New Year Tree CodeForces - 620E 线段树+dfs序+状态压缩

CodeForces 620E 新年树问题解析
本文介绍了一个CodeForces上的竞赛题620E,题目涉及在一棵有根树上进行颜色修改查询和不同颜色数量查询。通过离线线段树实现高效处理更新和查询操作,适用于算法竞赛和技术面试。

New Year Tree

CodeForces - 620E

The New Year holidays are over, but Resha doesn't want to throw away the New Year tree. He invited his best friends Kerim and Gural to help him to redecorate the New Year tree.

The New Year tree is an undirected tree with n vertices and root in the vertex 1.

You should process the queries of the two types:

  1. Change the colours of all vertices in the subtree of the vertex v to the colour c.
  2. Find the number of different colours in the subtree of the vertex v.
Input

The first line contains two integers n, m (1 ≤ n, m ≤ 4·105) — the number of vertices in the tree and the number of the queries.

The second line contains n integers ci (1 ≤ ci ≤ 60) — the colour of the i-th vertex.

Each of the next n - 1 lines contains two integers xj, yj (1 ≤ xj, yj ≤ n) — the vertices of the j-th edge. It is guaranteed that you are given correct undirected tree.

The last m lines contains the description of the queries. Each description starts with the integer tk (1 ≤ tk ≤ 2) — the type of the k-th query. For the queries of the first type then follows two integers vk, ck (1 ≤ vk ≤ n, 1 ≤ ck ≤ 60) — the number of the vertex whose subtree will be recoloured with the colour ck. For the queries of the second type then follows integer vk (1 ≤ vk ≤ n) — the number of the vertex for which subtree you should find the number of different colours.

Output

For each query of the second type print the integer a — the number of different colours in the subtree of the vertex given in the query.

Each of the numbers should be printed on a separate line in order of query appearing in the input.

Example
Input
7 10
1 1 1 1 1 1 1
1 2
1 3
1 4
3 5
3 6
3 7
1 3 2
2 1
1 4 3
2 1
1 2 5
2 1
1 6 4
2 1
2 2
2 3
Output
2
3
4
5
1
2
Input
23 30
1 2 2 6 5 3 2 1 1 1 2 4 5 3 4 4 3 3 3 3 3 4 6
1 2
1 3
1 4
2 5
2 6
3 7
3 8
4 9
4 10
4 11
6 12
6 13
7 14
7 15
7 16
8 17
8 18
10 19
10 20
10 21
11 22
11 23
2 1
2 5
2 6
2 7
2 8
2 9
2 10
2 11
2 4
1 12 1
1 13 1
1 14 1
1 15 1
1 16 1
1 17 1
1 18 1
1 19 1
1 20 1
1 21 1
1 22 1
1 23 1
2 1
2 5
2 6
2 7
2 8
2 9
2 10
2 11
2 4
Output
6
1
3
3
2
1
2
3
5
5
1
2
2
1
1
1
2
3

code:

#include <iostream>
#include <cstdio>
#include <cstring>
#include <vector>
using namespace std;

typedef long long ll;
const int maxn = 400010;
ll c[maxn];//记录颜色
vector<int>v[maxn];
struct node{
    ll num,lazy;
}tree[maxn<<2];
struct Node{
    int l,r;
}E[maxn];//每个点能管的范围,既包括自己的所以子树节点
ll val[maxn];
int time;
bool vis[maxn];
void dfs(int x){
    E[x].l = ++time;
    val[time] = c[x];
    vis[x] = true;
    for(int i = 0; i < v[x].size(); i++){
        int y = v[x][i];
        if(!vis[y])
            dfs(y);
    }
    E[x].r = time;
}
void pushup(int rt){
    tree[rt].num = tree[rt<<1].num | tree[rt<<1|1].num;
}
void pushdown(int rt){
    if(tree[rt].lazy){
        tree[rt<<1].num = tree[rt<<1|1].num = 1ll<<tree[rt].lazy;
        tree[rt<<1].lazy = tree[rt<<1|1].lazy = tree[rt].lazy;
        tree[rt].lazy = 0;
    }
}
void build(int rt,int l,int r){
    tree[rt].lazy = 0;
    if(l == r){
        tree[rt].num = 1ll<<val[l];
        return;
    }
    int mid = (l + r) >> 1;
    build(rt<<1,l,mid);
    build(rt<<1|1,mid+1,r);
    pushup(rt);
}
void update(int rt,int L,int R,int l,int r,ll co){
    if(l <= L && r >= R){
        tree[rt].lazy = co;
        tree[rt].num = 1ll<<co;
        return;
    }
    int mid = (L + R) >> 1;
    pushdown(rt);
    if(l <= mid) update(rt<<1,L,mid,l,r,co);
    if(r > mid) update(rt<<1|1,mid+1,R,l,r,co);
    pushup(rt);
}
ll ans;
void solve(int rt,int L,int R,int l,int r){
    if(l <= L && r >= R){
        ans |= tree[rt].num;
        return;
    }
    pushdown(rt);
    int mid = (L + R) >> 1;
    if(l <= mid) solve(rt<<1,L,mid,l,r);
    if(r > mid) solve(rt<<1|1,mid+1,R,l,r);
}
int main(){
    int n,m;
    scanf("%d%d",&n,&m);
    for(int i = 1; i <= n; i++)
        scanf("%lld",&c[i]);
    for(int i = 1; i < n; i++){
        int x,y;
        scanf("%d%d",&x,&y);
        v[x].push_back(y);
        v[y].push_back(x);
    }
    memset(vis,false,sizeof(vis));
    time = 0;
    dfs(1);
    build(1,1,n);
    for(int i = 1; i <= m; i++){
        int d,x;
        ll co;
        scanf("%d%d",&d,&x);
        if(d == 1){
            scanf("%lld",&co);
            update(1,1,n,E[x].l,E[x].r,co);
        }
        else{
            ans = 0;
            solve(1,1,n,E[x].l,E[x].r);
            int sum = 0;
            while(ans){
                if(ans&1) sum++;
                ans >>= 1;
            }
            printf("%d\n",sum);
        }
    }
    return 0;

}

引用\[1\]中提到了一种树形动态规划的方法来解决CodeForces - 982C问题。在这个问题中,subtree指的是子连通块,而不是子树。为了使cnt_white - cnt_black尽可能大,可以使用两次树形动态规划来求解。第一次是自底向上的过程,维护一个dp数组,表示以每个节点为根的子树中的最大连通块。第二次是自顶向下的过程,处理自底向上过程中无法包含的树链所代表的子树。在第二次遍历中,需要维护一个sum变量,用于存储树链所代表的子树的贡献。根据ans\[u\]的正负,决定是否能对相邻的子节点做出贡献。如果ans\[u\]为正,则减去dp\[v\]就是树链所代表的子树的权值。最终,ans\[u\]代表包含节点u在内的子连通块的最大权值。\[1\] 问题: CodeForces - 982C 树形DP是什么问题?如何解决? 回答: CodeForces - 982C是一个树形动态规划问题。在这个问题中,需要求解子连通块的最大权值和,使得cnt_white - cnt_black尽可能大。解决这个问题的方法是使用两次树形动态规划。第一次是自底向上的过程,维护一个dp数组,表示以每个节点为根的子树中的最大连通块。第二次是自顶向下的过程,处理自底向上过程中无法包含的树链所代表的子树。在第二次遍历中,需要维护一个sum变量,用于存储树链所代表的子树的贡献。根据ans\[u\]的正负,决定是否能对相邻的子节点做出贡献。最终,ans\[u\]代表包含节点u在内的子连通块的最大权值。\[1\] #### 引用[.reference_title] - *1* *2* [CodeForces - 1324F Maximum White Subtree(树形dp)](https://blog.csdn.net/qq_45458915/article/details/104831678)[target="_blank" data-report-click={"spm":"1018.2226.3001.9630","extra":{"utm_source":"vip_chatgpt_common_search_pc_result","utm_medium":"distribute.pc_search_result.none-task-cask-2~all~insert_cask~default-1-null.142^v91^koosearch_v1,239^v3^insert_chatgpt"}} ] [.reference_item] [ .reference_list ]
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