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 Kotlin (1.0.0) (judge verdict: Time Limit Exceeded)

import java.math.*

val MaxMatch = 499500

val Edges = Array(MaxMatch, { mutableListOf<Int>() })
val RevEdges = Array(MaxMatch, {mutableListOf<Int>() })

fun main(args: Array<String>) {
    val N = readInputLine().toInt()
    
    val A = Array(N, { readInputLine().split(" ").map { it.toInt() - 1 }})
    
    val match = Array(N, { IntArray(N - 1, { -1 }) })
    
    val matchMemo = Array(N, { IntArray(N, { -1 })})
    
    var id = 0
    
    for (i in 0 until N) {
        for (j in 0 until N - 1) {
            if (matchMemo[i][A[i][j]] == -1) {
                match[i][j] = id
                matchMemo[i][A[i][j]] = id
                matchMemo[A[i][j]][i] = id
                id++
            } else {
                match[i][j] = matchMemo[i][A[i][j]]
            }
        }
    }

    for (m in match) {
        for (i in 1 until N - 1) {
            val matchId = m[i]
            val befId = m[i - 1]
            
            Edges[befId].add(matchId)
            RevEdges[matchId].add(befId)
        }
    }

    if (existsCycle()) {
        println(-1)
        return
    }

    println(maxPathLength())
}

fun readInputLine(): String {
    return readLine()!!
}

val NewState = 1
val ActiveState = 2
val FinishState = 3
val nodeStatus = IntArray(MaxMatch, { NewState })

fun existsCycle(): Boolean {
    for (v in 0 until MaxMatch) {
        if (nodeStatus[v] == NewState) {
            if (existsCycleSub(v)) {
                return true
            }
        }
    }

    return false
}

fun existsCycleSub(source: Int): Boolean {
    nodeStatus[source] = ActiveState
    
    for (e in Edges[source]) {
        if (nodeStatus[e] == ActiveState) {
            return true
        } else if (nodeStatus[e] == NewState) {
            if (existsCycleSub(e)) {
                return true
            }
        }
    }

    nodeStatus[source] = FinishState
    
    return false
}

val lengthMemo = IntArray(MaxMatch, { -1 })

fun maxPathLength(): Int {
    var ans = 0

    for (v in 0 until MaxMatch) {
        ans = Math.max(ans, pathLengthSub(v) + 1)
    }

    return ans
}

fun pathLengthSub(id: Int): Int {
    if (lengthMemo[id] == -1) {
        var length = 0
        
        for (e in RevEdges[id]) {
            length = Math.max(length, pathLengthSub(e) + 1)
        }
        
        lengthMemo[id] = length
    }

    return lengthMemo[id]
}