Euler Tour Tree
コードについての説明
木のオイラーツアー(DFS で探索するときの頂点順)を動的に管理するデータ構造. 辺の追加, 削除, 頂点間の連結性判定,
根の付け替えおよび連結成分内の全ての頂点に対する演算もしくはある部分木内の全ての頂点に対する演算などが $\O (\log n)$ で行える. 辺の追加はもちろん閉路ができないような追加のみ許される.
LinkCutTree とは異なり, 部分木もしくは連結成分全体に対する処理が効率よく行え, 動的連結性クエリに対するアルゴリズムにも用いられている.
実装は平衡二分木の split, merge を用いてでき, 説明はこのスライドがわかりやすい.
以下の実装では平衡二分木として splay 木を用いているためならし $\O (\log n)$ となっている. (時間効率, メモリ効率の意味で merge-split ベースの赤黒木で実装するよりも splay 木の方が良いと思っている).
また頂点数を $n$, 辺数を $m$ とすると $n + 2m$ 個のノードからなる平衡二分探索木を保持するため定数倍が少々重い. 平衡二分探索木ではなく $\O (\log n)$-分木で保持することで計算量上では $\O (\log \log n)$ factor の改善が可能になる
そのため部分木内の頂点集合に対する演算は op(node v, node p, function f) の形($v$ が根なら $p$ は $-1$ とする) で与えることにしている.
ここで必ず $p$ は $v$ の親である必要があることに注意する(今, 森は根つき森として管理している). もし $p$ が $v$ の親であることが保証されない場合(1 回でも link 操作を行った場合) は query(v, p) を呼ぶ前に reroot$(p)$ を行う.
ヘルパー関数は例えば最初辺数 $0$ からスタートではなくある木や森からスタートというときにそのグラフに対応する完全平衡二分木を効率よく構築し, それを初めの splay 木としてやることでそれ以降の操作の高速化が望めそう(？)なので置いてある.
元論文は "Randomized Dynamic Graph Algorithms with Polylogarithmic Time per Operation" [Henzinger, King 1995]
(追記)
高速な自作 Unordered Map を用いた高速版も一応用意した.
時間計算量: 各クエリならし $\O (\log n)$ (本来は最悪 $\O (\log n)$)
コード
template<typename _Tp> class BSTNode {
public:
int from, to, sz;
_Tp value, al, lazy;
BSTNode *left, *right, *par;
BSTNode(const int _from, const int _to) noexcept
: from(_from), to(_to), sz(0), value(id1), al(id2), lazy(id1),
left(nullptr), right(nullptr), par(nullptr){}
BSTNode(const int _from, const int _to, const _Tp& _value) noexcept
: from(_from), to(_to), sz(1), value(_value), al(value), lazy(id1),
left(nullptr), right(nullptr), par(nullptr){}
static const _Tp id1 = (_Tp)0;
static const _Tp id2 = (_Tp)0;
static void opr1(_Tp& arg1, const _Tp arg2) noexcept { arg1 += arg2; }
static _Tp opr2(const _Tp arg1, const _Tp arg2) noexcept { return arg1 + arg2; }
inline bool IsRoot() const noexcept { return !par; }
inline bool IsVertex() const noexcept { return from == to; }
void push() noexcept {
if(lazy != id1){
if(IsVertex()) opr1(value, lazy);
al += lazy * sz;
if(left) opr1(left->lazy, lazy);
if(right) opr1(right->lazy, lazy);
lazy = id1;
}
}
void eval() noexcept {
sz = IsVertex(), al = value;
if(left) left->push(), sz += left->sz, al = opr2(left->al, al);
if(right) right->push(), sz += right->sz, al = opr2(al, right->al);
}
void rotate(const bool right_) noexcept {
BSTNode *p = par, *g = p->par;
if(right_){
if((p->left = right)) right->par = p;
right = p, p->par = this;
}else{
if((p->right = left)) left->par = p;
left = p, p->par = this;
}
p->eval(), eval();
if(!(par = g)) return;
if(g->left == p) g->left = this;
if(g->right == p) g->right = this;
g->eval();
}
};
template<typename _Tp>
BSTNode<_Tp>* splay(BSTNode<_Tp> *u) noexcept {
if(!u) return nullptr;
while(!(u->IsRoot())){
BSTNode<_Tp> *p = u->par, *gp = p->par;
if(p->IsRoot()){ // zig
p->push(), u->push();
u->rotate((u == p->left));
}else{
gp->push(), p->push(), u->push();
bool flag = (u == p->left);
if((u == p->left) == (p == gp->left)){ // zig-zig
p->rotate(flag), u->rotate(flag);
}else{ // zig-zag
u->rotate(flag), u->rotate(!flag);
}
}
}
u->push();
return u;
}
template<typename _Tp>
BSTNode<_Tp>* join(BSTNode<_Tp> *root1, BSTNode<_Tp> *root2) noexcept {
if(!root1 || !root2) return root1 ? root1 : root2;
BSTNode<_Tp> *cur = nullptr, *nx = root1;
do{ cur = nx, cur->push(), nx = cur->right; }while(nx);
BSTNode<_Tp> *ver = splay(cur);
ver->right = root2, ver->eval(), root2->par = ver;
return ver;
}
template<typename _Tp>
pair<BSTNode<_Tp>*, BSTNode<_Tp>*> split_lower_bond(BSTNode<_Tp> *ver) noexcept {
BSTNode<_Tp> *res = splay(ver)->left;
ver->left = nullptr, ver->eval();
if(res) res->par = nullptr;
return make_pair(res, ver);
}
template<typename _Tp>
pair<BSTNode<_Tp>*, BSTNode<_Tp>*> split_upper_bond(BSTNode<_Tp> *ver) noexcept {
BSTNode<_Tp> *res = splay(ver)->right;
ver->right = nullptr, ver->eval();
if(res) res->par = nullptr;
return make_pair(ver, res);
}
template<typename T> class EulerTourTree {
private:
struct pair_hash {
template <class T1, class T2>
size_t operator() (const pair<T1, T2>& p) const {
size_t lhs = hash<T1>()(p.first), rhs = hash<T2>()(p.second);
return lhs^(rhs+0x9e3779b9+(lhs<<6)+(lhs>>2));
}
};
BSTNode<T>** vertex_set;
unordered_map<pair<int, int>, pair<BSTNode<T>*, BSTNode<T>*>, pair_hash> edge_set;
BSTNode<T> *reroot(BSTNode<T> *ver) noexcept {
BSTNode<T> *res = splay(ver)->left;
if(!res) return ver;
ver->left = nullptr, ver->eval();
while(ver->right) ver->push(), ver = ver->right;
splay(ver), ver->right = res, ver->eval(), res->par = ver;
return ver;
}
void link(BSTNode<T> *ver1, BSTNode<T> *ver2) noexcept {
BSTNode<T>* e1 = new BSTNode<T>(ver1->from, ver2->from);
BSTNode<T>* e2 = new BSTNode<T>(ver2->from, ver1->from);
edge_set[{ver1->from, ver2->from}] = {e1, e2};
join(join(reroot(ver1), e1), join(reroot(ver2), e2));
}
void cut(BSTNode<T> *edge1, BSTNode<T> *edge2) noexcept {
splay(edge1), splay(edge2);
BSTNode<T> *p = edge1->par;
bool _right = (edge1 == edge2->right);
if(p != edge2){
_right = (p == edge2->right);
p->par = nullptr, edge1->rotate((edge1 == p->left));
}
if(edge1->left) edge1->left->par = nullptr;
if(edge1->right) edge1->right->par = nullptr;
if(_right){
if(edge2->left) edge2->left->par = nullptr;
join(edge2->left, edge1->right);
}else{
if(edge2->right) edge2->right->par = nullptr;
join(edge1->left, edge2->right);
}
// delete edge1; delete edge2;
}
bool connected(BSTNode<T> *ver1, BSTNode<T> *ver2) noexcept {
splay(ver1), splay(ver2);
return ver1->par;
}
T& get(BSTNode<T> *ver) noexcept { return splay(ver)->value; }
void point_update(BSTNode<T> *ver, const T& val){
BSTNode<T>::opr1(splay(ver)->value, val);
ver->al = BSTNode<T>::opr2(ver->al, val);
}
void range(BSTNode<T> *edge1, BSTNode<T> *edge2, const T& val) noexcept {
auto res1 = split_lower_bond(edge1);
auto res2 = split_upper_bond(edge2);
BSTNode<T>::opr1(res2.first->lazy, val);
join(join(res1.first,res2.first), res2.second);
}
T query(BSTNode<T> *edge1, BSTNode<T> *edge2) noexcept {
auto res1 = split_lower_bond(edge1);
auto res2 = split_upper_bond(edge2);
T res = res2.first->al;
return join(join(res1.first,res2.first), res2.second), res;
}
void dfs(const int u, const int p, const BSTNode<T> *cur,
bool *visit, vector<BSTNode<T>*>& nodes, const vector<vector<int> >& G) noexcept {
visit[u] = true;
nodes.push_back(vertex_set[u]);
for(auto& v : G[u]){
if(v != p){
BSTNode<T>* e1 = new BSTNode<T>(u, v);
nodes.push_back(e1);
dfs(v, u, cur, visit, nodes, G);
BSTNode<T>* e2 = new BSTNode<T>(v, u);
if(u < v) edge_set[{u, v}] = {e1, e2};
else edge_set[{v, u}] = {e2, e1};
nodes.push_back(e2);
}
}
}
void bst_build(vector<BSTNode<T>* >& nodes) noexcept {
int i, n = (int)nodes.size(), st = 2, isolate = ((n % 4 == 1) ? (n-1) : -1);
while(st <= n){
for(i = st-1; i < n; i += 2*st){
nodes[i]->left = nodes[i-st/2], nodes[i-st/2]->par = nodes[i];
if(i+st/2 < n) nodes[i]->right = nodes[i+st/2], nodes[i+st/2]->par = nodes[i];
else if(isolate >= 0) nodes[i]->right = nodes[isolate], nodes[isolate]->par = nodes[i];
nodes[i]->eval();
}
isolate = ((n % (4*st) >= st && (n % (4*st) < 2*st)) ? (i-2*st): isolate);
st <<= 1;
}
}
void build_forest(const vector<vector<int> >& forest) noexcept {
bool *visit = new bool[V]();
for(int i = 0; i < V; i++){
if(!visit[i]){
vector<BSTNode<T>* > nodes;
BSTNode<T> *cur = nullptr;
dfs(i, -1, cur, visit, nodes, forest);
bst_build(nodes);
}
}
// delete[] visit;
}
void build_tree(const int root, const vector<vector<int> >& tree) noexcept {
bool *visit = new bool[V]();
vector<BSTNode<T>* > nodes;
BSTNode<T> *cur = nullptr;
dfs(root, -1, cur, visit, nodes, tree);
bst_build(nodes);
// delete[] visit;
}
public:
const int V;
EulerTourTree(const vector<T>& ver_value) noexcept : V((int)ver_value.size()){
vertex_set = new BSTNode<T>*[V];
for(int i = 0; i < V; i++) vertex_set[i] = new BSTNode<T>(i, i, ver_value[i]);
}
EulerTourTree(const vector<T>& ver_value, const vector<vector<int> >& forest) noexcept
: EulerTourTree(ver_value){ build_forest(forest); }
EulerTourTree(const vector<T>& ver_value, const int root, const vector<vector<int> >& tree) noexcept
: EulerTourTree(ver_value){ build_tree(root, tree); }
// ~EulerTourTree(){
// for(auto it : edge_set){
// delete (it.second).first;
// delete (it.second).second;
// }
// for(int i = 0; i < V; ++i) delete vertex_set[i];
// delete[] vertex_set;
// }
// 根を node_id にする
void reroot(const int node_id) noexcept { reroot(vertex_set[node_id]); }
// 辺(node1_id, node2_id) を追加
void link(int node1_id, int node2_id) noexcept {
if(node1_id > node2_id) swap(node1_id, node2_id);
link(vertex_set[node1_id], vertex_set[node2_id]);
}
// 辺(node1_id, node2_id) を削除(逆向きでも問題ない)
void cut(int node1_id, int node2_id){
if(node1_id > node2_id) swap(node1_id, node2_id);
auto it = edge_set.find({node1_id, node2_id});
assert(it != edge_set.end());
BSTNode<T> *edge1 = (it->second).first, *edge2 = (it->second).second;
edge_set.erase(it);
cut(edge1, edge2);
}
// node1_id と node2_id が同じ木(連結成分)に属するか
bool connected(const int node1_id, const int node2_id) noexcept {
if(node1_id == node2_id) return true;
return connected(vertex_set[node1_id], vertex_set[node2_id]);
}
// 頂点 ver_id の値を取得
T& get(const int ver_id) noexcept { return get(vertex_set[ver_id]); }
// 頂点 ver_id に val を加える
void point_update(const int ver_id, const T& val) noexcept {
return point_update(vertex_set[ver_id], val);
}
// 頂点 ver_id の存在する連結成分内の頂点全体に val を加える
void component_range(const int ver_id, const T& val){ range(ver_id, -1, val); }
// 親が par_id であるような頂点 ver_id の部分木内に存在する頂点全体に val を加える
void range(const int ver_id, const int par_id, const T& val){
if(par_id < 0) return BSTNode<T>::opr1(splay(vertex_set[ver_id])->lazy, val);
if(ver_id < par_id){
auto it = edge_set.find({ver_id, par_id});
assert(it != edge_set.end());
range((it->second).second, (it->second).first, val);
}else{
auto it = edge_set.find({par_id, ver_id});
assert(it != edge_set.end());
range((it->second).first, (it->second).second, val);
}
}
// 頂点 ver_id の存在する連結成分内の頂点全体の総和を取得
T component_query(const int ver_id){ return query(ver_id, -1); }
// 親が par_id であるような頂点 ver_id の部分木内に存在する頂点全体の総和を取得
T query(const int ver_id, const int par_id){
if(par_id < 0) return splay(vertex_set[ver_id])->al;
if(ver_id < par_id){
auto it = edge_set.find({ver_id, par_id});
assert(it != edge_set.end());
return query((it->second).second, (it->second).first);
}else{
auto it = edge_set.find({par_id, ver_id});
assert(it != edge_set.end());
return query((it->second).first, (it->second).second);
}
}
// DEBUG 用
void check_dfs(const BSTNode<T>* cur) const noexcept {
cur->push();
if(cur->left) check_dfs(cur->left);
if(cur->IsVertex()) cout <<"{" << (cur->from) << "," << (cur->value) << "} ";
if(cur->right) check_dfs(cur->right);
}
};
コード(高速版)
template<class _Key, class _Tp, class _Hash, bool DOWNSIZE> class UnorderedMapIterator;
template<class _Key, class _Tp, class _Hash = hash<_Key>, bool DOWNSIZE = false>
class UnorderedMap
{
private:
using iterator = UnorderedMapIterator<_Key, _Tp, _Hash, DOWNSIZE>;
using value_type = _Tp;
using data_type = pair<_Key, _Tp>;
using aligned_pointer = typename aligned_storage<sizeof(value_type), alignof(value_type)>::type;
friend UnorderedMapIterator<_Key, _Tp, _Hash, DOWNSIZE>;
struct bucket {
_Key _key;
short int _dist;
bool _last, _end;
aligned_pointer _value_ptr;
bucket() noexcept : _dist(-1), _last(false), _end(false){}
bucket& operator=(const bucket& another) noexcept {
_key = another._key, _dist = another._dist, _last = another._last, _end = another._end;
if(!another.empty()){
new(&_value_ptr) value_type(*reinterpret_cast<const value_type*>(&another._value_ptr));
}
return *this;
}
~bucket(){ if(!empty()) _delete(); }
inline void clear() noexcept { _dist = -1; }
inline void _delete(){ _dist = -1, value_ptr()->~value_type(); }
inline bool empty() const noexcept { return (_dist == -1); }
inline value_type& value() noexcept {
return *reinterpret_cast<value_type*>(&_value_ptr);
}
inline value_type* value_ptr() noexcept {
return reinterpret_cast<value_type*>(&_value_ptr);
}
inline void new_value(value_type&& value){
new(&_value_ptr) value_type(move(value));
}
};
inline static unsigned int ceilpow2(unsigned int u) noexcept {
if(u == 0u) return 0u;
--u, u |= u >> 1, u |= u >> 2, u |= u >> 4, u |= u >> 8;
return (u | (u >> 16)) + 1u;
}
inline static bucket *increment(bucket *cur) noexcept {
for(++cur; !cur->_end; ++cur){
if(!cur->empty()) break;
}
return cur;
}
inline bucket *next_bucket(bucket *cur) const noexcept {
return cur->_last ? _buckets : cur + 1;
}
inline unsigned int make_hash(const _Key& key) const noexcept {
return _Hash()(key);
}
inline float load_rate() const noexcept {
return (float)_data_count / _bucket_count;
}
bucket *insert(bucket *cur, _Key&& key, short int dist, value_type&& value){
bucket *ret = cur;
bool flag = false;
while(true){
if(cur->empty()){
cur->_key = move(key), cur->_dist = dist, cur->new_value(move(value));
if(!flag) ret = cur, flag = true;
break;
}else if(dist > cur->_dist){
swap(key, cur->_key), swap(dist, cur->_dist), swap(value, cur->value());
if(!flag) ret = cur, flag = true;
}
++dist;
cur = next_bucket(cur);
}
return ret;
}
template<class Key>
bucket *_find(Key&& key, bool push = false){
unsigned int hash = make_hash(key);
bucket *cur = _buckets + (hash & _mask);
short int dist = 0;
while(dist <= cur->_dist){
if(key == cur->_key) return cur;
++dist, cur = next_bucket(cur);
}
if(!push) return _buckets + _bucket_count;
++_data_count;
if(rehash_check()){
cur = _buckets + (hash & _mask), dist = 0;
}
value_type new_value = value_type();
_Key new_key = forward<Key>(key);
return insert(cur, move(new_key), dist, move(new_value));
}
template<class Data>
bucket *find_insert(Data&& data){
const _Key& key = data.first;
unsigned int hash = make_hash(key);
bucket *cur = _buckets + (hash & _mask);
short int dist = 0;
while(dist <= cur->_dist){
if(key == cur->_key) return cur;
++dist, cur = next_bucket(cur);
}
++_data_count;
if(rehash_check()){
cur = _buckets + (hash & _mask), dist = 0;
}
data_type new_data = forward<Data>(data);
return insert(cur, move(new_data.first), dist, move(new_data.second));
}
template<typename... Args>
bucket *emplace(Args&&... args){
return find_insert(data_type(forward<Args>(args)...));
}
bucket *backward_shift(bucket *cur, bool next_ret){
bucket *next = next_bucket(cur), *ret = cur;
if(next->_dist < 1) return next_ret ? increment(cur) : cur;
do {
cur->_key = next->_key, cur->_dist = next->_dist - 1;
cur->new_value(move(next->value()));
cur = next, next = next_bucket(cur);
}while(next->_dist >= 1);
cur->clear();
return ret;
}
bucket *erase_impl(bucket *cur, bool next_ret){
assert(static_cast<size_t>(cur - _buckets) != _bucket_count);
cur->_delete();
--_data_count;
return backward_shift(cur, next_ret);
}
bucket *erase_itr(bucket *cur, bool next_ret = true){
const _Key key = cur->_key;
return erase_impl(rehash_check() ? _find(key) : cur, next_ret);
}
bucket *erase_key(const _Key& key, bool next_ret = true){
rehash_check();
return erase_impl(_find(key), next_ret);
}
bool rehash_check(){
if(_bucket_count == 0){
rehash(1u);
return true;
}else if(load_rate() >= MAX_LOAD_RATE){
rehash(_bucket_count * 2u);
return true;
}else if(DOWNSIZE){
if(load_rate() <= MIN_LOAD_RATE && _bucket_count >= DOWNSIZE_THRESHOLD){
rehash(_bucket_count / 2u);
return true;
}
}
return false;
}
void move_data(bucket *cur){
insert(_buckets + (make_hash(cur->_key) & _mask), move(cur->_key), 0, move(cur->value()));
}
void rehash(unsigned int new_bucket_count){
UnorderedMap new_unordered_map(new_bucket_count);
new_unordered_map._data_count = _data_count;
for(bucket *cur = _buckets; !cur->_end; ++cur){
if(!cur->empty()){
new_unordered_map.move_data(cur);
}
}
swap(*this, new_unordered_map);
}
friend void swap(UnorderedMap& ump1, UnorderedMap& ump2){
swap(ump1._bucket_count, ump2._bucket_count);
swap(ump1._mask, ump2._mask);
swap(ump1._data_count, ump2._data_count);
swap(ump1._buckets, ump2._buckets);
}
private:
unsigned int _bucket_count, _mask, _data_count;
bucket *_buckets;
public:
const float MAX_LOAD_RATE = 0.5f;
const float MIN_LOAD_RATE = 0.1f;
const unsigned int DOWNSIZE_THRESHOLD = 16u;
UnorderedMap(unsigned int bucket_size = 0u)
: _bucket_count(ceilpow2(bucket_size)), _mask(_bucket_count - 1),
_data_count(0u), _buckets(new bucket[_bucket_count + 1]){
if(_bucket_count > 0) _buckets[_bucket_count - 1]._last = true;
else _mask = 0;
_buckets[_bucket_count]._end = true;
}
UnorderedMap(const UnorderedMap& another)
: _bucket_count(another._bucket_count), _mask(another._mask), _data_count(another._data_count){
_buckets = new bucket[_bucket_count + 1u];
for(unsigned int i = 0u; i <= _bucket_count; ++i){
_buckets[i] = another._buckets[i];
}
}
UnorderedMap(UnorderedMap&& another)
: _bucket_count(move(another._bucket_count)), _mask(move(another._mask)),
_data_count(move(another._data_count)), _buckets(another._buckets){
another._buckets = nullptr;
}
UnorderedMap& operator=(const UnorderedMap& another){
delete[] _buckets;
_bucket_count = another._bucket_count;
_mask = another._mask;
_data_count = another._data_count;
_buckets = new bucket[_bucket_count + 1u];
for(unsigned int i = 0u; i <= _bucket_count; ++i){
_buckets[i] = another._buckets[i];
}
return *this;
}
UnorderedMap& operator=(UnorderedMap&& another){
delete[] _buckets;
_bucket_count = move(another._bucket_count);
_mask = move(another._mask);
_data_count = move(another._data_count);
_buckets = another._buckets;
another._buckets = nullptr;
return *this;
}
void allocate(unsigned int element_size){
rehash(ceilpow2(ceil(element_size / MAX_LOAD_RATE) + 1));
}
~UnorderedMap(){ delete[] _buckets; }
friend ostream& operator<< (ostream& os, UnorderedMap& ump) noexcept {
for(auto val : ump) os << '{' << val.first << ',' << val.second << "} ";
return os;
}
_Tp& operator[](const _Key& key){ return _find(key, true)->value(); }
_Tp& operator[](_Key&& key){ return _find(move(key), true)->value(); }
const _Tp& at(const _Key& key){
bucket *res = _find(key);
if(res == _buckets + _bucket_count) __throw_out_of_range("Unordered_Map::at");
return res->value();
}
void clear(){
UnorderedMap new_unordered_map(1u);
swap(*this, new_unordered_map);
}
size_t size() const noexcept { return _data_count; }
size_t bucket_count() const noexcept { return _bucket_count; }
bool empty() const noexcept { return (_data_count == 0); }
iterator begin() noexcept {
return (_buckets->empty() && _bucket_count > 0) ? iterator(increment(_buckets)) : iterator(_buckets);
}
iterator end() noexcept { return iterator(_buckets + _bucket_count); }
iterator find(const _Key& key){ return iterator(_find(key)); }
iterator insert(const data_type& data){ return iterator(find_insert(data)); }
iterator insert(data_type&& data){ return iterator(find_insert(move(data))); }
template<typename... Args>
iterator emplace(Args&&... args){ return iterator(_emplace(forward<Args>(args)...)); }
iterator erase(const _Key& key){ return iterator(erase_key(key)); }
iterator erase(const iterator& itr){ return iterator(erase_itr(itr.bucket_ptr)); }
void simple_erase(const _Key& key){ erase_key(key, false); }
void simple_erase(const iterator& itr){ erase_itr(itr.bucket_ptr, false); }
// DEBUG 用
short int maximum_distance() const noexcept {
short int ret = -1;
for(bucket *cur = _buckets; !cur->_end; ++cur){
ret = max(ret, cur->_dist);
}
return ret;
}
};
template<class _Key, class _Tp, class _Hash, bool DOWNSIZE>
class UnorderedMapIterator {
private:
friend UnorderedMap<_Key, _Tp, _Hash, DOWNSIZE>;
typename UnorderedMap<_Key, _Tp, _Hash, DOWNSIZE>::bucket *bucket_ptr;
using iterator_category = forward_iterator_tag;
using value_type = pair<const _Key, _Tp>;
using difference_type = ptrdiff_t;
using reference = pair<const _Key&, _Tp&>;
private:
UnorderedMapIterator(typename UnorderedMap<_Key, _Tp, _Hash, DOWNSIZE>::bucket *_bucket_ptr)
noexcept : bucket_ptr(_bucket_ptr){}
public:
UnorderedMapIterator() noexcept : bucket_ptr(){}
UnorderedMapIterator(const UnorderedMapIterator& itr) noexcept : bucket_ptr(itr.bucket_ptr){}
UnorderedMapIterator& operator=(const UnorderedMapIterator& itr)
& noexcept { return bucket_ptr = itr.bucket_ptr, *this; }
UnorderedMapIterator& operator=(const UnorderedMapIterator&& itr)
& noexcept { return bucket_ptr = itr.bucket_ptr, *this; }
reference operator*() const noexcept { return {bucket_ptr->_key, bucket_ptr->value()}; }
UnorderedMapIterator& operator++() noexcept {
return bucket_ptr = UnorderedMap<_Key, _Tp, _Hash, DOWNSIZE>::increment(bucket_ptr), *this;
}
UnorderedMapIterator operator++(int) const noexcept {
return UnorderedMapIterator(UnorderedMap<_Key, _Tp, _Hash, DOWNSIZE>::increment(this->bucket_ptr));
}
bool operator==(const UnorderedMapIterator& itr) const noexcept { return !(*this != itr); };
bool operator!=(const UnorderedMapIterator& itr) const noexcept { return bucket_ptr != itr.bucket_ptr; }
};
template<typename _Tp> class BSTNode {
public:
int from, to, sz;
_Tp value, al, lazy;
BSTNode *left, *right, *par;
BSTNode(const int _from, const int _to) noexcept
: from(_from), to(_to), sz(0), value(id1), al(id2), lazy(id1),
left(nullptr), right(nullptr), par(nullptr){}
BSTNode(const int _from, const int _to, const _Tp& _value) noexcept
: from(_from), to(_to), sz(1), value(_value), al(value), lazy(id1),
left(nullptr), right(nullptr), par(nullptr){}
static const _Tp id1 = (_Tp)0;
static const _Tp id2 = (_Tp)0;
static void opr1(_Tp& arg1, const _Tp arg2) noexcept { arg1 += arg2; }
static _Tp opr2(const _Tp arg1, const _Tp arg2) noexcept { return arg1 + arg2; }
inline bool IsRoot() const noexcept { return !par; }
inline bool IsVertex() const noexcept { return from == to; }
void push() noexcept {
if(lazy != id1){
if(IsVertex()) opr1(value, lazy);
al += lazy * sz;
if(left) opr1(left->lazy, lazy);
if(right) opr1(right->lazy, lazy);
lazy = id1;
}
}
void eval() noexcept {
sz = IsVertex(), al = value;
if(left) left->push(), sz += left->sz, al = opr2(left->al, al);
if(right) right->push(), sz += right->sz, al = opr2(al, right->al);
}
void rotate(const bool right_) noexcept {
BSTNode *p = par, *g = p->par;
if(right_){
if((p->left = right)) right->par = p;
right = p, p->par = this;
}else{
if((p->right = left)) left->par = p;
left = p, p->par = this;
}
p->eval(), eval();
if(!(par = g)) return;
if(g->left == p) g->left = this;
if(g->right == p) g->right = this;
g->eval();
}
};
template<typename _Tp>
BSTNode<_Tp>* splay(BSTNode<_Tp> *u) noexcept {
if(!u) return nullptr;
while(!(u->IsRoot())){
BSTNode<_Tp> *p = u->par, *gp = p->par;
if(p->IsRoot()){ // zig
p->push(), u->push();
u->rotate((u == p->left));
}else{
gp->push(), p->push(), u->push();
bool flag = (u == p->left);
if((u == p->left) == (p == gp->left)){ // zig-zig
p->rotate(flag), u->rotate(flag);
}else{ // zig-zag
u->rotate(flag), u->rotate(!flag);
}
}
}
u->push();
return u;
}
template<typename _Tp>
BSTNode<_Tp>* join(BSTNode<_Tp> *root1, BSTNode<_Tp> *root2) noexcept {
if(!root1 || !root2) return root1 ? root1 : root2;
BSTNode<_Tp> *cur = nullptr, *nx = root1;
do{ cur = nx, cur->push(), nx = cur->right; }while(nx);
BSTNode<_Tp> *ver = splay(cur);
ver->right = root2, ver->eval(), root2->par = ver;
return ver;
}
template<typename _Tp>
pair<BSTNode<_Tp>*, BSTNode<_Tp>*> split_lower_bond(BSTNode<_Tp> *ver) noexcept {
BSTNode<_Tp> *res = splay(ver)->left;
ver->left = nullptr, ver->eval();
if(res) res->par = nullptr;
return make_pair(res, ver);
}
template<typename _Tp>
pair<BSTNode<_Tp>*, BSTNode<_Tp>*> split_upper_bond(BSTNode<_Tp> *ver) noexcept {
BSTNode<_Tp> *res = splay(ver)->right;
ver->right = nullptr, ver->eval();
if(res) res->par = nullptr;
return make_pair(ver, res);
}
template<typename T> class EulerTourTree {
private:
struct murmur_hash64 {
unsigned long long operator()(unsigned long long p) const {
const unsigned long long m = 0xc6a4a7935bd1e995; p *= m;
unsigned long long h = (p^(p>>47))*m;
return h = (h^(h>>47))*m, (h^(h>>47));
}
};
BSTNode<T>** vertex_set;
UnorderedMap<unsigned long long, pair<BSTNode<T>*, BSTNode<T>*>, murmur_hash64> edge_set;
inline static unsigned long long pair_to_ll(const int u, const int v){
return ((unsigned long long)(u) << 32) | v;
}
BSTNode<T> *reroot(BSTNode<T> *ver) noexcept {
BSTNode<T> *res = splay(ver)->left;
if(!res) return ver;
ver->left = nullptr, ver->eval();
while(ver->right) ver->push(), ver = ver->right;
splay(ver), ver->right = res, ver->eval(), res->par = ver;
return ver;
}
void link(BSTNode<T> *ver1, BSTNode<T> *ver2) noexcept {
BSTNode<T>* e1 = new BSTNode<T>(ver1->from, ver2->from);
BSTNode<T>* e2 = new BSTNode<T>(ver2->from, ver1->from);
edge_set[pair_to_ll(ver1->from, ver2->from)] = {e1, e2};
join(join(reroot(ver1), e1), join(reroot(ver2), e2));
}
void cut(BSTNode<T> *edge1, BSTNode<T> *edge2) noexcept {
splay(edge1), splay(edge2);
BSTNode<T> *p = edge1->par;
bool _right = (edge1 == edge2->right);
if(p != edge2){
_right = (p == edge2->right);
p->par = nullptr, edge1->rotate((edge1 == p->left));
}
if(edge1->left) edge1->left->par = nullptr;
if(edge1->right) edge1->right->par = nullptr;
if(_right){
if(edge2->left) edge2->left->par = nullptr;
join(edge2->left, edge1->right);
}else{
if(edge2->right) edge2->right->par = nullptr;
join(edge1->left, edge2->right);
}
// delete edge1; delete edge2;
}
bool connected(BSTNode<T> *ver1, BSTNode<T> *ver2) noexcept {
splay(ver1), splay(ver2);
return ver1->par;
}
T& get(BSTNode<T> *ver) noexcept { return splay(ver)->value; }
void point_update(BSTNode<T> *ver, const T& val){
BSTNode<T>::opr1(splay(ver)->value, val);
ver->al = BSTNode<T>::opr2(ver->al, val);
}
void range(BSTNode<T> *edge1, BSTNode<T> *edge2, const T& val) noexcept {
auto res1 = split_lower_bond(edge1);
auto res2 = split_upper_bond(edge2);
BSTNode<T>::opr1(res2.first->lazy, val);
join(join(res1.first,res2.first), res2.second);
}
T query(BSTNode<T> *edge1, BSTNode<T> *edge2) noexcept {
auto res1 = split_lower_bond(edge1);
auto res2 = split_upper_bond(edge2);
T res = res2.first->al;
return join(join(res1.first,res2.first), res2.second), res;
}
void dfs(const int u, const int p, const BSTNode<T> *cur,
bool *visit, vector<BSTNode<T>*>& nodes, const vector<vector<int> >& G) noexcept {
visit[u] = true;
nodes.push_back(vertex_set[u]);
for(auto& v : G[u]){
if(v != p){
BSTNode<T>* e1 = new BSTNode<T>(u, v);
nodes.push_back(e1);
dfs(v, u, cur, visit, nodes, G);
BSTNode<T>* e2 = new BSTNode<T>(v, u);
if(u < v) edge_set[pair_to_ll(u, v)] = {e1, e2};
else edge_set[pair_to_ll(v, u)] = {e2, e1};
nodes.push_back(e2);
}
}
}
void bst_build(vector<BSTNode<T>* >& nodes) noexcept {
int i, n = (int)nodes.size(), st = 2, isolate = ((n % 4 == 1) ? (n-1) : -1);
while(st <= n){
for(i = st-1; i < n; i += 2*st){
nodes[i]->left = nodes[i-st/2], nodes[i-st/2]->par = nodes[i];
if(i+st/2 < n) nodes[i]->right = nodes[i+st/2], nodes[i+st/2]->par = nodes[i];
else if(isolate >= 0) nodes[i]->right = nodes[isolate], nodes[isolate]->par = nodes[i];
nodes[i]->eval();
}
isolate = ((n % (4*st) >= st && (n % (4*st) < 2*st)) ? (i-2*st): isolate);
st <<= 1;
}
}
void build_forest(const vector<vector<int> >& forest) noexcept {
bool *visit = new bool[V]();
for(int i = 0; i < V; i++){
if(!visit[i]){
vector<BSTNode<T>* > nodes;
BSTNode<T> *cur = nullptr;
dfs(i, -1, cur, visit, nodes, forest);
bst_build(nodes);
}
}
// delete[] visit;
}
void build_tree(const int root, const vector<vector<int> >& tree) noexcept {
bool *visit = new bool[V]();
vector<BSTNode<T>* > nodes;
BSTNode<T> *cur = nullptr;
dfs(root, -1, cur, visit, nodes, tree);
bst_build(nodes);
// delete[] visit;
}
public:
const int V;
EulerTourTree(const vector<T>& ver_value) noexcept : V((int)ver_value.size()){
vertex_set = new BSTNode<T>*[V];
for(int i = 0; i < V; i++) vertex_set[i] = new BSTNode<T>(i, i, ver_value[i]);
}
EulerTourTree(const vector<T>& ver_value, const vector<vector<int> >& forest) noexcept
: EulerTourTree(ver_value){
unsigned int element_size = 0;
for(int i = 0; i < V; ++i) element_size += forest[i].size();
edge_set.allocate(element_size);
build_forest(forest);
}
EulerTourTree(const vector<T>& ver_value, const int root, const vector<vector<int> >& tree) noexcept
: EulerTourTree(ver_value){
unsigned int element_size = 0;
for(int i = 0; i < V; ++i) element_size += tree[i].size();
edge_set.allocate(element_size);
build_tree(root, tree);
}
// ~EulerTourTree(){
// for(auto it : edge_set){
// delete (it.second).first;
// delete (it.second).second;
// }
// for(int i = 0; i < V; ++i) delete vertex_set[i];
// delete[] vertex_set;
// }
// 根を node_id にする
void reroot(const int node_id) noexcept { reroot(vertex_set[node_id]); }
// 辺(node1_id, node2_id) を追加
void link(int node1_id, int node2_id) noexcept {
if(node1_id > node2_id) swap(node1_id, node2_id);
link(vertex_set[node1_id], vertex_set[node2_id]);
}
// 辺(node1_id, node2_id) を削除(逆向きでも問題ない)
void cut(int node1_id, int node2_id){
if(node1_id > node2_id) swap(node1_id, node2_id);
auto it = edge_set.find(pair_to_ll(node1_id, node2_id));
assert(it != edge_set.end());
BSTNode<T> *edge1 = ((*it).second).first, *edge2 = ((*it).second).second;
edge_set.simple_erase(it);
cut(edge1, edge2);
}
// node1_id と node2_id が同じ木(連結成分)に属するか
bool connected(const int node1_id, const int node2_id) noexcept {
if(node1_id == node2_id) return true;
return connected(vertex_set[node1_id], vertex_set[node2_id]);
}
// 頂点 ver_id の値を取得
T& get(int ver_id) noexcept { return get(vertex_set[ver_id]); }
// 頂点 ver_id に val を加える
void point_update(const int ver_id, const T& val) noexcept {
return point_update(vertex_set[ver_id], val);
}
// 頂点 ver_id の存在する連結成分内の頂点全体に val を加える
void component_range(const int ver_id, const T& val){ range(ver_id, -1, val); }
// 親が par_id であるような頂点 ver_id の部分木内に存在する頂点全体に val を加える
void range(const int ver_id, const int par_id, const T& val){
if(par_id < 0) return BSTNode<T>::opr1(splay(vertex_set[ver_id])->lazy, val);
if(ver_id < par_id){
auto it = edge_set.find(pair_to_ll(ver_id, par_id));
assert(it != edge_set.end());
range(((*it).second).second, ((*it).second).first, val);
}else{
auto it = edge_set.find(pair_to_ll(par_id, ver_id));
assert(it != edge_set.end());
range(((*it).second).first, ((*it).second).second, val);
}
}
// 頂点 ver_id の存在する連結成分内の頂点全体の総和を取得
T component_query(const int ver_id){ return query(ver_id, -1); }
// 親が par_id であるような頂点 ver_id の部分木内に存在する頂点全体の総和を取得
T query(const int ver_id, const int par_id){
if(par_id < 0) return splay(vertex_set[ver_id])->al;
if(ver_id < par_id){
auto it = edge_set.find(pair_to_ll(ver_id, par_id));
assert(it != edge_set.end());
return query(((*it).second).second, ((*it).second).first);
}else{
auto it = edge_set.find(pair_to_ll(par_id, ver_id));
assert(it != edge_set.end());
return query(((*it).second).first, ((*it).second).second);
}
}
// DEBUG 用
void check_dfs(const BSTNode<T>* cur) const noexcept {
cur->push();
if(cur->left) check_dfs(cur->left);
if(cur->IsVertex()) cout <<"{" << (cur->from) << "," << (cur->value) << "} ";
if(cur->right) check_dfs(cur->right);
}
};
verify 用の問題
yukicoder : 全方位神童数
提出コード
提出コード(高速版)
yosupo さんの library checker : Dynamic Tree Vertex Add Subtree Sum
<提出コード
提出コード(高速版)