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: Time Limit Exceeded)

N,M = gets.split.map &:to_i
es = Hash[(1..N).map{ |x| [x,{}]}]
M.times {
  x,y = gets.split.map &:to_i
  es[x][y] = 1
}
dp = Array.new(N,0)

(1..N).each{ |x|
  dig_stack = [x]
  climb_stack = []
  while k = dig_stack.pop do
    ns = es[k].keys
    if ns.size == 0
      dp[k-1] = 0
    else
      climb_stack.push(k)
      dig_stack.concat(ns)
    end
  end
  while k = climb_stack.pop do
    dp[k-1] = es[k].keys.map{ |y| dp[y-1] }.max + 1
  end
}

p dp.max