重链剖分的总结与模板

目录

重链剖分的总结与模板

概述:

我们通常说的树链剖分指的是重链剖分。此外还有长链剖分，实链剖分。在学LCT时感觉需要对重剖来个总结。于是有了这一篇。一句话的概括。重链剖分是一种对树上结点进行编号。然后把链哈希成区间的一种方式。然后就可以把树上信息变成若干区间信息，通过线段树等数据结构进行高效的维护。如果你想学习树剖，网络上有很多资源。然后完成下面例题的点权和边权的维护，重链剖分就算入门了。

点权:

推荐例题HDU 3966 & FJUT OJ 2710 Aragorn's Story

///树剖 点权

#include <bits/stdc++.h>

using namespace std;
const int MAXN = 50010;
typedef long long LL;

struct Edge {
    int to, w, next;
} edge[MAXN * 2];

int first[MAXN], sign, tot;

int dep[MAXN], siz[MAXN], faz[MAXN], id[MAXN], son[MAXN], top[MAXN], tid[MAXN];

int n, m, q;

long long a[MAXN];

struct SegmentTree {

    struct Node {
        int l, r;
        LL sum, Lazy;
    } tree[MAXN * 4];

    inline void push_up(int rt) {
        tree[rt].sum = tree[rt << 1].sum + tree[rt << 1 | 1].sum;
    }

    inline void push_down(int rt) {
        if(tree[rt].Lazy) {
            tree[rt << 1].Lazy += tree[rt].Lazy;
            tree[rt << 1 | 1].Lazy += tree[rt].Lazy;
            tree[rt << 1].sum += (tree[rt << 1].r - tree[rt << 1].l + 1) * tree[rt].Lazy;
            tree[rt << 1 | 1].sum += (tree[rt << 1 | 1].r - tree[rt << 1 | 1].l + 1) * tree[rt].Lazy;
            tree[rt].Lazy = 0;
        }
    }

    void build(int rt, int l, int r) {
        tree[rt].l = l;
        tree[rt].r = r;
        tree[rt].sum = 0;
        tree[rt].Lazy = 0;
        if(l == r) {
            tree[rt].sum = a[ tid[l] ];
            return ;
        }
        int mid = (l + r) >> 1;
        build(rt << 1, l, mid);
        build(rt << 1 | 1, mid + 1, r);
        push_up(rt);
    }

    void update(int rt, int l, int r, LL val) {
        if(l <= tree[rt].l && tree[rt].r <= r) {
            tree[rt].sum += (tree[rt].r - tree[rt].l + 1) * val;
            tree[rt].Lazy += val;
            return ;
        }
        push_down(rt);
        int mid = (tree[rt].l + tree[rt].r) >> 1;
        if(r <= mid) {
            update(rt << 1, l, r, val);
        } else if(l > mid) {
            update(rt << 1 | 1, l, r, val);
        } else {
            update(rt << 1, l, mid, val);
            update(rt << 1 | 1, mid + 1, r, val);
        }
        push_up(rt);
    }

    LL query(int rt, int l, int r) {
        if(l <= tree[rt].l && tree[rt].r <= r) {
            return tree[rt].sum;
        }
        push_down(rt);
        int mid = (tree[rt].l + tree[rt].r) >> 1;
        if(r <= mid) {
            return query(rt << 1, l, r);
        } else if(l > mid) {
            return query(rt << 1 | 1, l, r);
        } else {
            return query(rt << 1, l, mid) + query(rt << 1 | 1, mid + 1, r);
        }
    }

} Seg;


void init() {
    sign = tot = 0;
    memset(first, -1, sizeof(first));
    memset(dep, 0, sizeof(dep));
    memset(siz, 0, sizeof(siz));
    memset(faz, 0, sizeof(faz));
    memset(id, 0, sizeof(id));
    memset(son, 0, sizeof(son));
    memset(top, 0, sizeof(top));
}

void add_edge(int u, int v, int w) {
    edge[sign].to = v;
    edge[sign].w = w;
    edge[sign].next = first[u];
    first[u] = sign++;
}

void dfs1(int now, int father, int depth) {
    siz[now] = 1;
    faz[now] = father;
    dep[now] = depth;
    son[now] = 0;
    for(int i = first[now]; ~i; i = edge[i].next) {
        int to = edge[i].to;
        if(to != father) {
            dfs1(to, now, depth + 1);
            siz[now] += siz[to];
            if(son[now] == 0 || siz[ son[now] ] < siz[to]) {
                son[now] = to;
            }
        }
    }
}

void dfs2(int now, int topf) {
    top[now] = topf;
    id[now] = ++tot;
    tid[id[now]] = now;
    if(son[now]) {
        dfs2(son[now], topf);
    }
    for(int i = first[now]; ~i; i = edge[i].next) {
        int to = edge[i].to;
        if(to == faz[now] || to == son[now]) {
            continue;
        }
        dfs2(to, to);
    }
}

void cutting(int u, int v, int val) {
    int fu = top[u], fv = top[v];
    while(fu != fv) {
        if(dep[fu] < dep[fv]) {
            swap(fu, fv);
            swap(u, v);
        }
        Seg.update(1, id[fu], id[u], val);
        u = faz[fu];
        fu = top[u];
    }
    if(dep[u] > dep[v]) {
        swap(u,v);
    }
    Seg.update(1, id[u], id[v], val);
}

long long query(int u, int v) {
    long long sum = 0;
    int fu = top[u], fv = top[v];
    while(fu != fv) {
        if(dep[fu] < dep[fv]) {
            swap(fu, fv);
            swap(u, v);
        }
        sum = sum + Seg.query(1, id[fu], id[u]);
        u = faz[fu];
        fu = top[u];
    }
    if(dep[u] > dep[v]) {
        swap(u, v);
    }
    return sum = sum + Seg.query(1, id[u], id[v]);
}

int main() {
    while(~scanf("%d %d %d", &n, &m, &q)) {
        init();
        for(int i = 1; i <= n; i++ ) {
            scanf("%d", &a[i]);
        }
        for(int i = 1; i <= m; i++ ) {
            int u, v;
            scanf("%d %d", &u, &v);
            add_edge(u, v, 1);
            add_edge(v, u, 1);
        }
        tot = 0;
        dfs1(1, 0, 0);
        dfs2(1, 1);
        Seg.build(1, 1, n);
        char opt[5];
        int x, y,z;
        for(int i = 1; i <= q; i++ ) {
            scanf("%s", opt);
            if(opt[0] == 'I') {
                scanf("%d %d %d", &x, &y, &z);
                cutting(x, y, z);
                continue;
            }
            if(opt[0] == 'D') {
                scanf("%d %d %d", &x, &y, &z);
                cutting(x, y, -z);
                continue;
            }
            if(opt[0] == 'Q') {
                scanf("%d", &x);
                printf("%I64d\n", query(x, x));
                continue;
            }
        }
    }
    return 0;
}

边权

推荐例题POJ2763 & FJUT OJ 2796 Housewife Wind

维护边权，只要把边权存到深度更深的结点即可。

这么长代码还好没写出bug。不然真不好办。。。

#include <cstdio>
#include <iostream>
#include <cstring>
#include <algorithm>

using namespace std;
const int maxn = 1e5 + 7;
const int INF = 0x7FFFFFF;

int dep[maxn], siz[maxn], faz[maxn], id[maxn], son[maxn], val[maxn], top[maxn];

struct Edge {
    int to, w, next;
} edge[maxn * 2];

int first[maxn], sign, n, m, s, tot;

struct Node {
    int from, to, cost;
} input[maxn];

struct TreeNode {
    int l, r, mx, mi, lazy;
} tree[maxn << 2];

inline void init() {
    memset(first, -1, sizeof(first));
    sign = 0;
}

inline void add_edge(int u, int v, int w) {
    edge[sign].to = v;
    edge[sign].w = w;
    edge[sign].next = first[u];
    first[u] = sign ++;
}

void dfs1(int now, int father, int depth) {
    siz[now] = 1;
    faz[now] = father;
    dep[now] = depth;
    son[now] = 0;
    for(int i = first[now]; ~i; i = edge[i].next) {
        int to = edge[i].to;
        if(to != father) {
            dfs1(to, now, depth + 1);
            siz[now] += siz[to];
            if(siz[ son[now] ] < siz[to]) {
                son[now] = to;
            }
        }
    }
}

void dfs2(int now, int topf) {
    top[now] = topf;
    id[now] = ++tot;
    if(son[now]) {
        dfs2(son[now], topf);
    }
    for(int i = first[now]; ~i; i = edge[i].next) {
        int to = edge[i].to;
        if(to == faz[now] || to == son[now]) {
            continue;
        }
        dfs2(to, to);
    }
}

void push_up(int rt) {
    tree[rt].mx = max(tree[rt << 1].mx, tree[rt << 1 | 1].mx);
    tree[rt].mi = min(tree[rt << 1].mi, tree[rt << 1 | 1].mi);
}

void push_down(int rt) {
    if(tree[rt].lazy) {
        tree[rt].lazy ^= 1;
        tree[rt << 1].lazy ^= 1;
        tree[rt << 1 | 1].lazy ^= 1;
        swap(tree[rt << 1].mx, tree[rt << 1].mi);
        tree[rt << 1].mx *= -1;
        tree[rt << 1].mi *= -1;
        swap(tree[rt << 1 | 1].mx, tree[rt << 1 | 1].mi);
        tree[rt << 1 | 1].mx *= -1;
        tree[rt << 1 | 1].mi *= -1;
    }
}

void build(int rt, int l, int r) {
    tree[rt].l = l, tree[rt].r = r;
    tree[rt].lazy = 0;
    if(l == r) {
        tree[rt].mx = tree[rt].mi = val[l];
        return ;
    }
    int mid = (l + r) >> 1;
    build(rt << 1, l, mid);
    build(rt << 1 | 1, mid + 1, r);
    push_up(rt);
}

void update(int rt, int l, int r) { ///区间取反
    if(l <= tree[rt].l && tree[rt].r <= r) {
        tree[rt].lazy ^= 1;
        swap(tree[rt].mx, tree[rt].mi);
        tree[rt].mx *= -1;
        tree[rt].mi *= -1;
        return ;
    }
    push_down(rt);
    int mid = (tree[rt].l + tree[rt].r) >> 1;
    if(r <= mid) {
        update(rt << 1, l, r);
    } else if(l > mid) {
        update(rt << 1 | 1, l, r);
    } else {
        update(rt << 1, l, mid);
        update(rt << 1 | 1, mid + 1, r);
    }
    push_up(rt);
}

void updatePos(int rt, int pos, int val) { ///单点修改
    if(tree[rt].l == tree[rt].r) {
        tree[rt].mx = tree[rt].mi = val;
        return ;
    }
    push_down(rt);
    int mid = (tree[rt].l + tree[rt].r) >> 1;
    if(pos <= mid) {
        updatePos(rt << 1, pos, val);
    } else {
        updatePos(rt << 1 | 1, pos, val);
    }
    push_up(rt);
}

int query(int rt, int l, int r) {
    if(l <= tree[rt].l && tree[rt].r <= r) {
        return tree[rt].mx;
    }
    push_down(rt);
    int mid = (tree[rt].l + tree[rt].r) >> 1;
    if(r <= mid) {
        return query(rt << 1, l, r);
    } else if(l > mid) {
        return query(rt << 1 | 1, l, r);
    } else {
        return max(query(rt << 1, l, mid), query(rt << 1 | 1, mid + 1, r));
    }
}

void treeUpdate(int x, int y) {
    while(top[x] != top[y]) {
        if(dep[top[x]] < dep[top[y]]) {
            swap(x,y);
        }
        update(1, id[top[x]], id[x]);
        x = faz[top[x]];
    }
    if(dep[x] > dep[y]) {
        swap(x,y);
    }
    if(x != y) {
        update(1, id[son[x]], id[y]);
    }
}

int treeQuery(int x, int y) {
    int ans = -INF;
    while(top[x] != top[y]) {
        if(dep[top[x]] < dep[top[y]]) {
            swap(x,y);
        }
        ans = max(ans, query(1, id[top[x]], id[x]));
        x = faz[top[x]];
    }
    if(dep[x] > dep[y]) {
        swap(x,y);
    }
    if(x != y) {
        ans = max(ans, query(1, id[son[x]], id[y]));
    }
    return ans;
}

int cutting(int x, int y) {
    int sum = 0;
    while(top[x] != top[y]) {
        if(dep[top[x]] < dep[top[y]]) {
            swap(x,y);
        }
        sum += query(1, id[top[x]], id[x]);
        x = faz[top[x]];
    }
    if(dep[x] > dep[y]) {
        swap(x,y);
    }
    if(x != y) {
        sum += query(1, id[son[x]], id[y]);
    }
    return sum;
}

int main() {
    int T, x, y;
    char opt[10];
    scanf("%d", &T);
    while(T--) {
        scanf("%d", &n);
        init();
        for(int i = 1; i <= n - 1; i++ ) {
            scanf("%d %d %d", &input[i].from, &input[i].to, &input[i].cost);
            add_edge(input[i].from, input[i].to, input[i].cost);
            add_edge(input[i].to, input[i].from, input[i].cost);
        }
        tot = 0;
        dfs1(1, 0, 1);
        dfs2(1, 1);
        for(int i = 1; i <= n - 1; i++ ) {
            if(dep[ input[i].from ] < dep[ input[i].to ]) {
                swap(input[i].from, input[i].to);
            }
            val[ id[ input[i].from ] ] = input[i].cost;
        }
        build(1, 1, n);
        while(~scanf("%s", opt) && strcmp(opt, "DONE")) {
            if(opt[0] == 'C') {
                scanf("%d %d", &x, &y);
                if(dep[ input[x].from ] < dep[ input[x].to ]) {
                    swap(input[x].from, input[x].to);
                }
                updatePos(1, id[ input[x].from ], y);
            }
            if(opt[0] == 'N') {
                scanf("%d %d", &x, &y);
                treeUpdate(x, y);
            }
            if(opt[0] == 'Q') {
                scanf("%d %d", &x, &y);
                printf("%d\n", treeQuery(x, y));
            }
        }
    }
    return 0;
}
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