HDU 6065 RXD, tree and sequence （LCA DP）

Posted 2020-10-01 贱人方

RXD, tree and sequence

Time Limit: 6000/3000 MS (Java/Others)    Memory Limit: 524288/524288 K (Java/Others)
Total Submission(s): 234    Accepted Submission(s): 82

Problem Description

RXD has a rooted tree T with size n, the root ID is 1, with the depth of 1
RXD has a permutation P with size n.
RXD wants to divide the permutaion into k continuous parts 
For each part, he would calculate the depth of the least common ancestor of the part. And finally accumulate them. 
He wants to make the final result minimized. 
Please calculate the minimal answer.
1≤k≤n≤3×105,n×k≤3×105

Input

There are several test cases, please keep reading until EOF.
For each test case, the first line consists of 2 integer n,k, which means the number of the tree nodes and the size of the permutaion, and k means the number of parts.
The next line consists of n different integers, which means the permutation P.
The next n−1 lines consists of 2 integers, a,b, means a tree edge.
It is guaranteed that the edges would form a tree.
There are 6 test cases.

Output

For each test case, output an integer, which means the answer.

Sample Input

6 3 4 6 2 5 1 3 1 2 2 3 3 4 4 5 4 6

Sample Output

6

Source

2017 Multi-University Training Contest - Team 3

 【题意】给你一棵树，然后给你树的节点编号的一种排列，然后要你将这个序列分成非空的k份，对于每一份求lca的深度，然后累加，   求和最小。

 【分析】首先得想到区间lca的一个性质，对于区间[i,j]，他们的lca肯定是lca[i,i+1],lca[i+1,i+2],...lca[j-1,j]中的某一个，也就是说，对于某一个区间，决定lca的只有某相邻的两个数。然后在这个序列中，对于某一个数，他有三种存在状态。第一，它存在于某个区间，但是这个区间的lca不是由它决定的。第二，它一个数本身作为一份。第三，它和它左边的数的lca作为某一份的lca。然后dp[i][j]表示前i个数分成j段的最小答案，对于上面三种情况分别对应三个方程

dp[i][j]=min(dp[i][j],dp[i-1][j]);
dp[i][j]=min(dp[i][j],dp[i-1][j-1]+dep[a[i]]);
dp[i][j]=min(dp[i][j],dp[i-2][j-1]+lca[i]);

#include <bits/stdc++.h>
#define inf 0x3f3f3f3f
#define met(a,b) memset(a,b,sizeof a)
#define pb push_back
#define mp make_pair
#define inf 0x3f3f3f3f
using namespace std;
typedef long long ll;
const int N = 5e5+50;
const int M = 16000009;
const int mod = 1e9+7;
const double pi= acos(-1.0);
typedef pair<int,int>pii;
int n,k,ans;
int dep[N],fa[N][20];
int a[N],lca[N];
vector<int>edg[N];
void dfs(int u,int f){
    fa[u][0]=f;
    for(int i=1;i<20;i++){
        fa[u][i]=fa[fa[u][i-1]][i-1];
    }
    for(int v : edg[u]){
        if(v==f)continue;
        dep[v]=dep[u]+1;
        dfs(v,u);
    }
}
int LCA(int u,int v){
    int U=u,V=v;
    if(dep[u]<dep[v])swap(u,v);
    for(int i=19;i>=0;i--){
        if(dep[fa[u][i]]>=dep[v]){
            u=fa[u][i];
        }
    }
    if(u==v)return (u);
    for(int i=19;i>=0;i--){
        if(fa[u][i]!=fa[v][i]){
            u=fa[u][i];v=fa[v][i];
        }
    }
    return (fa[u][0]);
}

int main(){
    while(~scanf("%d%d",&n,&k)){
        int dp[n+50][k+50];
        met(dp,inf);
        for(int i=1;i<=n;i++)scanf("%d",&a[i]),edg[i].clear();
        for (int i=1,u,v;i<n;i++){
            scanf("%d%d",&u,&v);
            edg[u].pb(v);
            edg[v].pb(u);
        }
        dep[1]=1;
        dfs(1,0);
        dp[0][0]=0;
        for(int i=2;i<=n;i++){
            int m = LCA(a[i-1],a[i]);
            lca[i]=dep[m];
        }
        for(int i=1;i<=n;i++){
            for(int j=0;j<=min(i,k);j++){
                if(i-1>=j)dp[i][j]=min(dp[i][j],dp[i-1][j]);
                if(j>0)dp[i][j]=min(dp[i][j],dp[i-1][j-1]+dep[a[i]]);
                if(j>0&&i>=2&&j-1<=i-2)dp[i][j]=min(dp[i][j],dp[i-2][j-1]+lca[i]);
            }
        }
        printf("%d\n",dp[n][k]);
    }
    return 0;
}