算法学习(14)：树链刨分(重链刨分)

树链刨分

代码

第一个 DFS 记录每个结点的父节点（father）、深度（deep）、子树大小（size）、重子节点（hson）。
第二个 DFS 记录所在链的链顶（top，应初始化为结点本身）、重边优先遍历时的 DFS 序（dfn）、DFS 序对应的节点编号（rank）。

void dfs1(int o) {
  son[o] = -1;
  siz[o] = 1;
  for (int j = h[o]; j; j = nxt[j])
    if (!dep[p[j]]) {
      dep[p[j]] = dep[o] + 1;
      fa[p[j]] = o;
      dfs1(p[j]);
      siz[o] += siz[p[j]];
      if (son[o] == -1 || siz[p[j]] > siz[son[o]]) son[o] = p[j];
    }
}
void dfs2(int o, int t) {
  top[o] = t;
  cnt++;
  dfn[o] = cnt;
  rnk[cnt] = o;
  if (son[o] == -1) return;
  dfs2(son[o], t);  // 优先对重儿子进行 DFS，可以保证同一条重链上的点 DFS 序连续
  for (int j = h[o]; j; j = nxt[j])
    if (p[j] != son[o] && p[j] != fa[o]) dfs2(p[j], p[j]);
}

应用

1. 树上两点路径权值和。
2. 维护子树上的信息，譬如将以x为根的子树的所有结点的权值增加v。
3. 求最近公共祖先。(附代码)

int lca(int u, int v) {
  while (top[u] != top[v]) {
    if (dep[top[u]] > dep[top[v]])
      u = fa[top[u]];
    else
      v = fa[top[v]];
  }
  return dep[u] > dep[v] ? v : u;
}

例题ZJOI树的统计

三个操作：

1. 单点修改
2. 查询区间最大值
3. 查询区间和

#include <algorithm>
#include <cstdio>
#include <cstring>
#define lc o << 1
#define rc o << 1 | 1
const int maxn = 60010;
const int inf = 2e9;
int n, a, b, w[maxn], q, u, v;
int cur, h[maxn], nxt[maxn], p[maxn];
int siz[maxn], top[maxn], son[maxn], dep[maxn], fa[maxn], dfn[maxn], rnk[maxn],
    cnt;
char op[10];
inline void add_edge(int x, int y) {
  cur++;
  nxt[cur] = h[x];
  h[x] = cur;
  p[cur] = y;
}
struct SegTree {
  int sum[maxn * 4], maxx[maxn * 4];
  void build(int o, int l, int r) {
    if (l == r) {
      sum[o] = maxx[o] = w[rnk[l]];
      return;
    }
    int mid = (l + r) >> 1;
    build(lc, l, mid);
    build(rc, mid + 1, r);
    sum[o] = sum[lc] + sum[rc];
    maxx[o] = std::max(maxx[lc], maxx[rc]);
  }
  int query1(int o, int l, int r, int ql, int qr) {  // max
    if (l > qr || r < ql) return -inf;
    if (ql <= l && r <= qr) return maxx[o];
    int mid = (l + r) >> 1;
    return std::max(query1(lc, l, mid, ql, qr), query1(rc, mid + 1, r, ql, qr));
  }
  int query2(int o, int l, int r, int ql, int qr) {  // sum
    if (l > qr || r < ql) return 0;
    if (ql <= l && r <= qr) return sum[o];
    int mid = (l + r) >> 1;
    return query2(lc, l, mid, ql, qr) + query2(rc, mid + 1, r, ql, qr);
  }
  void update(int o, int l, int r, int x, int t) {
    if (l == r) {
      maxx[o] = sum[o] = t;
      return;
    }
    int mid = (l + r) >> 1;
    if (x <= mid)
      update(lc, l, mid, x, t);
    else
      update(rc, mid + 1, r, x, t);
    sum[o] = sum[lc] + sum[rc];
    maxx[o] = std::max(maxx[lc], maxx[rc]);
  }
} st;
void dfs1(int o) {
  son[o] = -1;
  siz[o] = 1;
  for (int j = h[o]; j; j = nxt[j])
    if (!dep[p[j]]) {
      dep[p[j]] = dep[o] + 1;
      fa[p[j]] = o;
      dfs1(p[j]);
      siz[o] += siz[p[j]];
      if (son[o] == -1 || siz[p[j]] > siz[son[o]]) son[o] = p[j];
    }
}
void dfs2(int o, int t) {
  top[o] = t;
  cnt++;
  dfn[o] = cnt;
  rnk[cnt] = o;
  if (son[o] == -1) return;
  dfs2(son[o], t);
  for (int j = h[o]; j; j = nxt[j])
    if (p[j] != son[o] && p[j] != fa[o]) dfs2(p[j], p[j]);
}
int querymax(int x, int y) {
  int ret = -inf, fx = top[x], fy = top[y];
  while (fx != fy) {
    if (dep[fx] >= dep[fy])
      ret = std::max(ret, st.query1(1, 1, n, dfn[fx], dfn[x])), x = fa[fx];
    else
      ret = std::max(ret, st.query1(1, 1, n, dfn[fy], dfn[y])), y = fa[fy];
    fx = top[x];
    fy = top[y];
  }
  if (dfn[x] < dfn[y])
    ret = std::max(ret, st.query1(1, 1, n, dfn[x], dfn[y]));
  else
    ret = std::max(ret, st.query1(1, 1, n, dfn[y], dfn[x]));
  return ret;
}
int querysum(int x, int y) {
  int ret = 0, fx = top[x], fy = top[y];
  while (fx != fy) {
    if (dep[fx] >= dep[fy])
      ret += st.query2(1, 1, n, dfn[fx], dfn[x]), x = fa[fx];
    else
      ret += st.query2(1, 1, n, dfn[fy], dfn[y]), y = fa[fy];
    fx = top[x];
    fy = top[y];
  }
  if (dfn[x] < dfn[y])
    ret += st.query2(1, 1, n, dfn[x], dfn[y]);
  else
    ret += st.query2(1, 1, n, dfn[y], dfn[x]);
  return ret;
}
int main() {
  scanf("%d", &n);
  for (int i = 1; i < n; i++)
    scanf("%d%d", &a, &b), add_edge(a, b), add_edge(b, a);
  for (int i = 1; i <= n; i++) scanf("%d", w + i);
  dep[1] = 1;
  dfs1(1);
  dfs2(1, 1);
  st.build(1, 1, n);
  scanf("%d", &q);
  while (q--) {
    scanf("%s%d%d", op, &u, &v);
    if (!strcmp(op, "CHANGE")) st.update(1, 1, n, dfn[u], v);
    if (!strcmp(op, "QMAX")) printf("%d\n", querymax(u, v));
    if (!strcmp(op, "QSUM")) printf("%d\n", querysum(u, v));
  }
  return 0;
}