Fully Indexable Dictionary / 完備辞書
(data_structure/fully_indexable_dictionary.hpp)

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• Last update: 2020-02-28 15:08:46+09:00
• Include: #include "data_structure/fully_indexable_dictionary.hpp"

概要

長さ $N$ の bit 列 $b = (b_0, b_1, \dots, b _ {n - 1}) \in 2^N$ に対し、次が $O(1)$ で処理可能:

• $\mathtt{rank}(l, r)$: 区間 $\lbrack l, r)$ 中の値 $1$ の出現回数 \(\unicode{35} \lbrace i \in \lbrack l, r) \mid a_i = 1 \rbrace\) を計算する。
• $\mathtt{select}(k)$: 位置 $l$ 以降で $k \ge 0$ 番目に出現する値 $1$ の位置 $i$ (つまり $i \ge l$ かつ $a_i = 1$ かつ \(\unicode{35} \lbrace j \in \lbrack l, i) \mid a_j = 1 \rbrace = k\) を満たす $i$) を計算する。
  • ただし今回の実装では妥協の結果 $O(\log N)$ になっている
• $\mathtt{access}(i)$: 値 $b_i$ を計算する。

Depends on

• utils/macros.hpp

Required by

• Wavelet Matrix (data_structure/wavelet_matrix.hpp)

Verified with

• data_structure/wavelet_matrix.aoj2674.test.cpp
• data_structure/wavelet_matrix.range_kth_smallest.test.cpp
• data_structure/wavelet_matrix.rectangle_sum.test.cpp

Code

#pragma once
#include <algorithm>
#include <cassert>
#include <cstdint>
#include <vector>
#include "utils/macros.hpp"

/**
 * @brief Fully Indexable Dictionary / 完備辞書
 * @docs data_structure/fully_indexable_dictionary.md
 * @note space complexity $o(N)$. $1.5N$-bit consumed
 */
class fully_indexable_dictionary {
    static constexpr std::size_t block_size = 64;
    std::vector<uint64_t> block;
    std::vector<int32_t> rank_block;  // a blocked cumulative sum
public:
    std::size_t size;
    fully_indexable_dictionary() = default;
    template <typename T>
    fully_indexable_dictionary(const std::vector<T> & bits) {
        size = bits.size();
        std::size_t block_count = size / block_size + 1;
        block.resize(block_count);
        REP (i, size) if (bits[i]) {
            block[i / block_size] |= (1ull << (i % block_size));
        }
        rank_block.resize(block_count);
        rank_block[0] = 0;
        REP (i, block_count - 1) {
            rank_block[i + 1] = rank_block[i] + __builtin_popcountll(block[i]);
        }
    }

    /**
     * @brief count the number of value in $[0, r)$
     * @note $O(1)$
     */
    int rank(bool value, int r) const {
        assert (0 <= r and r <= size);
        uint64_t mask = (1ull << (r % block_size)) - 1;
        int rank_1 = rank_block[r / block_size] + __builtin_popcountll(block[r /block_size] & mask);
        return value ? rank_1 : r - rank_1;
    }
    int rank(bool value, int l, int r) const {
        assert (0 <= l and l <= r and r <= size);
        return rank(value, r) - rank(value, l);
    }

    /**
     * @brief find the index of the $k$-th occurrence of value
     * @note if there is no such index, returns size
     * @note $O(\log N)$
     */
    int select(bool value, int k) const {
        if (k >= rank(value, size)) return size;
        // binary search: max { i | rank_block[i] <= k }
        int l = 0, r = block.size();  // [l, r)
        while (r - l > 1) {
            int m = (l + r) / 2;
            int rank_block_m = (value ? rank_block[m] : m * block_size - rank_block[m]);
            (rank_block_m <= k ? l : r) = m;
        }
        int block_index = l;
        // binary search: max { i | rank(i) <= k }
        l = block_index * block_size;
        r = std::min<int>(size, (block_index + 1) * block_size);  // [l, r)
        while (r - l > 1) {
            int m = (l + r) / 2;
            (rank(value, m) <= k ? l : r) = m;
        }
        return l;
    }
    /**
     * @brief select(value, k) in [l, size)
     */
    int select(bool value, int k, int l) const {
        assert (0 <= l and l <= size);
        return select(value, k + rank(value, l));
    }

    /**
     * @brief get the $i$-th element
     * @note $O(1)$
     */
    bool access(int i) const {
        assert (0 <= i and i < size);
        return block[i / block_size] & (1ull << (i % block_size));
    }
};

#line 2 "data_structure/fully_indexable_dictionary.hpp"
#include <algorithm>
#include <cassert>
#include <cstdint>
#include <vector>
#line 2 "utils/macros.hpp"
#define REP(i, n) for (int i = 0; (i) < (int)(n); ++ (i))
#define REP3(i, m, n) for (int i = (m); (i) < (int)(n); ++ (i))
#define REP_R(i, n) for (int i = (int)(n) - 1; (i) >= 0; -- (i))
#define REP3R(i, m, n) for (int i = (int)(n) - 1; (i) >= (int)(m); -- (i))
#define ALL(x) std::begin(x), std::end(x)
#line 7 "data_structure/fully_indexable_dictionary.hpp"

/**
 * @brief Fully Indexable Dictionary / 完備辞書
 * @docs data_structure/fully_indexable_dictionary.md
 * @note space complexity $o(N)$. $1.5N$-bit consumed
 */
class fully_indexable_dictionary {
    static constexpr std::size_t block_size = 64;
    std::vector<uint64_t> block;
    std::vector<int32_t> rank_block;  // a blocked cumulative sum
public:
    std::size_t size;
    fully_indexable_dictionary() = default;
    template <typename T>
    fully_indexable_dictionary(const std::vector<T> & bits) {
        size = bits.size();
        std::size_t block_count = size / block_size + 1;
        block.resize(block_count);
        REP (i, size) if (bits[i]) {
            block[i / block_size] |= (1ull << (i % block_size));
        }
        rank_block.resize(block_count);
        rank_block[0] = 0;
        REP (i, block_count - 1) {
            rank_block[i + 1] = rank_block[i] + __builtin_popcountll(block[i]);
        }
    }

    /**
     * @brief count the number of value in $[0, r)$
     * @note $O(1)$
     */
    int rank(bool value, int r) const {
        assert (0 <= r and r <= size);
        uint64_t mask = (1ull << (r % block_size)) - 1;
        int rank_1 = rank_block[r / block_size] + __builtin_popcountll(block[r /block_size] & mask);
        return value ? rank_1 : r - rank_1;
    }
    int rank(bool value, int l, int r) const {
        assert (0 <= l and l <= r and r <= size);
        return rank(value, r) - rank(value, l);
    }

    /**
     * @brief find the index of the $k$-th occurrence of value
     * @note if there is no such index, returns size
     * @note $O(\log N)$
     */
    int select(bool value, int k) const {
        if (k >= rank(value, size)) return size;
        // binary search: max { i | rank_block[i] <= k }
        int l = 0, r = block.size();  // [l, r)
        while (r - l > 1) {
            int m = (l + r) / 2;
            int rank_block_m = (value ? rank_block[m] : m * block_size - rank_block[m]);
            (rank_block_m <= k ? l : r) = m;
        }
        int block_index = l;
        // binary search: max { i | rank(i) <= k }
        l = block_index * block_size;
        r = std::min<int>(size, (block_index + 1) * block_size);  // [l, r)
        while (r - l > 1) {
            int m = (l + r) / 2;
            (rank(value, m) <= k ? l : r) = m;
        }
        return l;
    }
    /**
     * @brief select(value, k) in [l, size)
     */
    int select(bool value, int k, int l) const {
        assert (0 <= l and l <= size);
        return select(value, k + rank(value, l));
    }

    /**
     * @brief get the $i$-th element
     * @note $O(1)$
     */
    bool access(int i) const {
        assert (0 <= i and i < size);
        return block[i / block_size] & (1ull << (i % block_size));
    }
};

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