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)

N,M = gets.split.map &:to_i
es = (0..N).map{ [] }
M.times {
  x,y = gets.split.map &:to_i
  es[x] << y
}
dp = Array.new(N,0)

(1..N).each{ |x|
  dig_stack = [x]
  climb_stack = []
  #p x
  xs = {}
  while k = dig_stack.pop do
    next if dp[k-1] > 0 || xs[k]
    ns = es[k]
    xs[k] = 1
    if ns.size > 0
      climb_stack.push(k)
      dig_stack.concat(ns)
    end
    #p [k,dig_stack,climb_stack]
  end
  climb_stack.reverse_each{ |k|
    v = 0
    es[k].each{ |y| v = dp[y-1] > v ? dp[y-1] : v }
    dp[k-1] = v + 1
  }
}

p dp.max