Hello lady/bros, I am struggling with 100551E.Disconnected Graph.
I have considered: (1) Online fully dynamic graph connectivity. I copied a piece of code here: https://www.luogu.com.cn/problem/solution/P5247. I passed LuoguP5247 and SPOJDYNACON2, however I could not pass this problem;
SPOJ Code
#include <iostream>
#include <unordered_set>
#include <stack>
#include <unordered_map>
#include <cstring>
#include <vector>
#define GUARANTEE_LEGAL 0
#define fastio std::cin.tie(0) -> sync_with_stdio(0)
struct LCT {
std::vector<std::array<int, 2>> c;
std::vector<int> fa, sta, subtree_size, subtree_size2;
std::vector<char> r;
struct Tag {
// Only stores edges of this level.
std::unordered_set<int> edges;
int tag;
std::unordered_set<int> tagged_non_preferred_children;
};
std::vector<Tag> tag_tree, tag_non_tree;
void update_tag(std::vector<Tag>& tags, int x) {
if (!tags[x].edges.empty()) {
tags[x].tag = x;
} else if (tags[ls(x)].tag) {
tags[x].tag = tags[ls(x)].tag;
} else if (tags[rs(x)].tag) {
tags[x].tag = tags[rs(x)].tag;
} else if (!tags[x].tagged_non_preferred_children.empty()) {
tags[x].tag = tags[*tags[x].tagged_non_preferred_children.begin()].tag;
} else {
tags[x].tag = 0;
}
}
void new_non_preferred_child(int x) {
if (fa[x] == 0)
return;
subtree_size2[fa[x]] += subtree_size[x];
if (tag_tree[x].tag)
tag_tree[fa[x]].tagged_non_preferred_children.insert(x);
if (tag_non_tree[x].tag)
tag_non_tree[fa[x]].tagged_non_preferred_children.insert(x);
}
void delete_non_preferred_child(int x) {
if (fa[x] == 0)
return;
subtree_size2[fa[x]] -= subtree_size[x];
if (tag_tree[x].tag)
tag_tree[fa[x]].tagged_non_preferred_children.erase(x);
if (tag_non_tree[x].tag)
tag_non_tree[fa[x]].tagged_non_preferred_children.erase(x);
}
inline int& ls(int rt) {
return c[rt][0];
}
inline int& rs(int rt) {
return c[rt][1];
}
inline bool not_splay_rt(int x) {
return ls(fa[x]) == x || rs(fa[x]) == x;
}
inline int side(int x) {
return x == rs(fa[x]);
}
void Init(int n) {
// Initially every node is a tree by itself.
// memset all to 0.
c.resize(n+2); fa.resize(n+2); sta.resize(n+2); subtree_size.resize(n+2); subtree_size2.resize(n+2);
r.resize(n+2);
tag_tree.resize(n+2); tag_non_tree.resize(n+2);
for (int i = 1; i <= n; ++i) {
subtree_size[i] = 1;
}
}
inline void pushr(int x) {
std::swap(ls(x), rs(x));
r[x] ^= 1;
}
inline void pushdown(int x) {
if (r[x]) {
if (ls(x))
pushr(ls(x));
if (rs(x))
pushr(rs(x));
r[x] = false;
}
}
inline void __pushup(int x) {
update_tag(tag_tree, x);
update_tag(tag_non_tree, x);
subtree_size[x] = subtree_size[ls(x)] + subtree_size[rs(x)] + 1 + subtree_size2[x];
}
// At first x is not in its tagged_non_preferred_children
inline void __pushup_splay_rt(int x) {
__pushup(x);
new_non_preferred_child(x);
// No need to update tag[fa[x]], because if it was in this subtree, then it is still in this subtree.
}
// tag[x] is not updated.
void __rotate_up(int x) {
int y = fa[x], z = fa[y], side_x = side(x), w = c[x][side_x ^ 1];
fa[x] = z;
if (not_splay_rt(y))
c[z][side(y)] = x;
if (w)
fa[w] = y;
c[y][side_x] = w;
fa[y] = x;
c[x][side_x ^ 1] = y;
__pushup(y);
}
// tag[x] is not updated.
// The original splay root is removed from its father's tagged_non_preferred_children.
void __splay(int x) {
int y = x, top = 0;
while(1) {
sta[++top] = y;
if (!not_splay_rt(y))
break;
y = fa[y];
}
int to = fa[y];
delete_non_preferred_child(y);
while (top)
pushdown(sta[top--]);
while (fa[x] != to) {
int y = fa[x];
if (fa[y] != to)
__rotate_up(side(x) == side(y) ? y : x);
__rotate_up(x);
}
}
void splay(int x) {
__splay(x);
__pushup_splay_rt(x);
}
void access(int x) {
int ori_x = x;
for (int w = 0; x; w = x, x = fa[x]) {
__splay(x);
delete_non_preferred_child(w);
new_non_preferred_child(rs(x));
rs(x) = w;
__pushup_splay_rt(x);
}
__splay(ori_x);
__pushup(ori_x);
}
int find_root(int x) {
access(x);
for (; ls(x); x = ls(x))
pushdown(x);
__splay(x);
__pushup(x);
return x;
}
inline void make_root(int x) {
access(x);
pushr(x);
}
void __link(int x, int y) {
// If simply fa[x] = y, the complexity might be wrong.
access(y);
pushdown(x);
fa[y] = x;
ls(x) = y;
__pushup(x); // Might be unnecessary
}
inline void link_new(int x, int y) {
make_root(x);
__link(x, y);
}
inline void link(int x, int y) {
make_root(x);
if (find_root(y) == x)
return;
__link(x, y);
}
inline void split(int x, int y) {
make_root(x);
access(y);
}
void cut_existing(int x, int y) {
split(x, y);
fa[x] = ls(y) = 0;
__pushup(y); // Might be unnecessary
}
void cut(int x, int y) {
split(x, y);
if (ls(y) != x || rs(x) != 0)
return; // No such edge (x, y)
fa[x] = ls(y) = 0;
__pushup(y); // Might be unnecessary
}
std::unordered_set<int> take_out_edges(std::vector<Tag>& type, int x) {
access(x);
auto tmp = std::unordered_set<int>();
swap(tmp, type[x].edges);
update_tag(type, x);
return std::move(tmp);
}
void add_directed_edge(std::vector<Tag>& type, int x, int y) {
if (type[x].edges.empty()) {
access(x);
type[x].edges.insert(y);
update_tag(type, x);
} else {
type[x].edges.insert(y);
}
}
void delete_directed_edge(std::vector<Tag>& type, int x, int y) {
if (type[x].edges.size() == 1) {
access(x);
type[x].edges.erase(y);
update_tag(type, x);
} else {
type[x].edges.erase(y);
}
}
void new_tree_edge(int x, int y) {
link_new(x, y);
add_directed_edge(tag_tree, x, y);
add_directed_edge(tag_tree, y, x);
}
};
struct DynamicConnectivity {
int c; //components
std::vector<LCT> F;
std::unordered_map<int, std::unordered_map<int, int> > level;
std::vector<std::unordered_set<int>> adj;
void Init(int n) {
c = n;
for (int i = 0; (1 << i) <= n; ++i)
F.push_back(LCT());
for(auto& f:F) f.Init(n);
adj.resize(n+2);
}
// Assume no duplicate edge
void link(int x, int y) {
#if !GUARANTEE_LEGAL
if(adj[x].count(y) || adj[y].count(x)) return;
adj[x].insert(y);
adj[y].insert(x);
#endif
level[x][y] = 0;
level[y][x] = 0;
if (F[0].find_root(x) == F[0].find_root(y)) {
F[0].add_directed_edge(F[0].tag_non_tree, y, x);
F[0].add_directed_edge(F[0].tag_non_tree, x, y);
} else {
c--;
F[0].new_tree_edge(x, y);
}
}
bool reconnect(int x, int y, int l) {
F[l].access(x);
F[l].access(y);
if (F[l].subtree_size[x] > F[l].subtree_size[y])
std::swap(x, y);
while (1) {
F[l].access(x);
int u = F[l].tag_tree[x].tag;
if (u == 0)
break;
auto tmp = F[l].take_out_edges(F[l].tag_tree, u);
for (int v : tmp) {
F[l].delete_directed_edge(F[l].tag_tree, v, u);
F[l+1].new_tree_edge(u, v);
++level[u][v];
++level[v][u];
}
}
y = F[l].find_root(y);
while (1) {
F[l].access(x);
int u = F[l].tag_non_tree[x].tag;
if (u == 0)
break;
auto tmp = F[l].take_out_edges(F[l].tag_non_tree, u);
do {
auto it = tmp.begin();
int v = *it;
tmp.erase(it);
F[l].delete_directed_edge(F[l].tag_non_tree, v, u);
if (F[l].find_root(v) == y) {
if (!tmp.empty()) {
F[l].access(u);
swap(tmp, F[l].tag_non_tree[u].edges);
F[l].update_tag(F[l].tag_non_tree, u);
}
for (int i = 0; i < l; ++i)
F[i].link_new(u, v);
F[l].new_tree_edge(u, v);
return true;
} else {
F[l+1].add_directed_edge(F[l+1].tag_non_tree, u, v);
F[l+1].add_directed_edge(F[l+1].tag_non_tree, v, u);
++level[u][v];
++level[v][u];
}
} while (!tmp.empty());
};
return false;
}
void cut(int x, int y) {
auto it1 = level[x].find(y);
#if !GUARANTEE_LEGAL
if(!adj[x].count(y) || !adj[y].count(x) || it1==level[x].end()) return;
adj[x].erase(y);
adj[y].erase(x);
#endif
int l = it1->second;
level[x].erase(it1);
level[y].erase(x);
auto& s = F[l].tag_non_tree[x].edges;
if (s.find(y) != s.end()) {
F[l].delete_directed_edge(F[l].tag_non_tree, x, y);
F[l].delete_directed_edge(F[l].tag_non_tree, y, x);
return;
}
F[l].delete_directed_edge(F[l].tag_tree, x, y);
F[l].delete_directed_edge(F[l].tag_tree, y, x);
for (int i = 0; i <= l; ++i)
F[i].cut_existing(x, y);
int reconnect_successful = 0;
while (1) {
if (reconnect(x, y, l)){
reconnect_successful = 1;
break;
}
if (l == 0)
break;
--l;
}
if(!reconnect_successful) c++;
}
bool is_connected(int x, int y) {
return F[0].find_root(x) == F[0].find_root(y);
}
int comp(){
return c;
}
};
#define fastio std::cin.tie(0) -> sync_with_stdio(0)
int main() {
fastio;
int n, m;
static DynamicConnectivity dc;
std::cin >> n >> m;
dc.Init(n);
int last = 0;
while (m--) {
std::string op;
int x, y;
std::cin >> op >> x >> y;
switch (op[0]) {
case 'a':
dc.link(x, y);
break;
case 'r':
dc.cut(x, y);
break;
case 'c':
if (dc.is_connected(x, y)) {
std::cout << "YES\n";
} else {
std::cout << "NO\n";
}
break;
}
}
return 0;
}
Codeforces submission is similar, but it always gets TLE on test15, due to a relatively large constant:
Codeforces Disconnected Graph Submission
#pragma GCC optimize("O3,unroll-loops")
#pragma GCC target("avx2,bmi,bmi2,lzcnt,popcnt")
#include <iostream>
#include <unordered_set>
#include <stack>
#include <unordered_map>
#include <cstring>
#include <vector>
#define GUARANTEE_LEGAL 0
#define fastio std::cin.tie(0) -> sync_with_stdio(0)
struct LCT {
std::vector<std::array<int, 2>> c;
std::vector<int> fa, sta, subtree_size, subtree_size2;
std::vector<char> r;
struct Tag {
// Only stores edges of this level.
std::unordered_set<int> edges;
int tag;
std::unordered_set<int> tagged_non_preferred_children;
};
std::vector<Tag> tag_tree, tag_non_tree;
void update_tag(std::vector<Tag>& tags, int x) {
if (!tags[x].edges.empty()) {
tags[x].tag = x;
} else if (tags[ls(x)].tag) {
tags[x].tag = tags[ls(x)].tag;
} else if (tags[rs(x)].tag) {
tags[x].tag = tags[rs(x)].tag;
} else if (!tags[x].tagged_non_preferred_children.empty()) {
tags[x].tag = tags[*tags[x].tagged_non_preferred_children.begin()].tag;
} else {
tags[x].tag = 0;
}
}
void new_non_preferred_child(int x) {
if (fa[x] == 0)
return;
subtree_size2[fa[x]] += subtree_size[x];
if (tag_tree[x].tag)
tag_tree[fa[x]].tagged_non_preferred_children.insert(x);
if (tag_non_tree[x].tag)
tag_non_tree[fa[x]].tagged_non_preferred_children.insert(x);
}
void delete_non_preferred_child(int x) {
if (fa[x] == 0)
return;
subtree_size2[fa[x]] -= subtree_size[x];
if (tag_tree[x].tag)
tag_tree[fa[x]].tagged_non_preferred_children.erase(x);
if (tag_non_tree[x].tag)
tag_non_tree[fa[x]].tagged_non_preferred_children.erase(x);
}
inline int& ls(int rt) {
return c[rt][0];
}
inline int& rs(int rt) {
return c[rt][1];
}
inline bool not_splay_rt(int x) {
return ls(fa[x]) == x || rs(fa[x]) == x;
}
inline int side(int x) {
return x == rs(fa[x]);
}
void Init(int n) {
// Initially every node is a tree by itself.
// memset all to 0.
c.resize(n+2); fa.resize(n+2); sta.resize(n+2); subtree_size.resize(n+2); subtree_size2.resize(n+2);
r.resize(n+2);
tag_tree.resize(n+2); tag_non_tree.resize(n+2);
for (int i = 1; i <= n; ++i) {
subtree_size[i] = 1;
}
}
inline void pushr(int x) {
std::swap(ls(x), rs(x));
r[x] ^= 1;
}
inline void pushdown(int x) {
if (r[x]) {
if (ls(x))
pushr(ls(x));
if (rs(x))
pushr(rs(x));
r[x] = false;
}
}
inline void __pushup(int x) {
update_tag(tag_tree, x);
update_tag(tag_non_tree, x);
subtree_size[x] = subtree_size[ls(x)] + subtree_size[rs(x)] + 1 + subtree_size2[x];
}
// At first x is not in its tagged_non_preferred_children
inline void __pushup_splay_rt(int x) {
__pushup(x);
new_non_preferred_child(x);
// No need to update tag[fa[x]], because if it was in this subtree, then it is still in this subtree.
}
// tag[x] is not updated.
void __rotate_up(int x) {
int y = fa[x], z = fa[y], side_x = side(x), w = c[x][side_x ^ 1];
fa[x] = z;
if (not_splay_rt(y))
c[z][side(y)] = x;
if (w)
fa[w] = y;
c[y][side_x] = w;
fa[y] = x;
c[x][side_x ^ 1] = y;
__pushup(y);
}
// tag[x] is not updated.
// The original splay root is removed from its father's tagged_non_preferred_children.
void __splay(int x) {
int y = x, top = 0;
while(1) {
sta[++top] = y;
if (!not_splay_rt(y))
break;
y = fa[y];
}
int to = fa[y];
delete_non_preferred_child(y);
while (top)
pushdown(sta[top--]);
while (fa[x] != to) {
int y = fa[x];
if (fa[y] != to)
__rotate_up(side(x) == side(y) ? y : x);
__rotate_up(x);
}
}
void splay(int x) {
__splay(x);
__pushup_splay_rt(x);
}
void access(int x) {
int ori_x = x;
for (int w = 0; x; w = x, x = fa[x]) {
__splay(x);
delete_non_preferred_child(w);
new_non_preferred_child(rs(x));
rs(x) = w;
__pushup_splay_rt(x);
}
__splay(ori_x);
__pushup(ori_x);
}
int find_root(int x) {
access(x);
for (; ls(x); x = ls(x))
pushdown(x);
__splay(x);
__pushup(x);
return x;
}
inline void make_root(int x) {
access(x);
pushr(x);
}
void __link(int x, int y) {
// If simply fa[x] = y, the complexity might be wrong.
access(y);
pushdown(x);
fa[y] = x;
ls(x) = y;
__pushup(x); // Might be unnecessary
}
inline void link_new(int x, int y) {
make_root(x);
__link(x, y);
}
inline void link(int x, int y) {
make_root(x);
if (find_root(y) == x)
return;
__link(x, y);
}
inline void split(int x, int y) {
make_root(x);
access(y);
}
void cut_existing(int x, int y) {
split(x, y);
fa[x] = ls(y) = 0;
__pushup(y); // Might be unnecessary
}
void cut(int x, int y) {
split(x, y);
if (ls(y) != x || rs(x) != 0)
return; // No such edge (x, y)
fa[x] = ls(y) = 0;
__pushup(y); // Might be unnecessary
}
std::unordered_set<int> take_out_edges(std::vector<Tag>& type, int x) {
access(x);
auto tmp = std::unordered_set<int>();
swap(tmp, type[x].edges);
update_tag(type, x);
return std::move(tmp);
}
void add_directed_edge(std::vector<Tag>& type, int x, int y) {
if (type[x].edges.empty()) {
access(x);
type[x].edges.insert(y);
update_tag(type, x);
} else {
type[x].edges.insert(y);
}
}
void delete_directed_edge(std::vector<Tag>& type, int x, int y) {
if (type[x].edges.size() == 1) {
access(x);
type[x].edges.erase(y);
update_tag(type, x);
} else {
type[x].edges.erase(y);
}
}
void new_tree_edge(int x, int y) {
link_new(x, y);
add_directed_edge(tag_tree, x, y);
add_directed_edge(tag_tree, y, x);
}
};
struct DynamicConnectivity {
int c; //components
std::vector<LCT> F;
std::unordered_map<int, std::unordered_map<int, int> > level;
std::vector<std::unordered_set<int>> adj;
void Init(int n) {
c = n;
for (int i = 0; (1 << i) <= n; ++i)
F.push_back(LCT());
for(auto& f:F) f.Init(n);
adj.resize(n+2);
}
// Assume no duplicate edge
void link(int x, int y) {
#if !GUARANTEE_LEGAL
if(adj[x].count(y) || adj[y].count(x)) return;
adj[x].insert(y);
adj[y].insert(x);
#endif
level[x][y] = 0;
level[y][x] = 0;
if (F[0].find_root(x) == F[0].find_root(y)) {
F[0].add_directed_edge(F[0].tag_non_tree, y, x);
F[0].add_directed_edge(F[0].tag_non_tree, x, y);
} else {
c--;
F[0].new_tree_edge(x, y);
}
}
bool reconnect(int x, int y, int l) {
F[l].access(x);
F[l].access(y);
if (F[l].subtree_size[x] > F[l].subtree_size[y])
std::swap(x, y);
while (1) {
F[l].access(x);
int u = F[l].tag_tree[x].tag;
if (u == 0)
break;
auto tmp = F[l].take_out_edges(F[l].tag_tree, u);
for (int v : tmp) {
F[l].delete_directed_edge(F[l].tag_tree, v, u);
F[l+1].new_tree_edge(u, v);
++level[u][v];
++level[v][u];
}
}
y = F[l].find_root(y);
while (1) {
F[l].access(x);
int u = F[l].tag_non_tree[x].tag;
if (u == 0)
break;
auto tmp = F[l].take_out_edges(F[l].tag_non_tree, u);
do {
auto it = tmp.begin();
int v = *it;
tmp.erase(it);
F[l].delete_directed_edge(F[l].tag_non_tree, v, u);
if (F[l].find_root(v) == y) {
if (!tmp.empty()) {
F[l].access(u);
swap(tmp, F[l].tag_non_tree[u].edges);
F[l].update_tag(F[l].tag_non_tree, u);
}
for (int i = 0; i < l; ++i)
F[i].link_new(u, v);
F[l].new_tree_edge(u, v);
return true;
} else {
F[l+1].add_directed_edge(F[l+1].tag_non_tree, u, v);
F[l+1].add_directed_edge(F[l+1].tag_non_tree, v, u);
++level[u][v];
++level[v][u];
}
} while (!tmp.empty());
};
return false;
}
void cut(int x, int y) {
auto it1 = level[x].find(y);
#if !GUARANTEE_LEGAL
if(!adj[x].count(y) || !adj[y].count(x) || it1==level[x].end()) return;
adj[x].erase(y);
adj[y].erase(x);
#endif
int l = it1->second;
level[x].erase(it1);
level[y].erase(x);
auto& s = F[l].tag_non_tree[x].edges;
if (s.find(y) != s.end()) {
F[l].delete_directed_edge(F[l].tag_non_tree, x, y);
F[l].delete_directed_edge(F[l].tag_non_tree, y, x);
return;
}
F[l].delete_directed_edge(F[l].tag_tree, x, y);
F[l].delete_directed_edge(F[l].tag_tree, y, x);
for (int i = 0; i <= l; ++i)
F[i].cut_existing(x, y);
int reconnect_successful = 0;
while (1) {
if (reconnect(x, y, l)){
reconnect_successful = 1;
break;
}
if (l == 0)
break;
--l;
}
if(!reconnect_successful) c++;
}
bool is_connected(int x, int y) {
return F[0].find_root(x) == F[0].find_root(y);
}
int comp(){
return c;
}
};
#define DEBUG 0
int main(void){
#if !DEBUG
freopen("disconnected.in", "r", stdin);
freopen("disconnected.out", "w", stdout);
fastio;
#else
freopen("test1.txt", "r", stdin);
#endif
int n, m;
std::cin >> n >> m;
DynamicConnectivity dc;
dc.Init(n);
std::vector<std::pair<int, int>> edgevec = {{0, 0}};
for(int i = 1, u, v; i <= m; ++i){
std::cin >> u >> v;
dc.link(u, v);
edgevec.push_back(std::make_pair(u, v));
}
int k;
std::cin >> k;
for(int i = 1; i <= k; ++i){
int c; std::cin >> c;
std::vector<int> removed(c+1);
for(int j = 1; j <= c; ++j){
std::cin >> removed[j];
dc.cut(edgevec[removed[j]].first, edgevec[removed[j]].second);
}
std::cout << (dc.comp() == 1 ? "Connected\n" : "Disconnected\n");
for(int j = 1; j <= c; ++j){
dc.link(edgevec[removed[j]].first, edgevec[removed[j]].second);
}
}
}
(2)Retractable DSU, however it seems that DSU only supports rolling back the add options, it cannot roll back delete operations...
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