in F$ holds for all $f, g \in F$.

Given an array $F$ of length $N$, it processes the following queries in $O(\log N)$ time.

• Compositing the function $f \in F$ for all the functions on an interval
• Calculating the composed function for an element

For simplicity, in this document, we assume that the oracles composition and id work in constant time. If these oracles work in $O(T)$ time, each time complexity appear in this document is multipled by $O(T)$.

License

• CC0

Author

• anqooqie

Constructor of buffer

persistent_dual_segtree<FM>::buffer buffer();

It creates an empty buffer for tools::persistent_dual_segtree<FM>. It defines $F$ by typename FM::T, $\mathrm{composition}$ by F FM::op(F f, F g) and $\mathrm{id}$ by F FM::e().

Constraints

• None

Time Complexity

• $O(1)$

Constructor

persistent_dual_segtree<FM> a(persistent_dual_segtree<FM>::buffer& buffer, long long l_star, long long r_star);

It creates an array $(a_{l^\ast}, a_{l^\ast + 1}, \ldots, a_{r^\ast - 1})$. All the elements are initialized to id(). The data will be stored on buffer.

Constraints

• $\forall f \in F. \forall g \in F. \forall h \in F. \mathrm{composition}(\mathrm{composition}(f, g), h) = \mathrm{composition}(f, \mathrm{composition}(g, h))$
• $\forall f \in F. \mathrm{composition}(\mathrm{id}(), f) = \mathrm{composition}(f, \mathrm{id}()) = f$
• buffer has not been used so far.
• $-4 \times 10^{18} \leq l^\ast \leq r^\ast \leq 4 \times 10^{18}$

Time Complexity

• $O(\log (r^\ast - l^\ast))$

lower_bound

long long a.lower_bound();

It returns $l^\ast$.

Constraints

• buffer is in its lifetime.

Time Complexity

• $O(1)$

upper_bound

long long a.upper_bound();

It returns $r^\ast$.

Constraints

• buffer is in its lifetime.

Time Complexity

• $O(1)$

get

F a.get(long long p);

It returns $a_p$.

Constraints

• buffer is in its lifetime.
• $l^\ast \leq p < r^\ast$

Time Complexity

• $O(\log (r^\ast - l^\ast))$

apply

(1) persistent_dual_segtree<FM> a.apply(long long p, F f);
(2) persistent_dual_segtree<FM> a.apply(long long l, long long r, F f);

• (1)
  • It creates $b$, a copy of $a$, assigns $\mathrm{composition}(f, b_p)$ to $b_p$ and returns $b$.
• (2)
  • It creates $b$, a copy of $a$, assigns $\mathrm{composition}(f, b_i)$ to $b_i$ for all $i$ such that $l \leq i < r$ and returns $b$.

Constraints

• buffer is in its lifetime.
• (1)
  • $l^\ast \leq p < r^\ast$
• (2)
  • $l^\ast \leq l \leq r \leq r^\ast$

Time Complexity

• $O(\log (r^\ast - l^\ast))$

rollback

persistent_dual_segtree<FM> a.rollback(persistent_dual_segtree<FM> s, long long l, long long r);

It creates $b$, a copy of $a$, assigns $s_i$ to $b_i$ for all $i$ such that $l \leq i < r$ and returns $b$.

Constraints

• buffer is in its lifetime.
• a and s shares the same buffer.
• $l^\ast \leq l \leq r \leq r^\ast$

Time Complexity

• $O(\log (r^\ast - l^\ast))$

Depends on

• Bit width required to represent a given integer (tools/bit_width.hpp)
• $\left\lceil \log_2(x) \right\rceil$ (tools/ceil_log2.hpp)
• Fixed point combinator (tools/fix.hpp)
• std::is_integral extended for tools::int128_t and tools::uint128_t (tools/is_integral.hpp)
• std::is_signed extended for tools::int128_t and tools::uint128_t (tools/is_signed.hpp)
• std::make_unsigned extended for tools::int128_t and tools::uint128_t (tools/make_unsigned.hpp)

Verified with

• tests/persistent_dual_segtree.test.cpp

Code

#ifndef TOOLS_PERSISTENT_DUAL_SEGTREE_HPP
#define TOOLS_PERSISTENT_DUAL_SEGTREE_HPP

#include <algorithm>
#include <array>
#include <cassert>
#include <functional>
#include <numeric>
#include <ranges>
#include <type_traits>
#include <utility>
#include <vector>
#include "tools/ceil_log2.hpp"
#include "tools/fix.hpp"

namespace tools {
  template <typename FM>
  class persistent_dual_segtree {
    using F = typename FM::T;

    struct node {
      F lazy;
      ::std::array<int, 2> children;
    };

  public:
    class buffer {
      ::std::vector<::tools::persistent_dual_segtree<FM>::node> m_nodes;
      long long m_offset;
      long long m_size;
      int m_height;

    public:
      friend ::tools::persistent_dual_segtree<FM>;
    };

  private:
    ::tools::persistent_dual_segtree<FM>::buffer *m_buffer;
    int m_root;

    long long capacity() const {
      return 1LL << this->m_buffer->m_height;
    }

  public:
    persistent_dual_segtree() = default;
    persistent_dual_segtree(::tools::persistent_dual_segtree<FM>::buffer& buffer, const long long l_star, const long long r_star) : m_buffer(&buffer) {
      assert(buffer.m_nodes.empty());
      assert(l_star <= r_star);
      buffer.m_offset = l_star;
      buffer.m_size = r_star - l_star;
      buffer.m_height = ::tools::ceil_log2(::std::max(1LL, r_star - l_star));
      buffer.m_nodes.push_back({FM::e(), {-1, -1}});
      for (int k = 1; k <= buffer.m_height; ++k) {
        buffer.m_nodes.push_back({FM::e(), {k - 1, k - 1}});
      }
      this->m_root = buffer.m_height;
    }

    long long lower_bound() const {
      return this->m_buffer->m_offset;
    }
    long long upper_bound() const {
      return this->m_buffer->m_offset + this->m_buffer->m_size;
    }
    F get(long long p) const {
      assert(this->lower_bound() <= p && p < this->upper_bound());
      auto& buffer = *this->m_buffer;
      p -= buffer.m_offset;

      return ::tools::fix([&](auto&& dfs, const int k, const long long kl, const long long kr, const F& lz) -> F {
        assert(kl < kr);
        if (p <= kl && kr <= p + 1) return FM::op(lz, buffer.m_nodes[k].lazy);
        const auto km = ::std::midpoint(kl, kr);
        const F next_lz = FM::op(lz, buffer.m_nodes[k].lazy);
        if (p < km) return dfs(buffer.m_nodes[k].children[0], kl, km, next_lz);
        else return dfs(buffer.m_nodes[k].children[1], km, kr, next_lz);
      })(this->m_root, 0, this->capacity(), FM::e());
    }
    ::tools::persistent_dual_segtree<FM> apply(const long long p, const F& f) const {
      return this->apply(p, p + 1, f);
    }
    ::tools::persistent_dual_segtree<FM> apply(long long l, long long r, const F& f) const {
      assert(this->lower_bound() <= l && l <= r && r <= this->upper_bound());
      if (l == r) return *this;
      auto& buffer = *this->m_buffer;
      l -= buffer.m_offset;
      r -= buffer.m_offset;

      auto res = *this;
      res.m_root = ::tools::fix([&](auto&& dfs, const int k, const long long kl, const long long kr, const F& lz) -> int {
        assert(kl < kr);
        if (l <= kl && kr <= r) {
          const F modified_lz = FM::op(f, lz);
          buffer.m_nodes.push_back({FM::op(modified_lz, buffer.m_nodes[k].lazy), buffer.m_nodes[k].children});
          return buffer.m_nodes.size() - 1;
        }
        if (kr <= l || r <= kl) {
          if (lz == FM::e()) return k;
          buffer.m_nodes.push_back({FM::op(lz, buffer.m_nodes[k].lazy), buffer.m_nodes[k].children});
          return buffer.m_nodes.size() - 1;
        }
        const auto km = ::std::midpoint(kl, kr);
        const F next_lz = FM::op(lz, buffer.m_nodes[k].lazy);
        const auto left_child = dfs(buffer.m_nodes[k].children[0], kl, km, next_lz);
        const auto right_child = dfs(buffer.m_nodes[k].children[1], km, kr, next_lz);
        buffer.m_nodes.push_back({FM::e(), {left_child, right_child}});
        return buffer.m_nodes.size() - 1;
      })(res.m_root, 0, res.capacity(), FM::e());
      return res;
    }
    ::tools::persistent_dual_segtree<FM> rollback(const ::tools::persistent_dual_segtree<FM>& s, long long l, long long r) const {
      assert(this->m_buffer == s.m_buffer);
      assert(this->lower_bound() <= l && l <= r && r <= this->upper_bound());
      if (l == r) return *this;
      if (l == this->lower_bound() && r == this->upper_bound()) return s;
      auto& buffer = *this->m_buffer;
      l -= buffer.m_offset;
      r -= buffer.m_offset;

      auto res = *this;
      res.m_root = ::tools::fix([&](auto&& dfs, const int k1, const int k2, const long long kl, const long long kr, const F& lz1, const F& lz2) -> int {
        assert(kl < kr);
        if (l <= kl && kr <= r) {
          if (lz2 == FM::e()) return k2;
          buffer.m_nodes.push_back({FM::op(lz2, buffer.m_nodes[k2].lazy), buffer.m_nodes[k2].children});
          return buffer.m_nodes.size() - 1;
        }
        if (kr <= l || r <= kl) {
          if (lz1 == FM::e()) return k1;
          buffer.m_nodes.push_back({FM::op(lz1, buffer.m_nodes[k1].lazy), buffer.m_nodes[k1].children});
          return buffer.m_nodes.size() - 1;
        }
        const auto km = ::std::midpoint(kl, kr);
        const F next_lz1 = FM::op(lz1, buffer.m_nodes[k1].lazy);
        const F next_lz2 = FM::op(lz2, buffer.m_nodes[k2].lazy);
        const auto left_child = dfs(buffer.m_nodes[k1].children[0], buffer.m_nodes[k2].children[0], kl, km, next_lz1, next_lz2);
        const auto right_child = dfs(buffer.m_nodes[k1].children[1], buffer.m_nodes[k2].children[1], km, kr, next_lz1, next_lz2);
        buffer.m_nodes.push_back({FM::e(), {left_child, right_child}});
        return buffer.m_nodes.size() - 1;
      })(res.m_root, s.m_root, 0, res.capacity(), FM::e(), FM::e());
      return res;
    }
  };
}

#endif

#line 1 "tools/persistent_dual_segtree.hpp"



#include <algorithm>
#include <array>
#include <cassert>
#include <functional>
#include <numeric>
#include <ranges>
#include <type_traits>
#include <utility>
#include <vector>
#line 1 "tools/ceil_log2.hpp"



#line 1 "tools/bit_width.hpp"



#include <bit>
#line 1 "tools/is_integral.hpp"



#line 5 "tools/is_integral.hpp"

namespace tools {
  template <typename T>
  struct is_integral : ::std::is_integral<T> {};

  template <typename T>
  inline constexpr bool is_integral_v = ::tools::is_integral<T>::value;
}


#line 1 "tools/is_signed.hpp"



#line 5 "tools/is_signed.hpp"

namespace tools {
  template <typename T>
  struct is_signed : ::std::is_signed<T> {};

  template <typename T>
  inline constexpr bool is_signed_v = ::tools::is_signed<T>::value;
}


#line 1 "tools/make_unsigned.hpp"



#line 5 "tools/make_unsigned.hpp"

namespace tools {
  template <typename T>
  struct make_unsigned : ::std::make_unsigned<T> {};

  template <typename T>
  using make_unsigned_t = typename ::tools::make_unsigned<T>::type;
}


#line 10 "tools/bit_width.hpp"

namespace tools {
  template <typename T>
  constexpr int bit_width(T) noexcept;

  template <typename T>
  constexpr int bit_width(const T x) noexcept {
    static_assert(::tools::is_integral_v<T> && !::std::is_same_v<::std::remove_cv_t<T>, bool>);
    if constexpr (::tools::is_signed_v<T>) {
      assert(x >= 0);
      return ::tools::bit_width<::tools::make_unsigned_t<T>>(x);
    } else {
      return ::std::bit_width(x);
    }
  }
}


#line 6 "tools/ceil_log2.hpp"

namespace tools {
  template <typename T>
  constexpr T ceil_log2(T x) noexcept {
    assert(x > 0);
    return ::tools::bit_width(x - 1);
  }
}


#line 1 "tools/fix.hpp"



#line 6 "tools/fix.hpp"

namespace tools {
  template <typename F>
  struct fix : F {
    template <typename G>
    fix(G&& g) : F({::std::forward<G>(g)}) {
    }

    template <typename... Args>
    decltype(auto) operator()(Args&&... args) const {
      return F::operator()(*this, ::std::forward<Args>(args)...);
    }
  };

  template <typename F>
  fix(F&&) -> fix<::std::decay_t<F>>;
}


#line 15 "tools/persistent_dual_segtree.hpp"

namespace tools {
  template <typename FM>
  class persistent_dual_segtree {
    using F = typename FM::T;

    struct node {
      F lazy;
      ::std::array<int, 2> children;
    };

  public:
    class buffer {
      ::std::vector<::tools::persistent_dual_segtree<FM>::node> m_nodes;
      long long m_offset;
      long long m_size;
      int m_height;

    public:
      friend ::tools::persistent_dual_segtree<FM>;
    };

  private:
    ::tools::persistent_dual_segtree<FM>::buffer *m_buffer;
    int m_root;

    long long capacity() const {
      return 1LL << this->m_buffer->m_height;
    }

  public:
    persistent_dual_segtree() = default;
    persistent_dual_segtree(::tools::persistent_dual_segtree<FM>::buffer& buffer, const long long l_star, const long long r_star) : m_buffer(&buffer) {
      assert(buffer.m_nodes.empty());
      assert(l_star <= r_star);
      buffer.m_offset = l_star;
      buffer.m_size = r_star - l_star;
      buffer.m_height = ::tools::ceil_log2(::std::max(1LL, r_star - l_star));
      buffer.m_nodes.push_back({FM::e(), {-1, -1}});
      for (int k = 1; k <= buffer.m_height; ++k) {
        buffer.m_nodes.push_back({FM::e(), {k - 1, k - 1}});
      }
      this->m_root = buffer.m_height;
    }

    long long lower_bound() const {
      return this->m_buffer->m_offset;
    }
    long long upper_bound() const {
      return this->m_buffer->m_offset + this->m_buffer->m_size;
    }
    F get(long long p) const {
      assert(this->lower_bound() <= p && p < this->upper_bound());
      auto& buffer = *this->m_buffer;
      p -= buffer.m_offset;

      return ::tools::fix([&](auto&& dfs, const int k, const long long kl, const long long kr, const F& lz) -> F {
        assert(kl < kr);
        if (p <= kl && kr <= p + 1) return FM::op(lz, buffer.m_nodes[k].lazy);
        const auto km = ::std::midpoint(kl, kr);
        const F next_lz = FM::op(lz, buffer.m_nodes[k].lazy);
        if (p < km) return dfs(buffer.m_nodes[k].children[0], kl, km, next_lz);
        else return dfs(buffer.m_nodes[k].children[1], km, kr, next_lz);
      })(this->m_root, 0, this->capacity(), FM::e());
    }
    ::tools::persistent_dual_segtree<FM> apply(const long long p, const F& f) const {
      return this->apply(p, p + 1, f);
    }
    ::tools::persistent_dual_segtree<FM> apply(long long l, long long r, const F& f) const {
      assert(this->lower_bound() <= l && l <= r && r <= this->upper_bound());
      if (l == r) return *this;
      auto& buffer = *this->m_buffer;
      l -= buffer.m_offset;
      r -= buffer.m_offset;

      auto res = *this;
      res.m_root = ::tools::fix([&](auto&& dfs, const int k, const long long kl, const long long kr, const F& lz) -> int {
        assert(kl < kr);
        if (l <= kl && kr <= r) {
          const F modified_lz = FM::op(f, lz);
          buffer.m_nodes.push_back({FM::op(modified_lz, buffer.m_nodes[k].lazy), buffer.m_nodes[k].children});
          return buffer.m_nodes.size() - 1;
        }
        if (kr <= l || r <= kl) {
          if (lz == FM::e()) return k;
          buffer.m_nodes.push_back({FM::op(lz, buffer.m_nodes[k].lazy), buffer.m_nodes[k].children});
          return buffer.m_nodes.size() - 1;
        }
        const auto km = ::std::midpoint(kl, kr);
        const F next_lz = FM::op(lz, buffer.m_nodes[k].lazy);
        const auto left_child = dfs(buffer.m_nodes[k].children[0], kl, km, next_lz);
        const auto right_child = dfs(buffer.m_nodes[k].children[1], km, kr, next_lz);
        buffer.m_nodes.push_back({FM::e(), {left_child, right_child}});
        return buffer.m_nodes.size() - 1;
      })(res.m_root, 0, res.capacity(), FM::e());
      return res;
    }
    ::tools::persistent_dual_segtree<FM> rollback(const ::tools::persistent_dual_segtree<FM>& s, long long l, long long r) const {
      assert(this->m_buffer == s.m_buffer);
      assert(this->lower_bound() <= l && l <= r && r <= this->upper_bound());
      if (l == r) return *this;
      if (l == this->lower_bound() && r == this->upper_bound()) return s;
      auto& buffer = *this->m_buffer;
      l -= buffer.m_offset;
      r -= buffer.m_offset;

      auto res = *this;
      res.m_root = ::tools::fix([&](auto&& dfs, const int k1, const int k2, const long long kl, const long long kr, const F& lz1, const F& lz2) -> int {
        assert(kl < kr);
        if (l <= kl && kr <= r) {
          if (lz2 == FM::e()) return k2;
          buffer.m_nodes.push_back({FM::op(lz2, buffer.m_nodes[k2].lazy), buffer.m_nodes[k2].children});
          return buffer.m_nodes.size() - 1;
        }
        if (kr <= l || r <= kl) {
          if (lz1 == FM::e()) return k1;
          buffer.m_nodes.push_back({FM::op(lz1, buffer.m_nodes[k1].lazy), buffer.m_nodes[k1].children});
          return buffer.m_nodes.size() - 1;
        }
        const auto km = ::std::midpoint(kl, kr);
        const F next_lz1 = FM::op(lz1, buffer.m_nodes[k1].lazy);
        const F next_lz2 = FM::op(lz2, buffer.m_nodes[k2].lazy);
        const auto left_child = dfs(buffer.m_nodes[k1].children[0], buffer.m_nodes[k2].children[0], kl, km, next_lz1, next_lz2);
        const auto right_child = dfs(buffer.m_nodes[k1].children[1], buffer.m_nodes[k2].children[1], km, kr, next_lz1, next_lz2);
        buffer.m_nodes.push_back({FM::e(), {left_child, right_child}});
        return buffer.m_nodes.size() - 1;
      })(res.m_root, s.m_root, 0, res.capacity(), FM::e(), FM::e());
      return res;
    }
  };
}

Back to top page