Problem:
Score : 300 points

Problem Statement

Snuke has an integer sequence A of length N.

He will freely choose an integer b.
Here, he will get sad if A_i and b+i are far from each other.
More specifically, the sadness of Snuke is calculated as follows:

abs(A_1 - (b+1)) + abs(A_2 - (b+2)) + ... + abs(A_N - (b+N))

Here, abs(x) is a function that returns the absolute value of x.

Find the minimum possible sadness of Snuke.

Constraints

1 \leq N \leq 2 \times 10^5

1 \leq A_i \leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 A_2 ... A_N

Output

Print the minimum possible sadness of Snuke.

Sample Input 1

5
2 2 3 5 5

Sample Output 1

2

If we choose b=0, the sadness of Snuke would be abs(2-(0+1))+abs(2-(0+2))+abs(3-(0+3))+abs(5-(0+4))+abs(5-(0+5))=2.
Any choice of b does not make the sadness of Snuke less than 2, so the answer is 2.

Sample Input 2

9
1 2 3 4 5 6 7 8 9

Sample Output 2

0

Sample Input 3

6
6 5 4 3 2 1

Sample Output 3

18

Sample Input 4

7
1 1 1 1 2 3 4

Sample Output 4

6

Solution in C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <iostream>
#include <cstdio>
#include <vector>
#include <algorithm>
#include <string>
#include <cstring>
#define mp make_pair
#define pb push_back
using namespace std;
const int N = 2e5 + 5;
const long long INF = 1e18;

int n;
long long a[N];

vector <long long> b;
long long ans = INF;

long long prefix[N];

long long labs(long long x)
{
    if(x < 0)
        return -x;
    return x;
}

long long check_negative(long long x)
{
    int low, high, mid;
    int first_nonneg = n + 1, first_positive = n + 1;

    low = 1;
    high = n;
    while(low <= high)
    {
        mid = (low + high) / 2;
        if(a[mid] >= 0)
        {
            first_nonneg = mid;
            high = mid - 1;
        }
        else
        {
            low = mid + 1;
        }
    }

    low = 1;
    high = n;
    while(low <= high)
    {
        mid = (low + high) / 2;
        if(a[mid] >= x)
        {
            first_positive = mid;
            high = mid - 1;
        }
        else
        {
            low = mid + 1;
        }
    }

    long long res = 0;

    res = prefix[n] - prefix[first_nonneg - 1] - 1LL * (n - first_nonneg + 1) * x;
    res += labs( prefix[first_nonneg - 1] - prefix[first_positive - 1] + 1LL * (first_nonneg - 1 - (first_positive) + 1) * x);
    res += prefix[first_positive - 1] + 1LL * (first_positive - 1) * x;

   // cerr << x << " " << res << " " << first_nonneg << " " << first_positive << endl;

    return res;
}

long long check_positive(long long x)
{
    int low, high, mid;
    int first_nonneg = 0, first_positive = n + 1;

    low = 1;
    high = n;
    while(low <= high)
    {
        mid = (low + high) / 2;
        if(a[mid] >= 0)
        {
            first_nonneg = mid;
            high = mid - 1;
        }
        else
        {
            low = mid + 1;
        }
    }

    low = 1;
    high = n;
    while(low <= high)
    {
        mid = (low + high) / 2;
        if(a[mid] >= x)
        {
            first_positive = mid;
            high = mid - 1;
        }
        else
        {
            low = mid + 1;
        }
    }

    long long res = 0;

    res = prefix[first_nonneg - 1] + 1LL * (first_nonneg - 1) * x;
    res += labs ( (prefix[first_positive - 1] - prefix[first_nonneg - 1]) - 1LL * (first_positive - first_nonneg) );
    if(first_positive < n + 1)
        res += (prefix[n] - prefix[first_positive - 1]) - 1LL * (n - first_positive + 1) * x;

  //  cerr << x << " " << res << endl;

    return res;
}

int main()
{
    ios_base::sync_with_stdio(false);
    cin.tie(nullptr); cout.tie(nullptr);

    int i;

    cin >> n;
    for(i = 1; i <= n; i++)
        cin >> a[i];

    for(i = 1; i <= n; i++)
    {
        b.pb(a[i] - i);
        a[i] -= i;
    }

    sort(a + 1, a + n + 1);
    sort(b.begin(), b.end());

    for(i = 1; i <= n; i++)
        prefix[i] = prefix[i - 1] + labs(a[i]);
    prefix[n + 1] = prefix[n];

    for(i = 1; i <= n; i++)
        cerr << a[i] << " ";
    cerr << endl;

    for(i = 0; i < b.size(); i++)
    {
        if(b[i] >= 0)
          ans = min(ans, check_positive(b[i]));
        else
          ans = min(ans, check_negative(b[i]));
    }

    cout << ans << endl;

    return 0;
}