树链剖分

P3379 【模板】最近公共祖先（LCA）

dfs1: 处理一个点的深度、父结点、子树大小，重儿子。

dfs2: 记录每个点的最顶部。

query: 哪个 top 的深度小跳哪个。

#include <bits/stdc++.h>

using namespace std;

const int N = 500010, M = 1000010;

struct edge {
    int to, next;
} e[M];

int head[N], idx = 1;

void add(int u, int v) {
    idx++, e[idx].to = v, e[idx].next = head[u], head[u] = idx;
}

int n, w[N];

int dep[N], fa[N], sz[N], son[N];
int top[N], rk[N], id[N], cnt;

void dfs1(int u, int f) {
    sz[u] = 1, fa[u] = f, dep[u] = dep[f] + 1;
    for (int i = head[u]; i; i = e[i].next) {
        int to = e[i].to;
        if (to == f) continue;
        dfs1(to, u);
        sz[u] += sz[to];
        if (sz[son[u]] < sz[to]) son[u] = to;
    }
}

void dfs2(int u, int t) {
    top[u] = t; id[u] = ++cnt; rk[cnt] = u;
    if (son[u]) dfs2(son[u], t);
    for (int i = head[u]; i; i = e[i].next) {
        int to = e[i].to;
        if (to == fa[u] || to == son[u]) continue;
        dfs2(to, to);
    }
}

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

int main() {
    ios::sync_with_stdio(false);
    cin.tie(nullptr);

    int q, root;
    cin >> n >> q >> root;
    for (int i = 1; i < n; i++) {
        int u, v;
        cin >> u >> v;
        add(u, v), add(v, u);
    }

    dfs1(root, root);
    dfs2(root, root);
    
    while (q--) {
        int x, y;
        cin >> x >> y;
        cout << query(x, y) << '\n';
    }
    return 0;
}

P2590 [ZJOI2008] 树的统计

按照访问的顺序为点赋值。

如果是重链剖分，一条链上的所有点编号是连续的，所以我们可以考虑使用数据结构进行维护，此题我们可以使用线段树维护 \(sum\) 和 \(max\)。

#include <bits/stdc++.h>

using namespace std;

const int N = 30010, M = 60010;

struct edge {
    int to, next;
} e[M];

int head[N], idx;

void add(int u, int v) {
    idx++, e[idx].to = v, e[idx].next = head[u], head[u] = idx;
}

int n, w[N];
int sz[N], son[N], fa[N], dep[N];
int top[N], rk[N], id[N], cnt;

void dfs1(int u, int f) {
    sz[u] = 1; dep[u] = dep[f] + 1; fa[u] = f;
    for (int i = head[u]; i; i = e[i].next) {
        int to = e[i].to;
        if (to == f) continue;
        dfs1(to, u);
        sz[u] += sz[to];
        if (sz[son[u]] < sz[to]) son[u] = to;
    }
}

void dfs2(int u, int t) {
    top[u] = t; id[u] = ++cnt, rk[cnt] = u;
    if (son[u]) dfs2(son[u], t);
    for (int i = head[u]; i; i = e[i].next) {
        int to = e[i].to;
        if (to == fa[u] || to == son[u]) continue;
        dfs2(to, to);
    }
}

struct node {
    int max, sum;
} tr[N << 2];

node pushup(node l, node r) {
    return {max(l.max, r.max), l.sum + r.sum};
}

void init(int u, int l, int r) {
    if (l == r) {
        tr[u].max = tr[u].sum = w[rk[l]];
        return;
    }
    int mid = l + r >> 1;
    init(u << 1, l, mid);
    init(u << 1 | 1, mid + 1, r);
    tr[u] = pushup(tr[u << 1], tr[u << 1 | 1]);
}

void modify(int u, int l, int r, int x, int v) {
    if (l == r) {
        tr[u].max = tr[u].sum = v;
        return;
    }
    int mid = l + r >> 1;
    if (x <= mid) modify(u << 1, l, mid, x, v);
    else modify(u << 1 | 1, mid + 1, r,  x, v);
    tr[u] = pushup(tr[u << 1], tr[u << 1 | 1]);
}

node query(int u, int l, int r, int pl, int pr) {
    if (pl <= l && r <= pr) return tr[u];
    int mid = l + r >> 1;
    if (pr <= mid) return query(u << 1, l, mid, pl, pr);
    else if (pl > mid) return query(u << 1 | 1, mid + 1, r, pl, pr);
    else return pushup(query(u << 1, l, mid, pl, pr),  query(u << 1 | 1, mid + 1, r, pl, pr));
}

node ask(int x, int y) {
    node ans = {-0x3f3f3f3f, 0};
    while (top[x] != top[y]) {
        if (dep[top[x]] < dep[top[y]]) swap(x, y);
        ans = pushup(ans, query(1, 1, n, id[top[x]], id[x]));
        x = fa[top[x]];
    }
    if (dep[x] > dep[y]) swap(x, y);
    ans = pushup(ans, query(1, 1, n, id[x], id[y]));
    return ans;
}

int main() {
    freopen("a.in", "r", stdin);
    ios::sync_with_stdio(false);
    cin.tie(nullptr);

    cin >> n;
    for (int i = 1; i < n; i++) {
        int u, v;
        cin >> u >> v;
        add(u, v), add(v, u);
    }

    dfs1(1, 1);
    dfs2(1, 1);

    for (int i = 1; i <= n; i++) cin >> w[i];
    init(1, 1, n);
    int q;
    cin >> q;
    string opt;
    int x, y;
    while (q--) {
        cin >> opt >> x >> y;
        if (opt == "CHANGE") modify(1, 1, n, id[x], y);
        else if (opt == "QMAX") cout << ask(x, y).max << '\n';
        else cout << ask(x, y).sum << '\n';
    }
    return 0;
}