//In the name of God
#include <bits/stdc++.h>
#include "books.h"
using namespace std;
typedef long long ll;
typedef pair<ll, ll> pll;
const ll maxn = 1e6 + 100;
const ll mod = 1e9 + 7;
const ll inf = 1e18;
#define pb push_back
#define Mp make_pair
#define F first
#define S second
#define Sz(x) ll((x).size())
#define all(x) (x).begin(), (x).end()
ll a[maxn], t, tt, c[maxn], par[maxn];
ll mn[maxn], mx[maxn], mnp[maxn], mxp[maxn];
bool vis[maxn];
vector<ll> vec, g[maxn], gp[maxn], gg[maxn];
vector<pll> v2;
void dfs(ll v){
vis[v] = 1;
for(ll u : g[v]){
if(!vis[u]) dfs(u);
}
vec.pb(v);
}
void sfd(ll v){
vis[v] = 1;
c[v] = tt;
mnp[tt] = min(mnp[tt], mn[v]);
mxp[tt] = max(mxp[tt], mx[v]);
for(ll u : gp[v]){
if(!vis[u]) sfd(u);
}
}
ll solve(ll v, ll s){
ll x = 0;
ll ans = 0;
ll i = 0;
for(ll u : gg[v]){
if(mnp[u] <= s && s <= mxp[u]) x = i;
ans += solve(u, s);
i++;
}
if(mnp[v] == -1){
return ans + Sz(gg[v]) - 1;
}
ans += max(0ll, min(x, Sz(gg[v]) - x - 1));
return ans;
}
long long minimum_walk(vector<int> p, int s){
ll n = Sz(p);
ll l = 0;
while(l < n && p[l] == l) l++;
ll r = n;
while(r && p[r - 1] == r - 1) r--;
l = min(l, (ll)s);
r = max(r, s + 1ll);
mn[0] = -1;
mx[0] = n + 1;
t++;
for(ll i = l; i < r; i++){
if(p[i] > i){
a[i]++;
a[p[i]]--;
}
if(!vis[i]){
ll v = i;
mn[t] = inf;
mx[t] = -inf;
do{
vis[v] = 1;
mx[t] = max(mx[t], v);
mn[t] = min(mn[t], v);
v = p[v];
}while(v != i);
t++;
}
}
ll ans = 0;
for(ll i = 1; i < n; i++){
ans += a[i - 1];
a[i] += a[i - 1];
}
for(ll i = 0; i < s; i++){
g[c[i]].pb(c[i + 1]);
gp[c[i + 1]].pb(c[i]);
}
for(ll i = s; i + 1 < n; i++){
g[c[i + 1]].pb(c[i]);
gp[c[i]].pb(c[i + 1]);
}
// SCC
fill(vis, vis + t, 0);
for(ll i = 0; i < t; i++){
if(!vis[i]) dfs(i);
}
reverse(all(vec));
fill(vis, vis + t, 0);
for(ll i = 0; i < t; i++){
if(!vis[vec[i]]){
mnp[tt] = inf;
mxp[tt] = -inf;
sfd(vec[i]);
tt++;
}
}
vec.clear();
ll root = -1;
for(ll i = 0; i < tt; i++){
if(mnp[i] == -1) root = i;
v2.pb(Mp(mnp[i], i));
}
sort(all(v2));
for(pll i : v2){
if(vec.empty()){
vec.pb(i.S);
continue;
}
while(mxp[vec.back()] < i.F) vec.pop_back();
gg[vec.back()].pb(i.S);
vec.pb(i.S);
}
ans += solve(root, s);
return ans * 2;
}
