【LOJ3077】「2019 集训队互测 Day 4」绝目编诗

【题目链接】

点击打开链接

【思路要点】

不难发现各个边双连通分量可以分开处理，桥边可以直接删除。

可以证明，对于每一个边双连通分量，当 M−NM-NM−N 超过 O(N)O(\sqrt{N})O(N) 级别，答案一定为 YesYesYes 。

那么，将剩余图中的度为 222 的点缩去，剩余的图是一张点数和边数均在 O(N)O(\sqrt{N})O(N) 级别的图。

选择一个起始点，找到包含其的所有环，再删去该起始点，每个环会被搜到恰好 222 次。

每走出一步均判断当前点与起始点的连通性以确保能够搜到一个环。

至多会搜到 O(N)O(N)O(N) 个环，每个环环长为 O(N)O(\sqrt{N})O(N) 个点，判断连通性需要 O(N)O(\sqrt{N})O(N) 的时间。

总时间复杂度 O(N2)O(N^2)O(N2) 。

互测当时笔者采用没有判断连通性的做法配合贪心和卡时输出 NoNoNo 通过了本题。

【代码】

// Standard Solution O(N ^ 2)
#include<bits/stdc++.h>
using namespace std;
const int MAXN = 1e5 + 5;
const int MAXM = 1e6 + 5;
typedef long long ll;
typedef long double ld;
typedef unsigned long long ull;
template <typename T> void chkmax(T &x, T y) {x = max(x, y); }
template <typename T> void chkmin(T &x, T y) {x = min(x, y); }
template <typename T> void read(T &x) {x = 0; int f = 1;char c = getchar();for (; !isdigit(c); c = getchar()) if (c == '-') f = -f;for (; isdigit(c); c = getchar()) x = x * 10 + c - '0';x *= f;
}
template <typename T> void write(T x) {if (x < 0) x = -x, putchar('-');if (x > 9) write(x / 10);putchar(x % 10 + '0');
}
template <typename T> void writeln(T x) {write(x);puts("");
}
int x[MAXM], y[MAXM], f[MAXN];
int n, m, em, v, e, ans[MAXN];
vector <int> a[MAXN], p; set <int> b[MAXN];
vector <pair <pair <int, int>, int>> c[MAXN];
bool vis[MAXN], vise[MAXN], iskey[MAXN];bool instack[MAXN];
int tot, belong[MAXN], top, Stack[MAXN];
int timer, dfn[MAXN], low[MAXN];
void tarjan(int pos, int fa) {low[pos] = dfn[pos] = ++timer;v++, e += a[pos].size();Stack[++top] = pos;instack[pos] = true;for (unsigned i = 0; i < a[pos].size(); i++) {if (a[pos][i] == fa) continue;if (dfn[a[pos][i]] == 0) {tarjan(a[pos][i], pos);low[pos] = min(low[pos], low[a[pos][i]]);} else if (instack[a[pos][i]]) low[pos] = min(low[pos], dfn[a[pos][i]]);}if (low[pos] == dfn[pos]) {int tmp = Stack[top--];belong[tmp] = ++tot;instack[tmp] = false;while (tmp != pos) {tmp = Stack[top--];belong[tmp] = tot;instack[tmp] = false;}}
}
void rebuild(int from, int fa, int pos, int len) {b[fa].erase(pos), b[pos].erase(fa);if (iskey[pos]) {if (from == pos) {ans[len] += 2;if (ans[len] >= 3) {puts("Yes");exit(0);}return;}c[from].emplace_back(make_pair(pos, len), ++em);c[pos].emplace_back(make_pair(from, len), em);return;}int dest = *b[pos].begin();rebuild(from, pos, dest, len + 1);
}
void dfs(int pos) {vis[pos] = true;v++, e += b[pos].size();for (auto x : b[pos])if (!vis[x]) dfs(x);
}
int F(int x) {if (f[x] == x) return x;else return f[x] = F(f[x]);
}
bool connected(int a, int b) {for (auto x : p)f[x] = x;for (auto x : p)for (auto y : c[x])if (!vise[y.second]) f[F(x)] = F(y.first.first);return F(a) == F(b);
}
void work(int pos, int from, int depth) {if (!connected(pos, from)) return;if (depth && pos == from) {if (++ans[depth] >= 3) {puts("Yes");exit(0);}return;}while (!c[pos].empty() && c[pos].back().first.first < from) c[pos].pop_back();for (unsigned i = 0; i < c[pos].size(); i++) {pair <pair <int, int>, int> x = c[pos][i];if (!vise[x.second] && !vis[x.first.first]) {vise[x.second] = vis[x.first.first] = true;work(x.first.first, from, depth + x.first.second);vise[x.second] = vis[x.first.first] = false;}}
}
int main() {read(n), read(m);for (int i = 1; i <= m; i++) {read(x[i]), read(y[i]);a[x[i]].push_back(y[i]);a[y[i]].push_back(x[i]);}for (int i = 1; i <= n; i++) {if (dfn[i] == 0) {v = 0, e = 0;tarjan(i, 0);if (e / 2 >= v + sqrt(v) + 2) {puts("Yes");return 0;}}}for (int i = 1; i <= m; i++)if (belong[x[i]] == belong[y[i]]) {b[x[i]].insert(y[i]);b[y[i]].insert(x[i]);}for (int i = 1; i <= n; i++) {iskey[i] = b[i].size() >= 3;if (!vis[i]) {v = 0, e = 0, dfs(i);if (e == 2 * v) {ans[v] += 2;if (ans[v] >= 3) {puts("Yes");return 0;}}}}for (int i = 1; i <= n; i++)if (iskey[i]) {p.push_back(i);while (!b[i].empty()) {int dest = *b[i].begin();rebuild(i, i, dest, 1);}}for (int i = 1; i <= n; i++) {sort(c[i].begin(), c[i].end());reverse(c[i].begin(), c[i].end());}memset(vis, false, sizeof(vis));for (int i = 1; i <= n; i++)if (iskey[i]) work(i, i, 0);puts("No");return 0;
}
// Cheating Solution O(Exp(N))
#include<bits/stdc++.h>
using namespace std;
const int MAXN = 1e5 + 5;
const int MAXM = 1e6 + 5;
typedef long long ll;
typedef long double ld;
typedef unsigned long long ull;
template <typename T> void chkmax(T &x, T y) {x = max(x, y); }
template <typename T> void chkmin(T &x, T y) {x = min(x, y); }
template <typename T> void read(T &x) {x = 0; int f = 1;char c = getchar();for (; !isdigit(c); c = getchar()) if (c == '-') f = -f;for (; isdigit(c); c = getchar()) x = x * 10 + c - '0';x *= f;
}
template <typename T> void write(T x) {if (x < 0) x = -x, putchar('-');if (x > 9) write(x / 10);putchar(x % 10 + '0');
}
template <typename T> void writeln(T x) {write(x);puts("");
}
int n, m, em, v, e, start, ans[MAXN];
int x[MAXM], y[MAXM], p[MAXN], rk[MAXN], rd[MAXN];
vector <int> a[MAXN]; set <int> b[MAXN];
vector <pair <pair <int, int>, int>> c[MAXN];
bool vis[MAXN], vise[MAXN], iskey[MAXN];bool instack[MAXN];
int tot, belong[MAXN], top, Stack[MAXN];
int timer, dfn[MAXN], low[MAXN];
bool cmp(int x, int y) {if (c[x].size() == c[y].size()) return rd[x] < rd[y];else return c[x].size() > c[y].size();
}
bool cnp(pair <pair <int, int>, int> x, pair <pair <int, int>, int> y) {if (x.first.first == y.first.first) {if (x.second == y.second) return x.first.second < y.first.second;else return x.second < y.second;} else return rk[x.first.first] < rk[y.first.first];
}
void tarjan(int pos, int fa) {low[pos] = dfn[pos] = ++timer;v++, e += a[pos].size();Stack[++top] = pos;instack[pos] = true;for (unsigned i = 0; i < a[pos].size(); i++) {if (a[pos][i] == fa) continue;if (dfn[a[pos][i]] == 0) {tarjan(a[pos][i], pos);low[pos] = min(low[pos], low[a[pos][i]]);} else if (instack[a[pos][i]]) low[pos] = min(low[pos], dfn[a[pos][i]]);}if (low[pos] == dfn[pos]) {int tmp = Stack[top--];belong[tmp] = ++tot;instack[tmp] = false;while (tmp != pos) {tmp = Stack[top--];belong[tmp] = tot;instack[tmp] = false;}}
}
void rebuild(int from, int fa, int pos, int len) {b[fa].erase(pos), b[pos].erase(fa);if (iskey[pos]) {if (from == pos) {ans[len] += 2;if (ans[len] >= 3) {puts("Yes");exit(0);}return;}c[from].emplace_back(make_pair(pos, len), ++em);c[pos].emplace_back(make_pair(from, len), em);return;}int dest = *b[pos].begin();rebuild(from, pos, dest, len + 1);
}
void dfs(int pos) {vis[pos] = true;v++, e += b[pos].size();for (auto x : b[pos])if (!vis[x]) dfs(x);
}
void work(int pos, int from, int depth) {if (depth && pos == from) {if (++ans[depth] >= 3) {puts("Yes");exit(0);}return;}if ((clock() - start) >= 1.5 * CLOCKS_PER_SEC) {puts("No");exit(0);}while (!c[pos].empty() && rk[c[pos].back().first.first] < rk[from]) c[pos].pop_back();for (unsigned i = 0; i < c[pos].size(); i++) {pair <pair <int, int>, int> x = c[pos][i];if (!vise[x.second] && !vis[x.first.first]) {vise[x.second] = vis[x.first.first] = true;work(x.first.first, from, depth + x.first.second);vise[x.second] = vis[x.first.first] = false;}}
}
int main() {read(n), read(m), start = clock();for (int i = 1; i <= m; i++) {read(x[i]), read(y[i]);a[x[i]].push_back(y[i]);a[y[i]].push_back(x[i]);}for (int i = 1; i <= n; i++) {if (dfn[i] == 0) {v = 0, e = 0;tarjan(i, 0);if (e / 2 >= v + sqrt(v) + 2) {puts("Yes");return 0;}}}for (int i = 1; i <= m; i++)if (belong[x[i]] == belong[y[i]]) {b[x[i]].insert(y[i]);b[y[i]].insert(x[i]);}for (int i = 1; i <= n; i++) {iskey[i] = b[i].size() >= 3;if (!vis[i]) {v = 0, e = 0, dfs(i);if (e == 2 * v) {ans[v] += 2;if (ans[v] >= 3) {puts("Yes");return 0;}}}}for (int i = 1; i <= n; i++)if (iskey[i]) {while (!b[i].empty()) {int dest = *b[i].begin();rebuild(i, i, dest, 1);}}for (int i = 1; i <= n; i++)p[i] = i, rd[i] = i;random_shuffle(rd + 1, rd + n + 1);sort(p + 1, p + n + 1, cmp);for (int i = 1; i <= n; i++)rk[p[i]] = i;for (int i = 1; i <= n; i++) {sort(c[i].begin(), c[i].end(), cnp);reverse(c[i].begin(), c[i].end());}memset(vis, false, sizeof(vis));for (int i = 1; i <= n; i++)work(p[i], p[i], 0);puts("No");return 0;
}

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