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binary-trie
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‎cpp/binary-trie.hpp

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#pragma once
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#include <assert.h>
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#include <memory>
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#include <utility>
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/**
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* @brief 符号なし整数の多重集合を管理する
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* @tparam `d` 扱う整数値のビット幅。64以下であることを要請
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*/
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template <unsigned int d> class BinaryTrie {
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static_assert(d <= 64, "d must be 64 or less");
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struct BinaryTrieNode {
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std::shared_ptr<BinaryTrieNode> children[2] = {nullptr, nullptr};
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unsigned int level, subcnt = 0;
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unsigned long long xval = 0;
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BinaryTrieNode(int lvl) : level(lvl) {}
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bool get_bit(unsigned long long v) const { return (v >> (level - 1)) & 1; }
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bool is_leaf() const { return level == 0; }
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// 子の状態がvalidかどうか
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// 0: xx, 1: xo, 2: ox, 3: oo
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int state_children() const {
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return ((bool)children[1] << 1) | (bool)children[0];
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}
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// xorの値を子ノードに伝播する
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void affect_xor() {
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if (get_bit(xval)) {
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std::swap(children[0], children[1]);
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}
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if (children[0])
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children[0]->xval ^= xval;
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if (children[1])
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children[1]->xval ^= xval;
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xval = 0;
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}
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};
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using NodePtr = std::shared_ptr<BinaryTrieNode>;
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NodePtr root_ptr = std::make_shared<BinaryTrieNode>(d);
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public:
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/**
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* @brief 集合にnを追加 (O(d))
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*/
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void insert(unsigned long long n) {
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NodePtr cur_ptr = root_ptr;
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while (!cur_ptr->is_leaf()) {
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cur_ptr->affect_xor();
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cur_ptr->subcnt += 1;
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NodePtr &nxt_ptr = cur_ptr->children[cur_ptr->get_bit(n)];
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if (!nxt_ptr) {
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nxt_ptr = std::make_shared<BinaryTrieNode>(cur_ptr->level - 1);
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}
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cur_ptr = nxt_ptr;
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}
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assert(cur_ptr->is_leaf());
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cur_ptr->subcnt += 1;
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}
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int size() const { return root_ptr->subcnt; }
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/**
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* @brief 集合からnを検索し、見つかった数を求める (O(d))
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*/
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int count(unsigned long long n) const {
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NodePtr cur_ptr = root_ptr;
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while (!cur_ptr->is_leaf()) {
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cur_ptr->affect_xor();
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NodePtr &nxt_ptr = cur_ptr->children[cur_ptr->get_bit(n)];
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if (!nxt_ptr) {
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return nullptr;
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}
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cur_ptr = nxt_ptr;
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}
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return (cur_ptr ? cur_ptr->subcnt : 0);
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}
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/**
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* @brief 集合からnを削除 (O(d))
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* @note 存在しない要素を指定したとき、何も起こらない
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*/
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void erase(unsigned long long n) const {
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int cnt = count(n);
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if (cnt == 0)
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return;
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NodePtr cur_ptr = root_ptr;
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while (true) {
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cur_ptr->affect_xor();
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cur_ptr->subcnt -= cnt;
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NodePtr &nxt_ptr = cur_ptr->children[cur_ptr->get_bit(n)];
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if (nxt_ptr->subcnt == cnt) {
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nxt_ptr = nullptr;
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return;
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}
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cur_ptr = nxt_ptr;
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}
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}
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/**
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* @brief 集合からnを一つだけ削除 (O(d))
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* @note 存在しない要素を指定したとき、何も起こらない
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*/
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void erase_one_element(unsigned long long n) const {
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if (count(n) == 0)
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return;
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NodePtr cur_ptr = root_ptr;
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while (!cur_ptr->is_leaf()) {
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cur_ptr->affect_xor();
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cur_ptr->subcnt -= 1;
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NodePtr &nxt_ptr = cur_ptr->children[cur_ptr->get_bit(n)];
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if (nxt_ptr->subcnt == 1) {
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nxt_ptr = nullptr;
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return;
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}
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cur_ptr = nxt_ptr;
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}
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cur_ptr->subcnt -= 1;
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}
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/**
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* @brief 昇順でn番目の要素を探索 (O(d))
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* @note nがtrie木のサイズ以上な場合、assert
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*/
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unsigned long long nth_element(int n) const {
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assert(n < size());
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unsigned long long ret = 0;
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NodePtr cur_ptr = root_ptr;
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while (!cur_ptr->is_leaf()) {
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cur_ptr->affect_xor();
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ret <<= 1;
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int state = cur_ptr->state_children();
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NodePtr &z_ptr = cur_ptr->children[0];
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NodePtr &o_ptr = cur_ptr->children[1];
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assert(state > 0);
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if (state == 1 || (state == 3 && n < z_ptr->subcnt)) {
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cur_ptr = z_ptr;
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} else {
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n -= (state & 1 ? z_ptr->subcnt : 0);
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ret |= 1;
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cur_ptr = o_ptr;
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}
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}
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return ret;
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}
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/**
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* @brief n以上の要素を探索 (O(d))
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* @return 探索した値が昇順で何番目か (0-indexed)。該当する要素がなければtrie木のサイズが返る
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*/
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int lower_bound(unsigned long long n) const {
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int ret = 0;
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NodePtr cur_ptr = root_ptr;
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while (!cur_ptr->is_leaf()) {
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cur_ptr->affect_xor();
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bool b = cur_ptr->get_bit(n);
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NodePtr &nxt_ptr = cur_ptr->children[b];
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NodePtr &z_ptr = cur_ptr->children[0];
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if (b && z_ptr) {
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ret += z_ptr->subcnt;
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}
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if (!nxt_ptr) {
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break;
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}
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cur_ptr = nxt_ptr;
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}
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return ret;
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}
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/**
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* @brief nより大きな要素を探索 (O(d))
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* @return 探索した値が昇順で何番目か (0-indexed)。該当する要素がなければtrie木のサイズが返る
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*/
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int upper_bound(unsigned long long n) const {
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return (n < UINT64_MAX ? lower_bound(n + 1) : size());
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}
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/**
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* @brief 集合のすべての要素にxorを作用
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*/
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void apply_xor(unsigned long long n) { root_ptr->xval ^= n; }
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/**
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* @brief 要素をすべて削除する。確保したメモリ領域も削除される
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*/
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void clear() {
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root_ptr = std::make_shared<BinaryTrieNode>(d);
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}
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};

‎test/yosupo-set-xor-min.test.cpp

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#define PROBLEM "https://judge.yosupo.jp/problem/set_xor_min"
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#include <iostream>
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#include "../cpp/binary-trie.hpp"
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int main(void) {
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int Q;
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std::cin >> Q;
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BinaryTrie<30> A;
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while (Q--) {
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int q, x;
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std::cin >> q >> x;
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switch (q) {
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case 0:
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if (A.count(x) == 0) {
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A.insert(x);
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}
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break;
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case 1:
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A.erase(x);
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break;
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case 2:
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A.apply_xor(x);
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std::cout << A.nth_element(0) << std::endl;
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A.apply_xor(x);
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break;
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}
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}
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}

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