Problem:
Score : 500 points

Problem Statement

N players will participate in a tennis tournament. We will call them Player 1, Player 2, \ldots, Player N.

The tournament is round-robin format, and there will be N(N-1)/2 matches in total.
Is it possible to schedule these matches so that all of the following conditions are satisfied? If the answer is yes, also find the minimum number of days required.

Each player plays at most one matches in a day.

Each player i (1 \leq i \leq N) plays one match against Player A_{i, 1}, A_{i, 2}, \ldots, A_{i, N-1} in this order.

Constraints

3 \leq N \leq 1000

1 \leq A_{i, j} \leq N

A_{i, j} \neq i

A_{i, 1}, A_{i, 2}, \ldots, A_{i, N-1} are all different.

Input

Input is given from Standard Input in the following format:

N
A_{1, 1} A_{1, 2} \ldots A_{1, N-1}
A_{2, 1} A_{2, 2} \ldots A_{2, N-1}
:
A_{N, 1} A_{N, 2} \ldots A_{N, N-1}

Output

If it is possible to schedule all the matches so that all of the conditions are satisfied, print the minimum number of days required; if it is impossible, print -1.

Sample Input 1

3
2 3
1 3
1 2

Sample Output 1

3

All the conditions can be satisfied if the matches are scheduled for three days as follows:

Day 1: Player 1 vs Player 2

Day 2: Player 1 vs Player 3

Day 3: Player 2 vs Player 3

This is the minimum number of days required.

Sample Input 2

4
2 3 4
1 3 4
4 1 2
3 1 2

Sample Output 2

4

All the conditions can be satisfied if the matches are scheduled for four days as follows:

Day 1: Player 1 vs Player 2, Player 3 vs Player 4

Day 2: Player 1 vs Player 3

Day 3: Player 1 vs Player 4, Player 2 vs Player 3

Day 4: Player 2 vs Player 4

This is the minimum number of days required.

Sample Input 3

3
2 3
3 1
1 2

Sample Output 3

-1

Any scheduling of the matches violates some condition.

Solution in OCaml (4.02.3) (judge verdict: Time Limit Exceeded)

let () = Scanf.scanf "%d\n" @@ fun n ->
  let ass =
    Array.init n @@ fun _ ->
      Array.init (n - 1) @@ fun _ ->
        Scanf.scanf "%d " @@ fun a -> a - 1 in
  let es = Array.make_matrix n n [] in
  for i = 0 to n - 1 do
    for j = 0 to n - 3 do
      es.(min i ass.(i).(j)).(max i ass.(i).(j)) <-
        (min i ass.(i).(j + 1), max i ass.(i).(j + 1)) ::
        es.(min i ass.(i).(j)).(max i ass.(i).(j))
    done
  done;
  let dp = Array.make_matrix n n (-1) in
  let vs = Array.make_matrix n n false in
  let rec visit acc (i, j) =
    if vs.(i).(j)
    then raise Not_found
    else begin
      vs.(i).(j) <- true;
      let ans = List.fold_left visit ((i, j) :: acc) es.(i).(j) in
      vs.(i).(j) <- false;
      ans
    end in
  try
    let sort =
      Array.fold_left visit [] @@
      Array.init n @@ fun i ->
        (min i ass.(i).(0), max i ass.(i).(0)) in
    let rec update prev (i, j) =
      if dp.(i).(j) <= prev
      then begin
        dp.(i).(j) <- prev + 1;
        List.iter (update dp.(i).(j)) es.(i).(j)
      end in
    List.iter (update 0) sort;
    Printf.printf "%d\n" @@
    Array.fold_left (Array.fold_left max) min_int dp
  with Not_found -> print_endline "-1"