珂朵莉树
简要
珂朵莉树的核心操作：split
实际很简单，一个集合中，有一部分需要修改，而另一部分不需要修改，就把集合拆开，拆成两部分。
珂朵莉树的推平操作：assign
珂朵莉树的复杂度是由assign保证的。
由于数据随机，有14\frac{1}{4}41的操作为assign。
set的大小快速下降，最终趋于logn\log nlogn ，使得这种看似暴力无比的数据结构复杂度接近mlognm\log nmlogn 。
struct node {int l, r;mutable ll v;node(int _l, int _r = -1, ll _v = 0) : l(_l), r(_r), v(_v) {}bool operator < (const node &t) const {return l < t.l;}
};set<node>::iterator split(int pos) {set<node>::iterator it = st.lower_bound(node(pos));if (it != st.end() && it->l == pos) {return it;}it--;int l = it->l, r = it->r;ll v = it->v;st.erase(it);st.insert(node(l, pos - 1, v));return st.insert(node(pos, r, v)).first;
}void assign(int l, int r, ll value) {set<node>::iterator itr = split(r + 1), itl = split(l);st.erase(itl, itr);st.insert(node(l, r, value));
}
下面是一些可用珂朵莉树acacac题目，好多题目都卡珂朵莉树了，所以只能迫不得已写毒瘤线段树。
CF896C Willem, Chtholly and Seniorious
#include <bits/stdc++.h>
#define int long longusing namespace std;typedef long long ll;
typedef pair<int, int> pii;struct node {int l, r;mutable ll v;node(int _l, int _r = -1, ll _v = 0) : l(_l), r(_r), v(_v) {}bool operator < (const node &t) const {return l < t.l;}
};set<node> st;set<node>::iterator split(int pos) {set<node>::iterator it = st.lower_bound(node(pos));if (it != st.end() && it->l == pos) {return it;}it--;int l = it->l, r = it->r;ll v = it->v;st.erase(it);st.insert(node(l, pos - 1, v));return st.insert(node(pos, r, v)).first;
}void add(int l, int r, ll value) {set<node>::iterator itr = split(r + 1), itl = split(l);for (; itl != itr; itl++) {itl->v += value;}
}void assign(int l, int r, ll value) {set<node>::iterator itr = split(r + 1), itl = split(l);st.erase(itl, itr);st.insert(node(l, r, value));
}ll rank_k(int l, int r, int k, bool reversed = false) {if (reversed) {k = r - l + 2 - k;}set<node>::iterator itr = split(r + 1), itl = split(l);vector<pii> ans;for (; itl != itr; itl++) {ans.push_back(make_pair(itl->v, itl->r - itl->l + 1));}sort(ans.begin(), ans.end());for (auto it : ans) {k -= it.second;if (k <= 0) {return it.first;}}return -1;
}ll quick_pow(ll a, int n, int mod) {ll ans = 1;a %= mod;while (n) {if (n & 1) {ans = ans * a % mod;}a = a * a % mod;n >>= 1;}return ans;
}ll sum(int l, int r, int n, int mod) {set<node>::iterator itr = split(r + 1), itl = split(l);ll ans = 0;for (; itl != itr; itl++) {ans = (ans + 1ll * (itl->r - itl->l + 1) * quick_pow(itl->v, n, mod) % mod) % mod;}return ans;
}int n, m, seed, vmax;const int mod = 1e9 + 7;int rnd() {int ret = seed;seed = (1ll * seed * 7 + 13) % mod;return ret;
}signed main() {// freopen("in.txt", "r", stdin);// freopen("out.txt", "w", stdout);// ios::sync_with_stdio(false), cin.tie(0), cout.tie(0);scanf("%lld %lld %lld %lld", &n, &m, &seed, &vmax);for (int i = 1; i <= n; i++) {st.insert(node(i, i, rnd() % vmax + 1));}for (int i = 1; i <= m; i++) {int op = rnd() % 4 + 1, l = rnd() % n + 1, r = rnd() % n + 1, x, y;if (l > r) {swap(l, r);}if (op == 3) {x = rnd() % (r - l + 1) + 1;}else {x = rnd() % vmax + 1;}if (op == 4) {y = rnd() % vmax + 1;}if (op == 1) {add(l, r, x);}else if (op == 2) {assign(l, r, x);}else if (op == 3) {printf("%lld\n", rank_k(l, r, x));}else {printf("%lld\n", sum(l, r, x, y));}}return 0;
}
CF915E Physical Education Lessons
#include <bits/stdc++.h>using namespace std;struct node {int l, r;mutable int v;node(int _l, int _r = -1, int _v = 0) : l(_l), r(_r), v(_v) {}bool operator < (const node &t) const {return l < t.l;}
};int n, m;set<node> st;set<node>::iterator split(int pos) {set<node>::iterator it = st.lower_bound(node(pos));if (it != st.end() && it->l == pos) {return it;}it--;int l = it->l, r = it->r, v = it->v;st.erase(it);st.insert(node(l, pos - 1, v));return st.insert(node(pos, r, v)).first;
}void assign(int l, int r, int value) {set<node>::iterator itr = split(r + 1), itl = split(l), it = itl;for (; itl != itr; itl++) {n -= (itl->r - itl->l + 1) * itl->v;}st.erase(it, itr);st.insert(node(l, r, value));n += (r - l + 1) * value;
}int main() {// freopen("in.txt", "r", stdin);// freopen("out.txt", "w", stdout);// ios::sync_with_stdio(false), cin.tie(0), cout.tie(0);scanf("%d %d", &n, &m);st.insert(node(1, n, 1));for (int i = 1; i <= m; i++) {int l, r, v;scanf("%d %d %d", &l, &r, &v);assign(l, r, v & 1 ^ 1);printf("%d\n", n);}return 0;
}
CF343D Water Tree
#include <bits/stdc++.h>using namespace std;const int N = 5e5 + 10;int head[N], to[N << 1], nex[N << 1], cnt = 1;int n, m, tot, son[N], sz[N], fa[N], top[N], rk[N], id[N], dep[N];struct node {int l, r;mutable int v;node(int _l, int _r = -1, int _v = 0) : l(_l), r(_r), v(_v) {}bool operator < (const node &t) const {return l < t.l;}
};void add(int x, int y) {to[cnt] = y;nex[cnt] = head[x];head[x] = cnt++;
}void dfs1(int rt, int f) {sz[rt] = 1, fa[rt] = f, dep[rt] = dep[f] + 1;for (int i = head[rt]; i; i = nex[i]) {if (to[i] == f) {continue;}dfs1(to[i], rt);sz[rt] += sz[to[i]];if (!son[rt] || sz[to[i]] > sz[son[rt]]) {son[rt] = to[i];}}
}void dfs2(int rt, int tp) {top[rt] = tp, rk[++tot] = rt, id[rt] = tot;if (!son[rt]) {return ;}dfs2(son[rt], tp);for (int i = head[rt]; i; i = nex[i]) {if (to[i] == fa[rt] || to[i] == son[rt]) {continue;}dfs2(to[i], to[i]);}
}set<node> st;auto split(int pos) {auto it = st.lower_bound(node(pos));if (it != st.end() && it->l == pos) {return it;}it--;int l = it->l, r = it->r, v = it->v;st.erase(it);st.insert(node(l, pos - 1, v));return st.insert(node(pos, r, v)).first;
}void assign(int l, int r, int value) {auto itr = split(r + 1), itl = split(l);st.erase(itl, itr);st.insert(node(l, r, value));
}int main() {// freopen("in.txt", "r", stdin);// freopen("out.txt", "w", stdout);// ios::sync_with_stdio(false), cin.tie(0), cout.tie(0);scanf("%d", &n);for (int i = 1; i < n; i++) {int x, y;scanf("%d %d", &x, &y);add(x, y);add(y, x);}dfs1(1, 0);dfs2(1, 1);st.insert(node(1, n, 0));scanf("%d", &m);for (int i = 1; i <= m; i++) {int op, u;scanf("%d %d", &op, &u);if (op == 1) {assign(id[u], id[u] + sz[u] - 1, 1);}else if (op == 2) {int x = 1, y = u;while (top[x] != top[y]) {if (dep[top[x]] < dep[top[y]]) {swap(x, y);}assign(id[top[x]], id[x], 0);x = fa[top[x]];}if (dep[x] > dep[y]) {swap(x, y);}assign(id[x], id[y], 0);}else {auto it = st.lower_bound(node(id[u]));if (it->l > id[u]) {it--;}printf("%d\n", it->v);}}return 0;
}
实例------定制菜单栏显示用户姓名)
