P4175 [CTSC2008]網絡管理（樹狀數組套主席樹）

P4175 [CTSC2008]網絡管理

每次詢問兩點間的第 k k k大，考慮建立 n n n棵主席樹，每棵主席樹記錄的是從根節點到目前節點的狀态，

由于主席樹是一種類似字首和的資料結構，可以使用差分的方式得到 ( u , v ) (u, v) (u,v)點對之間的主席樹為 u + v − l c a ( u , v ) − f a ( l c a ( u , v ) ) u + v - lca(u, v) - fa(lca(u, v)) u+v−lca(u,v)−fa(lca(u,v))，

#include <bits/stdc++.h>

using namespace std;

const int N = 8e4 + 10, maxn = 100000000;

int a[N], n, m;

int head[N], to[N << 1], nex[N << 1], cnt = 1;

int son[N], sz[N], l[N], r[N], dep[N], top[N], fa[N], tot;

int root[N], sum[N << 8], ls[N << 8], rs[N << 8], num;

int A[N], B[N], cnt1, cnt2;

inline int lowbit(int x) {
  return x & (-x);
}

inline void add(int x, int y) {
  to[cnt] = y;
  nex[cnt] = head[x];
  head[x] = cnt++;
}

void update(int &rt, int l, int r, int x, int v) {
  if (!rt) {
    rt = ++num;
  }
  sum[rt] += v;
  if (l == r) {
    return ;
  }
  int mid = l + r >> 1;
  if (x <= mid) {
    update(ls[rt], l, mid, x, v);
  }
  else {
    update(rs[rt], mid + 1, r, x, v);
  }
}

void dfs1(int rt, int f) {
  fa[rt] = f, l[rt] = ++tot, sz[rt] = 1, dep[rt] = dep[f] + 1;
  for (int i = l[rt]; i <= n; i += lowbit(i)) {
    update(root[i], 0, maxn, a[rt], 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[son[rt]] < sz[to[i]]) {
      son[rt] = to[i];
    }
  }
  r[rt] = tot;
  for (int i = r[rt] + 1; i <= n; i += lowbit(i)) {
    update(root[i], 0, maxn, a[rt], -1);
  }
}

void dfs2(int rt, int tp) {
  top[rt] = tp;
  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]);
  }
}

int lca(int x, int y) {
  while (top[x] != top[y]) {
    if (dep[top[x]] < dep[top[y]]) {
      swap(x, y);
    }
    x = fa[top[x]];
  }
  return dep[x] < dep[y] ? x : y;
}

int query(int l, int r, int k) {
  if (l == r) {
    return l;
  }
  int cur = 0, mid = l + r >> 1;
  for (int i = 1; i <= cnt1; i++) {
    cur += sum[ls[A[i]]];
  }
  for (int i = 1; i <= cnt2; i++) {
    cur -= sum[ls[B[i]]];
  }
  if (cur >= k) {
    for (int i = 1; i <= cnt1; i++) {
      A[i] = ls[A[i]];
    }
    for (int i = 1; i <= cnt2; i++) {
      B[i] = ls[B[i]];
    }
    return query(l, mid, k);
  }
  else {
    for (int i = 1; i <= cnt1; i++) {
      A[i] = rs[A[i]];
    }
    for (int i = 1; i <= cnt2; i++) {
      B[i] = rs[B[i]];
    }
    return query(mid + 1, r, k - cur);
  }
}

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);
  for (int i = 1; i <= n; i++) {
    scanf("%d", &a[i]);
  }
  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);
  for (int i = 1; i <= m; i++) {
    int k, u, v;
    scanf("%d %d %d", &k, &u, &v);
    if (!k) {
      for (int i = l[u]; i <= n; i += lowbit(i)) {
        update(root[i], 0, maxn, a[u], -1);
      }
      for (int i = r[u] + 1; i <= n; i += lowbit(i)) {
        update(root[i], 0, maxn, a[u], 1);
      }
      a[u] = v;
      for (int i = l[u]; i <= n; i += lowbit(i)) {
        update(root[i], 0, maxn, a[u], 1);
      }
      for (int i = r[u] + 1; i <= n; i += lowbit(i)) {
        update(root[i], 0, maxn, a[u], -1);
      }
    }
    else {
      int f = lca(u, v), ff = fa[f], total = dep[u] + dep[v] - dep[f] - dep[ff];
      if (total < k) {
        puts("invalid request!");
      }
      else {
        cnt1 = cnt2 = 0;
        for (int i = l[u]; i; i -= lowbit(i)) {
          A[++cnt1] = root[i];
        }
        for (int i = l[v]; i; i -= lowbit(i)) {
          A[++cnt1] = root[i];
        }
        for (int i = l[f]; i; i -= lowbit(i)) {
          B[++cnt2] = root[i];
        }
        for (int i = l[ff]; i; i -= lowbit(i)) {
          B[++cnt2] = root[i];
        }
        k = total - k + 1;
        printf("%d\n", query(0, maxn, k));
      }
    }
  }
  return 0;
}