# [jloi2016 / show2016] reconnaissance guard (tree DP)

Posted by Canabalooza on Thu, 06 Jan 2022 04:43:05 +0100

Considering the tree DP, suppose we have considered the placement of listening points in the current subtree, and the root is u u u. Consider what status we want to record: u u u how far can the monitoring point in the subtree be monitored outside the subtree, u u How far is the UN monitored point farthest from the root in the u subtree.

It is found that when the second state exists, the first state is useless because if u u There is an un monitored point in the u subtree v v v. Set it to u u The distance of u is d d d. Then:

• u u u the farthest listening distance from the listening point in the subtree to the outside of the subtree will not exceed d d d. Otherwise v v v can be monitored.
• v v v must finally be u u A point outside the u subtree x x x monitor, then all distances u u u less than or equal to d d All points of d can be x x I heard it.

That is, when v v When v exists, we don't need to consider it u u Listening points in u subtree u u For the monitoring outside the u subtree, the first state is useless.

Then this simplifies the DP state: let's set f u , i f_{u,i} fu,i ， means considered u u The placement of monitoring points in u subtree, u u u subtree inner separation u u u the distance of the farthest point not monitored is i i i. Minimum cost required. set up g u , i g_{u,i} gu,i ， indicates that consideration has been completed u u The placement of monitoring points in u subtree, u u There are no un monitored points in the u subtree, and u u The farthest listening distance from the listening point in the u subtree to the outside of the subtree i i i. Minimum cost required.

Merge on transfer u u u currently considered subtrees and enumerated sons v v Subtree of v, min f u , f v f_u,f_v fu​,fv​, f u , g v f_u,g_v fu​,gv​, g u , f v g_u,f_v gu​,fv​, g u , g v g_u,g_v gu, gv# four cases are combined:
f u , i + f v , j → f u , max ⁡ ( i , j + 1 ) f u , i + g v , j → { g u , j − 1 if  j − 1 ≥ i f u , i if  j − 1 < i g u , i + f v , j → { g u , i if  i ≥ j + 1 f u , j + 1 if  i < j + 1 g u , i + g v , j → g u , max ⁡ ( i , j − 1 ) \begin{aligned} f_{u,i}+f_{v,j}&\to f_{u,\max(i,j+1)}\\ f_{u,i}+g_{v,j}&\to \begin{cases}g_{u,j-1}&\text{if }j-1\geq i\\f_{u,i}&\text{if }j-1<i\end{cases}\\ g_{u,i}+f_{v,j}&\to \begin{cases}g_{u,i}&\text{if }i\geq j+1\\f_{u,j+1}&\text{if }i<j+1\end{cases}\\ g_{u,i}+g_{v,j}&\to g_{u,\max(i,j-1)} \end{aligned} fu,i​+fv,j​fu,i​+gv,j​gu,i​+fv,j​gu,i​+gv,j​​→fu,max(i,j+1)​→{gu,j−1​fu,i​​if j−1≥iif j−1<i​→{gu,i​fu,j+1​​if i≥j+1if i<j+1​→gu,max(i,j−1)​​
This can be done using prefixes and / or suffixes and optimizations O ( n D ) O(nD) O(nD).

Initially, each point forms a separate subtree, which is set for all points g u , D ← w u g_{u,D}\gets w_u gu,D ← wu, set the point to be monitored f u , 0 = 0 f_{u,0}=0 fu,0 = 0, set for points that do not need to be monitored g u , 0 = 0 g_{u,0}=0 gu,0​=0.

#include<bits/stdc++.h>

#define D 25
#define N 500010
#define INF 0x7fffffff

using namespace std;

inline void upmin(int &x,int y){if(y<x) x=y;}

{
int x=0,f=1;
char ch=getchar();
while(ch<'0'||ch>'9')
{
if(ch=='-') f=-1;
ch=getchar();
}
while(ch>='0'&&ch<='9')
{
x=(x<<1)+(x<<3)+(ch^'0');
ch=getchar();
}
return x*f;
}

int n,d,w[N];
int f[N][D],g[N][D],pref[N][D],preg[N][D];

{
to[++cnt]=v;
}

inline void premin(int *f,int *pre)
{
for(int i=0;i<=d;i++)
pre[i]=min(i?pre[i-1]:INF,f[i]);
}

void dfs(int u,int fa)
{
static int ff[D],gg[D];
premin(f[u],pref[u]),premin(g[u],preg[u]);
{
int v=to[i];
if(v==fa) continue;
dfs(v,u);
memset(ff,0x3f,sizeof(ff));
memset(gg,0x3f,sizeof(gg));
for(int i=1;i<=d;i++)
upmin(ff[i],f[u][i]+pref[v][i-1]);
for(int i=0;i<d;i++)
upmin(ff[i+1],pref[u][i+1]+f[v][i]);
for(int i=1;i<=d;i++)
upmin(gg[i-1],pref[u][i-1]+g[v][i]);
for(int i=0;i<=d;i++)
upmin(ff[i],f[u][i]+preg[v][i]);
for(int i=1;i<=d;i++)
upmin(gg[i],g[u][i]+pref[v][i-1]);
for(int i=0;i<d;i++)
upmin(ff[i+1],preg[u][i]+f[v][i]);
for(int i=0;i<=d;i++)
upmin(gg[i],g[u][i]+preg[v][min(i+1,d)]);
for(int i=1;i<=d;i++)
upmin(gg[i-1],preg[u][i-1]+g[v][i]);
memcpy(f[u],ff,sizeof(f[u]));
memcpy(g[u],gg,sizeof(g[u]));
premin(f[u],pref[u]),premin(g[u],preg[u]);
}
}

int main()
{
memset(f,0x3f,sizeof(f));
memset(g,0x3f,sizeof(g));
for(int i=1;i<=n;i++) if(f[i][0]) g[i][0]=0;
for(int i=1;i<n;i++)
{