Problem:
Score : 700 points

Problem Statement

Tonight, in your favourite cinema they are giving the movie Joker and all seats are occupied. In the cinema there are N rows with N seats each, forming an N\times N square. We denote with 1, 2,\dots, N the viewers in the first row (from left to right); with N+1, \dots, 2N the viewers in the second row (from left to right); and so on until the last row, whose viewers are denoted by N^2-N+1,\dots, N^2.

At the end of the movie, the viewers go out of the cinema in a certain order: the i-th viewer leaving her seat is the one denoted by the number P_i. The viewer P_{i+1} waits until viewer P_i has left the cinema before leaving her seat.
To exit from the cinema, a viewer must move from seat to seat until she exits the square of seats (any side of the square is a valid exit). A viewer can move from a seat to one of its 4 adjacent seats (same row or same column).
While leaving the cinema, it might be that a certain viewer x goes through a seat currently occupied by viewer y; in that case viewer y will hate viewer x forever. Each viewer chooses the way that minimizes the number of viewers that will hate her forever.

Compute the number of pairs of viewers (x, y) such that y will hate x forever.

Constraints

2 \le N \le 500

The sequence P_1, P_2, \dots, P_{N^2} is a permutation of \{1, 2, \dots, N^2\}.

Input

The input is given from Standard Input in the format

N
P_1 P_2 \cdots P_{N^2}

Output

If ans is the number of pairs of viewers described in the statement, you should print on Standard Output

ans

Sample Input 1

3
1 3 7 9 5 4 8 6 2

Sample Output 1

1

Before the end of the movie, the viewers are arranged in the cinema as follows:

1 2 3
4 5 6
7 8 9

The first four viewers leaving the cinema (1, 3, 7, 9) can leave the cinema without going through any seat, so they will not be hated by anybody.

Then, viewer 5 must go through one of the seats where viewers 2, 4, 6, 8 are currently seated while leaving the cinema; hence he will be hated by at least one of those viewers.

Finally the remaining viewers can leave the cinema (in the order 4, 8, 6, 2) without going through any occupied seat (actually, they can leave the cinema without going through any seat at all).

Sample Input 2

4
6 7 1 4 13 16 10 9 5 11 12 14 15 2 3 8

Sample Output 2

3

Sample Input 3

6
11 21 35 22 7 36 27 34 8 20 15 13 16 1 24 3 2 17 26 9 18 32 31 23 19 14 4 25 10 29 28 33 12 6 5 30

Sample Output 3

11

Solution in C++ (GCC 9.2.1) (judge verdict: Wrong Answer)

#include <bits/stdc++.h>

using namespace std;
typedef long long ll;

int main()
{
  ll n;
  cin >> n;
  vector<ll> ps;
  ll n_pow = n * n;

  for (auto i = 0; i < n_pow; i++)
  {
    ll p;
    cin >> p;
    ps.push_back(p);
  }

  // 初期化
  vector<vector<ll>> mat(n + 2, vector<ll>(n + 2, 0));
  ll mat_len = n + 2;
  for (ll x = 0; x < mat_len; x++)
  {
    for (ll y = 0; y < mat_len; y++)
    {
      mat[x][y] = min(min(x - 1, mat_len - x - 2), min(y - 1, mat_len - y - 2));
    }
  }
  unordered_set<ll> called;

  auto print_mat = [&mat, mat_len]() {
    for (ll x = 0; x < mat_len; x++)
    {
      for (ll y = 0; y < mat_len; y++)
      {
        printf("%2lld", mat[y][x]);
      }
      cout << endl;
    }
  };
  // print_mat();

  auto update_mat = [&mat, mat_len, &called, n](ll mat_x, ll mat_y) {
    queue<pair<ll, ll>> q;
    set<pair<ll, ll>> visited;
    int patterns[4][2] = {{1, 0}, {-1, 0}, {0, 1}, {0, -1}};

    // 抜けたやつの周囲の最小値がそこにうまる.
    ll curmin = 100000000;
    for (auto i = 0; i < 4; i++)
    {
      //最小値計算
      auto pattern = patterns[i];
      curmin = min(curmin, mat[mat_x + pattern[0]][mat_y + pattern[1]]);

      //周辺は最初の更新対象になる
      auto pp = make_pair(mat_x + pattern[0], mat_y + pattern[1]);
      if (pp.first >= 1 && pp.first <= mat_len - 1 && pp.second >= 1 && pp.second <= mat_len - 1)
      {
        q.push(pp);
        visited.insert(pp);
      }
      else
      {
        // printf("skipped x: %lld, y: %lld\n", pp.first, pp.second);
      }
    }

    mat[mat_x][mat_y] = curmin;
    auto p = make_pair(mat_x, mat_y);
    visited.insert(p);

    // 今回抜けたものの周囲を更新する
    while (!q.empty())
    {
      auto size = q.size();

      for (auto i = 0; i < size; i++)
      {
        auto p = q.front();
        q.pop();

        if (p.first == 0 || p.first == mat_len - 1 || p.second == 0 || p.second == mat_len - 1)
        {
          continue;
        }

        // 最初のやつがどのポジションにいるのかを計算しないといけない
        // 上下左右の最小値+1をすればいい。仮に自分自身が更新されていれば、その周囲のやつらもqueueに積む
        ll cur_min = 100000000;
        for (auto j = 0; j < 4; j++)
        {
          auto pattern = patterns[j];
          if ((p.first + pattern[0]) >= 0 && (p.first + pattern[0]) <= mat_len - 1 && (p.second + pattern[1]) >= 0 && (p.second + pattern[1]) <= mat_len - 1)
          {
            cur_min = min(cur_min, mat[p.first + pattern[0]][p.second + pattern[1]]);
          }
        }

        ll call_number = p.first + n * (p.second - 1);
        bool has_called = called.count(call_number) > 0;
        ll updated = has_called ? cur_min : cur_min + 1;

        // すでに呼ばれているならば、なんであれ更新する
        if (has_called)
        {
          mat[p.first][p.second] = updated;
          for (auto j = 0; j < 4; j++)
          {
            auto pattern = patterns[j];
            auto pp = make_pair(p.first + pattern[0], p.second + pattern[1]);

            // 更新の対象にするのは外側のやつだけ
            if (visited.count(pp) == 0 && pp.first >= 1 && pp.first < mat_len - 1 && pp.second >= 1 && pp.second < mat_len - 1)
            {
              q.push(pp);
              visited.insert(pp);
            }
          }
        }
        else // 呼ばれていないならば、自分に変化があったときだけ更新する
        {
          if (mat[p.first][p.second] != updated)
          {
            mat[p.first][p.second] = updated;
            for (auto j = 0; j < 4; j++)
            {
              auto pattern = patterns[j];
              auto pp = make_pair(p.first + pattern[0], p.second + pattern[1]);

              // 更新の対象にするのは外側のやつだけ
              if (visited.count(pp) == 0 && pp.first >= 1 && pp.first < mat_len - 1 && pp.second >= 1 && pp.second < mat_len - 1)
              {
                q.push(pp);
                visited.insert(pp);
              }
            }
          }
        }
      }
    }

    return;
  };

  ll cnt = 0;
  for (auto i = 0; i < ps.size(); i++)
  {
    auto p = ps[i];
    called.insert(p);

    // mat内のindex.
    ll mat_x = (p - 1) % n + 1;
    ll mat_y = ((p - 1) / n) + 1;
    // cout << "p: " << p << " mat_x: " << mat_x << " mat_y: " << mat_y << endl;

    cnt += mat[mat_x][mat_y];
    // matを更新する
    update_mat(mat_x, mat_y);
    // print_mat();
    // printf("cnt: %lld\n", cnt);
  }

  cout << cnt << endl;

  return 0;
}