#include "books.h"
#include<bits/stdc++.h>
using namespace std;
const long long inf = (long long) 1e18 + 10;
const int inf1 = (int) 1e9 + 10;
#define int long long
#define dbl long double
#define endl '\n'
#define sc second
#define fr first
#define mp make_pair
#define pb push_back
#define all(x) x.begin(), x.end()
const int maxn = 1e6+10;
int n;
int ds[maxn], dsz[maxn], dsl[maxn], dsr[maxn];
int nxr[maxn], costr[maxn];
int find(int v) {
if(v == ds[v]) return v;
return ds[v] = find(ds[v]);
}
void join(int u, int v) {
u = find(u);
v = find(v);
if(u == v) return;
if(dsz[u] < dsz[v]) swap(u,v);
dsz[u]+= dsz[v];
dsl[u] = min(dsl[u],dsl[v]);
dsr[u] = max(dsr[u],dsr[v]);
ds[v] = u;
}
int L,R;
map<int,int> dp[maxn];
int sol(int l, int r) {
if(dp[l].count(r)) return dp[l][r];
if(l == L && r == R) return dp[l][r] = 0;
dp[l][r] = inf;
if(l > L) dp[l][r] = min(dp[l][r],2+sol(dsl[find(l-1)],max(r,dsr[find(l-1)])));
if(r < R) dp[l][r] = min(dp[l][r],2*costr[find(r)]+sol(min(l,dsl[find(nxr[find(r)])]),dsr[find(nxr[find(r)])]));
return dp[l][r];
}
long long minimum_walk(vector<int32_t> p, int32_t s) {
// return 0;
n = p.size();
for(int i = 0; i < n; i++) {
ds[i] = i;
dsz[i] = 1;
dsl[i] = i;
dsr[i] = i;
}
for(int i = 0; i < n; i++) {
join(i,p[i]);
}
int itl = s;
while(itl != n) {
int itr = itl;
int cur = itl;
while(cur <= itr) {
itr = max(itr,dsr[find(cur)]);
cur++;
}
for(int i = itl; i < itr; i++) {
join(i,i+1);
}
itl = itr+1;
}
int itr = s;
while(itr >= 0) {
int itl = itr;
int cur = itr;
while(cur >= itl) {
itl = min(itl,dsl[find(cur)]);
cur--;
}
for(int i = itl; i < itr; i++) {
join(i,i+1);
}
itr = itl-1;
}
L = 0;
R = n-1;
while(L < s && dsz[find(L)] == 1) L++;
while(R > s && dsz[find(R)] == 1) R--;
nxr[find(R)] = R;
costr[find(R)] = 0;
for(int i = R-1; i >= s; i--) {
if(find(i) == find(i+1)) {
continue;
}
else if(true or dsl[find(i+1)] < s or dsr[find(i+1)] == R) {
nxr[find(i)] = i+1;
costr[find(i)] = 1;
}
else {
nxr[find(i)] = nxr[find(i+1)];
costr[find(i)] = costr[find(i+1)]+1;
}
}
int ans = 0;
for(int i = 0; i < n; i++) ans+= abs(p[i]-i);
return ans+sol(dsl[find(s)],dsr[find(s)]);
}
