#include<bits/stdc++.h> typedef long long ll; #define pb push_back #define fr first #define sc second #define endl '\n' using namespace std; #define mid ((left+right)>>1) int s=317; int n,k; int arr[100023]; vector<int>renk[100023],deps[100023]; vector<int>child[100023]; int in[100023],out[100023]; int dep[100023],h[100023]; int tim=0; pair<int,int>res[100023]; pair<int,int>ans; void dfs(int pos){ in[pos]=++tim; deps[dep[pos]].pb(pos); for(int x:child[pos]){ dep[x]=dep[pos]+1; dfs(x); h[pos]=max(h[pos],h[x]+1); } out[pos]=tim; } int main(){ ios_base::sync_with_stdio(23^23);cin.tie(NULL); cin>>n>>k; ans={-1,0}; for(int i=1;i<=n;i++){ res[i]={-1,0}; cin>>arr[i]; arr[i]++; renk[arr[i]].pb(i); } for(int i=2;i<=n;i++){ int x;cin>>x;x++; child[x].pb(i); } dfs(1); for(int o=1;o<=k;o++){ if(!renk[o].size())continue; if(renk[o].size()>s){ sort(renk[o].begin(),renk[o].end(),[&](const int &x,const int &y){ return in[x]<in[y]; }); vector<int>roots; vector<int>pot; int mxout=0; for(int x:renk[o]){ if(in[x]>mxout){ pot.pb(x); } mxout=max(mxout,out[x]); } sort(pot.begin(),pot.end(),[&](const int &x,const int &y){ return dep[x]+h[x]<dep[y]+h[y]; }); for(int i=n-1;i>0;i--){ int cnt=0; vector<int>v; for(int x:deps[i]){ if(arr[x]==o){ cnt++; v.pb(x); } } while(pot.size()&&dep[pot.back()]+h[pot.back()]>=i){ roots.pb(pot.back()); pot.pop_back(); } vector<int>roots2; for(int j=0;j<roots.size();j++){ int x=roots[j]; if(dep[x]<i)roots2.pb(x); } swap(roots,roots2); for(int j=0;j<roots.size();j++){ int var=0; int top=0; int l2=0,r2=deps[i].size(); while(l2<r2){ int mi=(l2+r2)/2; int x=deps[i][mi]; if(in[x]>=in[roots[j]])r2=mi; else l2=mi+1; } top-=l2; l2=0;r2=deps[i].size(); while(l2<r2){ int mi=(l2+r2)/2; int x=deps[i][mi]; if(in[x]>out[roots[j]])r2=mi; else l2=mi+1; } top+=l2; l2=0;r2=v.size(); while(l2<r2){ int mi=(l2+r2)/2; int x=v[mi]; if(in[x]>=in[roots[j]])r2=mi; else l2=mi+1; } var-=l2; l2=0;r2=v.size(); while(l2<r2){ int mi=(l2+r2)/2; int x=v[mi]; if(in[x]>out[roots[j]])r2=mi; else l2=mi+1; } var+=l2; res[roots[j]].fr-=min(top,cnt); res[roots[j]].sc+=max(0,min(top-var,cnt-var)); ans=min(ans,res[roots[j]]); } } } else{ sort(renk[o].begin(),renk[o].end(),[&](const int &x,const int &y){ return dep[x]<dep[y]; }); vector<int>v; for(int i=0;i<renk[o].size();i++){ for(int j=i+1;j<renk[o].size();j++){ if(dep[renk[o][j]]<=dep[renk[o][i]])continue; v.pb(renk[o][j]); if(j+1==renk[o].size()||dep[renk[o][j+1]]!=dep[renk[o][j]]){ int var=0,top=0; int cnt=v.size(); for(int x:v){ if(in[x]>=in[renk[o][i]]&&in[x]<=out[renk[o][i]])var++; } int l=0,r=deps[dep[v.back()]].size(); while(l<r){ int mi=(l+r)/2; int x=deps[dep[v.back()]][mi]; if(in[x]>=in[renk[o][i]])r=mi; else l=mi+1; } top-=l; l=0;r=deps[dep[v.back()]].size(); while(l<r){ int mi=(l+r)/2; int x=deps[dep[v.back()]][mi]; if(in[x]>out[renk[o][i]])r=mi; else l=mi+1; } top+=l; res[renk[o][i]].fr-=min(top,cnt); res[renk[o][i]].sc+=max(0,min(top-var,cnt-var)); v.clear(); } } //cout<<renk[o][i]<<": "<<-res[renk[o][i]].fr<<" "<<res[renk[o][i]].sc<<endl; ans=min(ans,res[renk[o][i]]); } } } cout<<-ans.fr<<" "<<ans.sc<<endl; }