Mount Allison Programming Showdown 2020

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2020-03-28 08:00 AKDT

Mount Allison Programming Showdown 2020

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2020-03-28 13:00 AKDT
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Problem L
The Wrath of Kahn

/problems/thewrathofkahn/file/statement/en/img-0001.png
Image by Liam Keliher, CC BY-SA

Topologically sorting the nodes of a directed graph (digraph) $G$ means putting the nodes in linear order

\[ v_0, v_1, v_2, v_3, \ldots \]

such that whenever there is an edge from node $x$ to node $y$ in $G$, $x$ always precedes $y$ in the linear ordering. It is not hard to show that the nodes of a digraph can be topologically sorted if and only if the graph is acyclic (does not contain any directed cycles).

Kahn’s Algorithm is a well-known topological sorting algorithm. Beginning with a digraph $G$ and an empty list $L$, it repeatedly executes the following three steps:

  1. Let $S$ be the set of source nodes, i.e., the nodes with no incoming edges. If $S$ is empty, terminate the algorithm.

  2. Let $\alpha $ be any node in $S$. Remove $\alpha $ and all its outgoing edges from $G$. If this removal of edges creates new source nodes, add these to $S$.

  3. Insert $\alpha $ at the end of $L$.

Once Kahn’s Algorithm terminates, if $G$ is now empty, then $L$ contains the nodes of the initial graph in topologically sorted order (such an ordering may not be unique). Otherwise, $G$ has one or more cycles, and therefore topological sorting is impossible.

Regardless of the outcome of Kahn’s Algorithm, consider any iteration of Step $1$ above. If $S$ contains more than one node, then $\alpha $ can be any one of them, and the choice of $\alpha $ affects the composition of $S$ in later iterations, when further choices may be made that affect the composition of $S$ in yet later iterations, and so on. Taking into account all possible sequences of choices of $\alpha $ during the execution of Kahn’s Algorithm, what is the largest $S$ can ever be at the beginning of Step $1$?

Input

The input specifies a single directed graph. The first line of input contains two space-separated integers, $n$ and $m$ ($1 \leq n \leq 500$,$0 \leq m \leq n(n-1)$), where $n$ is the number of nodes and $m$ is the number of edges. This is followed by $m$ lines, each containing two space-separated integers, $x$ and $y$ ($0 \leq x,y \leq (n-1)$, $x \neq y$), indicating that there is a directed edge from node $x$ to node $y$, where nodes are indexed $0, 1, 2, \ldots , (n-1)$. No edge will be listed more than once.

Output

Output a line containing a single integer: the largest possible size of $S$ at the beginning of any iteration of Step $1$ in the execution of Kahn’s Algorithm.

Sample Input 1 Sample Output 1
4 3
0 1
1 2
2 3
1
Sample Input 2 Sample Output 2
5 5
0 4
1 2
1 3
2 4
3 4
3