Problem:
Score : 100 points

Problem Statement

There is a directed graph G with N vertices and M edges.
The vertices are numbered 1, 2, \ldots, N, and for each i (1 \leq i \leq M), the i-th directed edge goes from Vertex x_i to y_i.
G does not contain directed cycles.

Find the length of the longest directed path in G.
Here, the length of a directed path is the number of edges in it.

Constraints

All values in input are integers.

2 \leq N \leq 10^5

1 \leq M \leq 10^5

1 \leq x_i, y_i \leq N

All pairs (x_i, y_i) are distinct.

G does not contain directed cycles.

Input

Input is given from Standard Input in the following format:

N M
x_1 y_1
x_2 y_2
:
x_M y_M

Output

Print the length of the longest directed path in G.

Sample Input 1

4 5
1 2
1 3
3 2
2 4
3 4

Sample Output 1

3

The red directed path in the following figure is the longest:

Sample Input 2

6 3
2 3
4 5
5 6

Sample Output 2

2

The red directed path in the following figure is the longest:

Sample Input 3

5 8
5 3
2 3
2 4
5 2
5 1
1 4
4 3
1 3

Sample Output 3

3

The red directed path in the following figure is one of the longest:

Solution in Ruby (2.3.3) (judge verdict: Wrong Answer)

# 要提出！
# 例外捕捉（適当にputs ** exit）を２分探索で特定！
# エラーの原因はスタックオーバーフロー？？
# 2 1
# 1 2

# 4 5
# 1 2
# 1 3
# 3 2
# 2 4
# 3 4
#=> 3

# 6 3
# 2 3
# 4 5
# 5 6
#=> 2

# 5 8
# 5 3
# 2 3
# 2 4
# 5 2
# 5 1
# 1 4
# 4 3
# 1 3
#=> 3

# require 'pry'
begin
N, M = gets.split.map(&:to_i)
@lines = {}
M.times do
  s, t = gets.split.map(&:to_i)
  (@lines[s-1] ||= []) << t-1
end
# p lines
@dp = Array.new(N, 0)
# binding.pry
@check2 = Array.new(N, false)
def dfs(i)
  if @check2[i]
    return @dp[i]
  end
  test 8 / 0
  @check2[i] = true
  res = 0
  if @lines[i]
    @lines[i].each do |to|
      pre = dfs(to) + 1
      res = pre if res < pre
    end
  end
  return @dp[i] = res
end

ans = 0
N.times do |i|
  if !@check2[i]
    ans = dfs(i) if ans < dfs(i)
  end
end
# puts @dp.max
puts ans
rescue
  puts 2525
  exit
end