Problem:
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Maximum Flow

A flow network is a directed graph which has a $source$ and a $sink$. In a flow network, each edge $(u, v)$ has a capacity $c(u, v)$. Each edge receives a flow, but the amount of flow on the edge can not exceed the corresponding capacity. Find the maximum flow from the $source$ to the $sink$.

Input

A flow network is given in the following format.

$|V|\;|E|$

$u_0\;v_0\;c_0$

$u_1\;v_1\;c_1$

:

$u_{|E|-1}\;v_{|E|-1}\;c_{|E|-1}$

$|V|$, $|E|$ is the number of vertices and edges of the flow network respectively. The vertices in $G$ are named with the numbers 0, 1,..., $|V|-1$. The source is 0 and the sink is $|V|-1$.

$u_i$, $v_i$, $c_i$ represent $i$-th edge of the flow network.
A pair of $u_i$ and $v_i$ denotes that there is an edge from $u_i$ to $v_i$ and $c_i$ is the capacity of $i$-th edge.

Output

Print the maximum flow.

Constraints

$2 \leq |V| \leq 100$

$1 \leq |E| \leq 1000$

$0 \leq c_i \leq 10000$

Sample Input 1

4 5
0 1 2
0 2 1
1 2 1
1 3 1
2 3 2

Sample Output 1

3

Solution in Ruby (judge verdict: Accepted)

MAX = 1.0 / 0.0
class Node
  attr_accessor :cf
  def initialize
    @cf = {}
  end
end

def dfs(s, t, g)
  stack = [[s, MAX]]
  done = {}
  done[s] = true
  while stack.size > 0
    u = stack[-1][0]
    return stack if u == t
    flag = true
    g[u].cf.each{|v, c|
      next if done[v]
      done[v] = true
      stack << [v, c]
      flag = false
      break
    }
    if flag
      stack.pop
    end
  end
  return nil
end

vn, e = gets.split.map(&:to_i)
g = Array.new(vn){Node.new}
e.times{
  u, v, c = gets.split.map(&:to_i)
  g[u].cf[v] = c
}

sum = 0
loop do
  #detect path from s to t
  path = dfs(0, vn-1, g)

  if path
    #maximum flow by the path is
    max = MAX
    path.size.times{|i|
      max = path[i][1] if max > path[i][1]
    }
    sum += max

    #subtract the flow
    (path.size-1).times{|i|
      s = path[i][0]
      t = path[i+1][0]
      g[s].cf[t] -= max
      g[t].cf[s] ||= 0
      g[t].cf[s] += max
      if g[s].cf[t] == 0
        g[s].cf.delete(t)
      end
    }
  else
    #there is no path
    break
  end
end

p sum