Problem:
Score : 500 points

Problem Statement

You are given a sequence with N integers: A = \{ A_1, A_2, \cdots, A_N \}.
For each of these N integers, we will choose a color and paint the integer with that color. Here the following condition must be satisfied:

If A_i and A_j (i < j) are painted with the same color, A_i < A_j.

Find the minimum number of colors required to satisfy the condition.

Constraints

1 \leq N \leq 10^5

0 \leq A_i \leq 10^9

Input

Input is given from Standard Input in the following format:

N
A_1
:
A_N

Output

Print the minimum number of colors required to satisfy the condition.

Sample Input 1

5
2
1
4
5
3

Sample Output 1

2

We can satisfy the condition with two colors by, for example, painting 2 and 3 red and painting 1, 4, and 5 blue.

Sample Input 2

4
0
0
0
0

Sample Output 2

4

We have to paint all the integers with distinct colors.

Solution in C# (Mono 4.6.2.0) (judge verdict: Time Limit Exceeded)

using static System.Math;
using static System.Console;
using System.Collections.Generic;
using System.Linq;
using System.Text;

namespace AtCoder
{
    class Program
    {
        static void Main(string[] args)
        {
            var n = int.Parse(ReadLine());
            var numbers = new int[n];
            var isColored = new bool[n];
            for (var i = 0; i < n; i++)
            {
                numbers[i] = int.Parse(ReadLine());
            }

            var count = 0;
            while (isColored.Any(value => !value))
            {
                count++;
                var dp = new int[n, 2];
                int[,] backupDp = null;
                var length = 0;
                for (var i = 0; i < n; i++)
                {
                    for (var j = 0; j < 2; j++)
                    {
                        dp[i, j] = int.MaxValue;
                    }
                }

                for (var i = 0; i < n; i++)
                {
                    if(isColored[i])
                    {
                        continue;
                    }

                    var ok = n - 1;
                    var ng = -1;

                    while (Abs(ok - ng) > 1)
                    {
                        var index = (ok + ng) / 2;

                        if (numbers[i] <= dp[index, 0])
                        {
                            ok = index;
                        }
                        else
                        {
                            ng = index;
                        }
                    }

                    if (dp[ok, 0] > numbers[i])
                    {
                        dp[ok, 0] = numbers[i];
                        dp[ok, 1] = i;

                        if(ok >= length)
                        {
                            backupDp = (int[,])dp.Clone();
                        }
                    }
                }

                dp = backupDp;

                for (var i = 0; i < n; i++)
                {
                    if (dp[i, 0] == int.MaxValue)
                    {
                        break;
                    }
                    isColored[dp[i, 1]] = true;
                }
            }

            WriteLine(count);
        }
    }
}