Question

I need to generate a Deterministic Finite Automata (DFA), selected from all possible DFAs that satisfy the properties below. The DFA must be selected with uniform distribution.

The DFA must have the following four properties:

• The DFA has N nodes.
• Each node has 2 outgoing transitions.
• Each node is reachable from every other node.
• The DFA is chosen with perfectly uniform randomness from all possibilities.

I am not considering labeling of nodes or transitions. If two DFAs have the same unlabeled directed graph they are considered the same.

Here are three algorithms that don’t work:

Algorithm #1

1. Start with a set of N nodes called A.
2. Choose a node from A and put it in set B.

3. While there are nodes left in set A

     -  3.1 Choose a node x from set A
     -  3.2 Choose a node y from set B with less than two outgoing transitions.
     -  3.3 Choose a node z from set B
     -  3.4 Add a transition from y to x.
     -  3.5 Add a transition from x to z
     -  3.6 Move x to set B
   

4. For each node n in B

     -  4.1 While n has less than two outgoing transitions
     -  4.2 Choose a node m in B
     -  4.3 Add a transition from n to m
   

Algorithm #2

1. Start with a directed graph with N vertices and no arcs.
2. Generate a random permutation of the N vertices to produce a random Hamiltonian cycle, and add it to the graph.
3. For each vertex add one outgoing arc to a randomly chosen vertex.

Algorithm #3

1. Start with a directed graph with N vertices and no arcs.
2. Generate a random directed cycle of some length between N and 2N and add it to the graph.
3. For each vertex add one outgoing arc to a randomly chosen vertex.

I created algorithm #3 based off of algorithm #2, however, I don’t know how to select the random directed cycle to create a uniform distribution. I don’t even know if it’s possible.

Any help would be greatly appreciated.

Answer 1

If N is small (there are N^(2N) possible sets of arcs that meet the first two conditions, so you would want this to be less than the range of your random number generator) you can generate random DFAs and discard the ones that don’t satisfy the reachability condition.