树链剖分模板

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本文详细介绍了树链剖分在处理点权和边权问题时的模板应用。点权模板中，操作包括M a b c用于将节点a到b路径上所有点染为颜色c，而Q a b则用于查询a到b路径上颜色段的数量。边权模板部分的内容未给出，但可以推测涉及到基于树链剖分的边权重处理技巧。

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点权模板：

1、M a b c将节点a到节点b路径上所有点都染成颜色c；
2、Q a b询问节点a到节点b路径上的颜色段数量（连续相同颜色被认为是同一段），如“112221”由3段组成：“11”、“222”和“1”。

const int N = 100100;
struct edge
{
    int to, next;
}g[N*2];
int cnt, head[N];
int dep[N], siz[N], son[N], fat[N], id[N], top[N];
int color[N], d[N][2], pos[N];
int n, m, num;
struct node
{
    int l, r, lc, rc, val, mark; /*lc, rc记录区间左端点和右端点颜色，*/
}tr[N*4];
void init()
{
    cnt = 0;
    memset(head, -1, sizeof head);
    num = 0;
}
void add_edge(int v, int u)
{
    g[cnt].to = u, g[cnt].next = head[v], head[v] = cnt++;
}
void dfs1(int v, int fa, int d)
{
    dep[v] = d, son[v] = 0, siz[v] = 1, fat[v] = fa;
    for(int i = head[v]; i != -1; i = g[i].next)
    {
        int u = g[i].to;
        if(u != fa)
        {
            dfs1(u, v, d + 1);
            siz[v] += siz[u];
            if(siz[son[v]] < siz[u]) son[v] = u;
        }
    }
}
void dfs2(int v, int tp)
{
    top[v] = tp, id[v] = ++num;
    if(son[v]) dfs2(son[v], top[v]);
    for(int i = head[v]; i != -1; i = g[i].next)
    {
        int u = g[i].to;
        if(u != fat[v] && u != son[v]) dfs2(u, u);
    }
}
void push_up(int k)
{ /*区间合并，判断左儿子的右端点和右儿子的左端点是否相同，更新父节点左右端点颜色*/
    tr[k].lc = tr[k<<1].lc, tr[k].rc = tr[k<<1|1].rc;
    if(tr[k<<1].rc == tr[k<<1|1].lc)
        tr[k].val = tr[k<<1].val + tr[k<<1|1].val - 1;
    else tr[k].val = tr[k<<1].val + tr[k<<1|1].val;
}
void push_down(int k)
{
    if(tr[k].mark)
    {
        tr[k<<1].mark = tr[k<<1|1].mark = tr[k].mark;
        tr[k<<1].lc = tr[k<<1].rc = tr[k<<1|1].lc = tr[k<<1|1].rc = tr[k].mark;
        tr[k<<1].val = tr[k<<1|1].val = 1;
        tr[k].mark = 0;
    }
}
void build(int l, int r, int k)
{
    tr[k].l = l, tr[k].r = r, tr[k].val = 1, tr[k].mark = 0;
    if(l == r)
    {
        tr[k].lc = tr[k].rc = pos[l];
        return;
    }
    int mid = (l + r) >> 1;
    build(l, mid, k << 1);
    build(mid + 1, r, k << 1|1);
    push_up(k);
}
void update(int l, int r, int c, int k)
{
    if(l <= tr[k].l && tr[k].r <= r)
    {
        tr[k].val = 1;
        tr[k].lc = tr[k].rc = c;
        tr[k].mark = c;
        return;
    }
    push_down(k);
    int mid = (tr[k].l + tr[k].r) >> 1;
    if(l <= mid) update(l, r, c, k << 1);
    if(r > mid) update(l, r, c, k << 1|1);
    push_up(k);
}
int query(int l, int r, int k)
{
    if(l <= tr[k].l && tr[k].r <= r) return tr[k].val;
    push_down(k);
    int mid = (tr[k].l + tr[k].r) >> 1;
    int ans = 0, f1 = 0, f2 = 0;
    if(l <= mid)
        ans += query(l, r, k << 1), f1 = 1;
    if(r > mid)
        ans += query(l, r, k << 1|1), f2 = 1;
    if(f1 && f2) /*查询时注意判断相邻两个区间相接处颜色是否相同*/
        if(tr[k<<1].rc == tr[k<<1|1].lc) ans -= 1;
    return ans;
    //注释掉的另一种查询
//    if(r <= mid) return query(l, r, k << 1);
//    else if(l > mid) return query(l, r, k << 1|1);
//    else
//    {
//        int ans = 0;
//        ans = query(l, r, k << 1) + query(l, r, k << 1|1);
//        if(tr[k<<1].rc == tr[k<<1|1].lc) ans -= 1;
//        return ans;
//    }
}
int query_node(int x, int k)
{
    if(tr[k].l == tr[k].r) return tr[k].rc;
    push_down(k);
    int mid = (tr[k].l + tr[k].r) >> 1;
    int ans = 0;
    if(x <= mid) ans = query_node(x, k << 1);
    else ans = query_node(x, k << 1|1);
    push_up(k);
    return ans;
}
void Update(int v, int u, int c)
{
    int t1 = top[v], t2 = top[u];
    while(t1 != t2)
    {
        if(dep[t1] < dep[t2])
            swap(t1, t2), swap(v, u);
        update(id[t1], id[v], c, 1);
        v = fat[t1], t1 = top[v];
    }
    //和边权不同的地方
    if(dep[v] > dep[u]) swap(v, u);
    update(id[v], id[u], c, 1);
}
int Query(int v, int u)
{
    int t1 = top[v], t2 = top[u];
    int ans = 0;
    while(t1 != t2)
    {
        if(dep[t1] < dep[t2])
            swap(t1, t2), swap(v, u);
        ans += query(id[t1], id[v], 1);
        /*判断相邻两个区间相接处颜色是否相同*/
        if(query_node(id[t1], 1) == query_node(id[fat[t1]], 1)) ans--;
        v = fat[t1], t1 = top[v];
    }
    //和边权不同的地方
    if(dep[v] > dep[u]) swap(v, u);
    return ans += query(id[v], id[u], 1);
}
int main()
{
    while(~ scanf("%d%d", &n, &m))
    {
        init();
        for(int i = 1; i <= n; i++)
            scanf("%d", &color[i]);
        for(int i = 1; i <= n - 1; i++)
        {
            scanf("%d%d", &d[i][0], &d[i][1]);
            add_edge(d[i][0], d[i][1]);
            add_edge(d[i][1], d[i][0]);
        }
        dfs1(1, 0, 1);
        dfs2(1, 1);
        for(int i = 1; i <= n; i++) pos[id[i]] = color[i];
        build(1, num, 1);
        char ch;
        int a, b, c;
        while(m--)
        {
            scanf(" %c", &ch);
            if(ch == 'Q')
            {
                scanf("%d%d", &a, &b);
                printf("%d\n", Query(a, b));
            }
            else
            {
                scanf("%d%d%d", &a, &b, &c);
                Update(a, b, c);
            }
        }
    }
    return 0;
}

边权模板：

const int N = 100100;
struct edge
{
    int to, next;
}g[N*2];
int cnt, head[N];
int dep[N], siz[N], son[N], fat[N], id[N], top[N];
int d[N][3], pos[N];
int n, m, num;
struct node
{
    int l, r, lc, rc, val, mark;
}tr[N*4];
void add_edge(int v, int u)
{
    g[cnt].to = u, g[cnt].next = head[v], head[v] = cnt++;
}
void dfs1(int v, int fa, int d)
{
    dep[v] = d, son[v] = 0, siz[v] = 1, fat[v] = fa;
    for(int i = head[v]; i != -1; i = g[i].next)
    {
        int u = g[i].to;
        if(u != fa)
        {
            dfs1(u, v, d + 1);
            siz[v] += siz[u];
            if(siz[son[v]] < siz[u]) son[v] = u;
        }
    }
}
void dfs2(int v, int tp)
{
    top[v] = tp, id[v] = ++num;
    if(son[v]) dfs2(son[v], top[v]);
    for(int i = head[v]; i != -1; i = g[i].next)
    {
        int u = g[i].to;
        if(u != fat[v] && u != son[v]) dfs2(u, u);
    }
}
void push_up(int k)
{
    tr[k].lc = tr[k<<1].lc, tr[k].rc = tr[k<<1|1].rc;
    if(tr[k<<1].rc == tr[k<<1|1].lc)
        tr[k].val = tr[k<<1].val + tr[k<<1|1].val - 1;
    else tr[k].val = tr[k<<1].val + tr[k<<1|1].val;
}
void push_down(int k)
{
    if(tr[k].mark)
    {
        tr[k<<1].mark = tr[k<<1|1].mark = tr[k].mark;
        tr[k<<1].lc = tr[k<<1].rc = tr[k<<1|1].lc = tr[k<<1|1].rc = tr[k].mark;
        tr[k<<1].val = tr[k<<1|1].val = 1;
        tr[k].mark = 0;
    }
}
void build(int l, int r, int k)
{
    tr[k].l = l, tr[k].r = r, tr[k].val = 1, tr[k].mark = 0;
    if(l == r)
    {
        tr[k].lc = tr[k].rc = pos[l];
        return;
    }
    int mid = (l + r) >> 1;
    build(l, mid, k << 1);
    build(mid + 1, r, k << 1|1);
    push_up(k);
}
void update(int l, int r, int c, int k)
{
    if(l <= tr[k].l && tr[k].r <= r)
    {
        tr[k].val = 1;
        tr[k].lc = tr[k].rc = c;
        tr[k].mark = c;
        return;
    }
    push_down(k);
    int mid = (tr[k].l + tr[k].r) >> 1;
    if(l <= mid) update(l, r, c, k << 1);
    if(r > mid) update(l, r, c, k << 1|1);
    push_up(k);
}
int query(int l, int r, int k)
{
    if(l <= tr[k].l && tr[k].r <= r) return tr[k].val;
    push_down(k);
    int mid = (tr[k].l + tr[k].r) >> 1;
    int ans = 0, f1 = 0, f2 = 0;
    if(l <= mid)
        ans += query(l, r, k << 1), f1 = 1;
    if(r > mid)
        ans += query(l, r, k << 1|1), f2 = 1;
    if(f1 && f2)
        if(tr[k<<1].rc == tr[k<<1|1].lc) ans -= 1;
    return ans;
}
int query_node(int x, int k)
{
    if(tr[k].l == tr[k].r) return tr[k].rc;
    push_down(k);
    int mid = (tr[k].l + tr[k].r) >> 1;
    int ans = 0;
    if(x <= mid) ans = query_node(x, k << 1);
    else ans = query_node(x, k << 1|1);
    return ans;
}
void Update(int v, int u, int c)
{
    int t1 = top[v], t2 = top[u];
    while(t1 != t2)
    {
        if(dep[t1] < dep[t2])
            swap(t1, t2), swap(v, u);
        update(id[t1], id[v], c, 1);
        v = fat[t1], t1 = top[v];
    }
    if(v == u) return;
    if(dep[v] > dep[u]) swap(v, u);
    update(id[son[v]], id[u], c, 1);
}
int Query(int v, int u)
{//Query函数和Update函数基本是相同的，Query函数里面多出来的是此题的代码，不是通用代码，参照Update函数
    int t1 = top[v], t2 = top[u];
    int ans = 0;
    int cv = 0, cu = 0, x;
    while(t1 != t2)
    {
        if(dep[t1] < dep[t2])
            swap(t1, t2), swap(v, u), swap(cu, cv);
        ans += query(id[t1], id[v], 1);
        x = query_node(id[v], 1);//查询当前链的尾端颜色
        if(x == cv) ans--; //如果和当前链的前一段链的首端颜色一样，则-1
        cv = query_node(id[t1], 1);//查询并记录当前链的首端颜色
        v = fat[t1], t1 = top[v];
    }
    if(v == u)
    {
        if(cv == cu) ans--;
        return ans;
    }
    /*此处特别判断，比如u是根节点，v到根节点路径是一段连续的重链*/
    if(dep[v] > dep[u]) swap(v, u), swap(cv, cu);
    ans += query(id[son[v]], id[u], 1);
    //最后查询的这条链和两条链相接，所以都要判断相接处颜色是否相同
    x = query_node(id[u], 1);
    if(x == cu) ans--;
    x = query_node(id[son[v]], 1);
    if(x == cv) ans--;
    return ans;
}  
int main()
{
    char str[100];
    int a, b, c;
    while(~ scanf("%d%d", &n, &m))
    {
        cnt = num = 0;
        memset(head, -1, sizeof head);
        for(int i = 1; i <= n - 1; i++)
        {
            scanf("%d%d%d", &d[i][0], &d[i][1], &d[i][2]);
            add_edge(d[i][0], d[i][1]), add_edge(d[i][1], d[i][0]);
        }
        dfs1(1, 0, 1);
        dfs2(1, 1);
        for(int i = 1; i <= n - 1; i++)
        {//把边关联到深度较深的端点上，更新第i条边时，即更新id[d[i][1]];
            if(dep[d[i][0]] > dep[d[i][1]]) swap(d[i][0], d[i][1]);
            pos[id[d[i][1]]] = d[i][2];
        }
        build(1, num, 1);
        for(int i = 1; i <= m; i++)
        {
            scanf(" %s", str);
            if(str[0] == 'Q')
            {
                scanf("%d%d", &a, &b);
                printf("%d\n", Query(a, b));
            }
            else
            {
                scanf("%d%d%d", &a, &b, &c);
                Update(a, b, c);
            }
        }
    }
    return 0;
}