Problem:
Longest Increasing Subsequence

For a given sequence A = {a0, a1, ... , an-1}, find the length of the longest increasing subsequnece (LIS) in A.

An increasing subsequence of A is defined by a subsequence {ai0, ai1, ... , aik} where 0 ≤ i0 < i1 < ... < ik < n and ai0 < ai1 < ... < aik.

Input

n
a0
a1
:
an-1

In the first line, an integer n is given. In the next n lines, elements of A are given.

Output

The length of the longest increasing subsequence of A.

Constraints

1 ≤ n ≤ 100000

0 ≤ ai ≤ 109

Sample Input 1

5
5
1
3
2
4

Sample Output 1

3

Sample Input 2

3
1
1
1

Sample Output 2

1

Solution in C# (judge verdict: Wrong Answer)

using System;
using System.Linq;
using System.Collections.Generic;

public class Program{
    public static int N;
    public static int[] A;
    public static void Main(){
        N = int.Parse(Console.ReadLine());
        A = new int[N];
        for(int i=0; i<N; i++){
            A[i] = int.Parse(Console.ReadLine());
        }
        //???????¢??????¨??????????????????????¨?????????¨???????????????????????§?§£???
        var L = Enumerable.Repeat(int.MaxValue, N).ToArray();
        int length = 1;
        L[0] = A[0];
        for(int i=1; i<N; i++){
            if(L[length-1]<A[i]){
                L[length] = A[i];
                length++;
            }else{
                //A[i]???L??????????????´????????\????????????????????????
                //???????????´?????¨??????L???????????§A[i]?????????????????§?????°????????????????????????????????§?????????
                var alterIdx = LowerBound(L, A[i]);
                L[alterIdx] = A[i];
            }
        }
        Console.WriteLine(length);
    }
    
    //array???????????§target?????????????????§????????????????????????????????????
    public static int LowerBound(int[] array, int target){
        int left = 0;
        int right = array.Count();
        int mid = 0;
        while(left<right){
            mid = (right+left)/2;
            if(array[mid]==target) break;
            if(target<array[mid])right = mid;
            else left = mid+1;
        }
        return left;
    }
}