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NC236183. Dynamic Diameter [CEOI 2019 day 1]

描述


You are given a weighted undirected tree on n vertices and a list of q updates. Each update changes the weight of one edge. The task is to output the diameter of the tree after each update.

(The distance between two vertices is the sum of the weights on the unique simple path that connects them. The diameter is the largest of all those distances.)


输入描述

The first line contains three space-separated integers n, q and w (, ) – the number of vertices in the tree, the number of updates and the limit on the weights of edges. The vertices are numbered 1 through n.

Next, n-1 lines describing the initial tree follow. The i-th of these lines contains three space-separated integers a_i, b_i, c_i (, ) meaning that initially, there is an edge between vertices a_i and b_i with weight c_i. It is guaranteed that these n-1 lines describe a tree.

Finally, q lines describing queries follow. The j-th of these lines contains two space-separated integers d_j, e_j (). These two integers are then transformed according to the following scheme:

    ·
    ·
where last is the result of the last query (initially ). Tuple (d'_j, e'_j) represents a query which takes the -th edge from the input and sets its weight to e'_j.

输出描述

Output q lines. For each i, line i should contain the diameter of the tree after the i-th update.

示例1

输入:

4 3 2000
1 2 100
2 3 1000
2 4 1000
2 1030
1 1020
1 890

输出:

2030
2080
2050

示例2

输入:

10 10 10000
1 9 1241
5 6 1630
10 5 1630
2 6 853
10 1 511
5 3 760
8 3 1076
4 10 1483
7 10 40
8 2051
5 6294
5 4168
7 1861
0 5244
6 5156
3 3001
8 5267
5 3102
8 3623

输出:

6164
7812
8385
6737
6738
7205
6641
7062
6581
5155

说明:

The first sample is depicted in the figure below. The left-most picture shows the initial state of the graph. Each following picture depicts the situation after an update. The weight of the updated edge is painted green, and the diameter is red.



The first query changes the weight of the 3rd edge, i.e. \{2, 4\}, to 1030. The largest distance between any pair of vertices is 2030 – the distance between 3 and 4.

As the answer is 2030, the second query is d'_2 = (1 + 2030) \bmod 3 = 0 e'_2 = (1020 + 2030) \bmod 2000 = 1050 Hence the weight of the edge \{1, 2\} is changed to 1050. This causes the pair \{1, 4\} to be the pair with the greatest distance, namely 2080.

The third query is decoded as d'_3 = (1 + 2080) \bmod 3 = 2 e'_3 = (890 + 2080) \bmod 2000 = 970 As the weight of the edge \{2, 4\} decreases to 970, the most distant pair is suddenly \{1, 3\} with 2050.

原站题解

上次编辑到这里,代码来自缓存 点击恢复默认模板

C++(g++ 7.5.0) 解法, 执行用时: 283ms, 内存消耗: 48596K, 提交时间: 2022-10-11 16:51:27

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

using namespace std ;

using ll = long long ;
using pii = pair<int,int> ;

// 当维护直径的时候,其实可以看作求解两个点之间距离最大值
// 那么就是 dis[u] + dis[v] - 2 * dis[lca(u,v)] ; 
// 对于欧拉序来说,这里 dis[u] + dis[v] - 2 * dis[lca(u,v)] ;
// u,v 两个点的 lca 就是欧拉序列中 dis 最小的一个
// 那么我们就可以使用线段树来进行维护
// 发现,我们维护 线段的 mx,mi,res 
// 然后为了快速求解出 res ,我们还需要维护 
// fmx = dis[u] - 2 * dis[j] (j >= u) gmx = dis[u] - 2 * dis[j] (j <= u) ;
// 然后之后的每次修改一条边,可以看作对当前的子树进行修改

const int N = 3e5 + 100 ;

int n,m ;
int h[N],e[N],ne[N],idx ;
struct Edge{
    int a,b ;
    ll c ;
}edge[N] ;
ll w[N],wmx,dis[N] ;
struct Node{
    int l,r ;
    ll mx,mi,res,fmx,gmx,add ;

    void work(ll tot){
        mx = tot,mi = tot ;
        res = 0 ;
        fmx = -tot,gmx = -tot ;
    }

}tr[N * 4] ;
int ou[N],time_stamp ;
int fi[N],se[N],dep[N] ; // 记录欧拉序第一次出现和第二次出现的位置

void add(int a,int b,ll c){
    e[idx] = b,w[idx] = c,ne[idx] = h[a],h[a] = idx ++ ;
}

void dfs(int u,int fa,int depth,ll distance){  // 用于求解出欧拉序
    ou[++time_stamp] = u ;
    fi[u] = time_stamp ;
    dep[u] = depth ;
    dis[u] = distance ;

    for(int i = h[u] ; ~ i ; i = ne[i]){
        int j = e[i] ;
        if(j == fa) continue ;
        dfs(j,u,depth+1,distance + w[i]) ;
        ou[++time_stamp] = u ;
    }

    se[u] = time_stamp ;
}

void pushup(int u){
    tr[u].mx = max(tr[u<<1].mx,tr[u<<1|1].mx) ;
    tr[u].mi = min(tr[u<<1].mi,tr[u<<1|1].mi) ;
    tr[u].res = max(max(tr[u<<1].res,tr[u<<1|1].res),max(tr[u<<1].fmx+tr[u<<1|1].mx,tr[u<<1|1].gmx+tr[u<<1].mx)) ;
    tr[u].fmx = max(max(tr[u<<1].fmx,tr[u<<1|1].fmx),tr[u<<1].mx - 2 * tr[u<<1|1].mi) ;
    tr[u].gmx = max(max(tr[u<<1].gmx,tr[u<<1|1].gmx),tr[u<<1|1].mx - 2 * tr[u<<1].mi) ;
}

void pushchange(int u,ll add){
    tr[u].mx += add,tr[u].mi += add ;
    tr[u].fmx -= add,tr[u].gmx -= add ;
    tr[u].add += add ;
}

void pushdown(int u){
    if(tr[u].add){
        pushchange(u<<1,tr[u].add) ;
        pushchange(u<<1|1,tr[u].add) ;
        tr[u].add = 0 ;
    }
}

void build(int u,int l,int r){
    tr[u] = {l,r} ;

    if(l == r){
        tr[u].work(dis[ou[l]]) ;
        return ;
    }

    int mid = l + r >> 1 ;
    build(u<<1,l,mid),build(u<<1|1,mid+1,r) ;
    pushup(u) ;
}

void modify(int u,int l,int r,ll add){
    if(l <= tr[u].l &&  tr[u].r <= r){
        pushchange(u,add) ;
        return ;
    }

    pushdown(u) ;
    int mid = tr[u].l + tr[u].r >> 1 ;
    if(l <= mid) modify(u<<1,l,r,add) ;
    if(r >= mid + 1) modify(u<<1|1,l,r,add) ; 
    pushup(u) ;
}

int main(){
    scanf("%d%d%lld",&n,&m,&wmx) ; 
    memset(h,-1,sizeof h) ;
    for(int i = 1 ; i <= n - 1 ; i ++){
        scanf("%d%d%lld",&edge[i].a,&edge[i].b,&edge[i].c) ;
        add(edge[i].a,edge[i].b,edge[i].c),add(edge[i].b,edge[i].a,edge[i].c) ;
    }
    dfs(1,-1,1,0) ;
    // for(int i = 1 ; i <= 2 * n ; i ++) cout << ou[i] << " " ;

    build(1,1,2 * n) ;
    ll last = 0 ;
    while(m--){
        ll d,e ;
        scanf("%lld%lld",&d,&e) ;
        d = (d + last) % (n - 1) + 1 ;
        e = (e + last) % wmx ;
        int u ;
        if(dep[edge[d].a] > dep[edge[d].b]) u = edge[d].a ;
        else u = edge[d].b ;

        // cout << u << " " << fi[u] << " " << se[u] << "\n" ;
        modify(1,fi[u],se[u],e-edge[d].c) ;

        last = tr[1].res ;
        printf("%lld\n",last) ;

        edge[d].c = e ;
    }
    return 0 ;
}

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