Deterministic finite automaton(DFA) is a finite state machine that accepts/rejects finite strings of symbols and only produces a unique computation (or run) of the automation for each input string.
DFAs can be represented using state diagrams. For example, in the automaton shown below, there are three states: S0, S1, and S2 (denoted graphically by circles). The automaton takes a finite sequence of 0s and 1s as input. For each state, there is a transition arrow leading out to a next state for both 0 and 1. Upon reading a symbol, a DFA jumps deterministically from a state to another by following the transition arrow. For example, if the automaton is currently in state S0 and current input symbol is 1 then it deterministically jumps to state S1. A DFA has a start state (denoted graphically by an arrow coming in from nowhere) where computations begin, and a set of accept states (denoted graphically by a double circle) which help define when a computation is successful.
These are some strings above DFA accepts,
0 00 000 11 110 1001
You are given a DFA in input and an integer N. You have to tell how many distinct strings of length N the given DFA accepts. Return answer modulo 109+7.
- Assume each state has two outgoing edges(one for 0 and one for 1). Both outgoing edges won’t go to the same state.
- There could be multiple accept states, but only one start state.
- A start state could also be an accept state.
- States are numbered from 0 to K-1, where K is total number of states in DFA.
- You are given three arrays A, B, C and two integers D and N.
- Array A denotes a 0 edge from state numbered i to state A[i], for all 0 ≤ i ≤ K-1
- Array B denotes a 1 edge from state numbered i to state B[i], for all 0 ≤ i ≤ K-1
- Array C contains indices of all accept states.
- Integer D denotes the start state.
- Integer N denotes you have to count how many distinct strings of length N the given DFA accepts.
1 ≤ K ≤ 50
1 ≤ N ≤ 104
For the DFA shown in image, input is A = [0, 2, 1] B = [1, 0, 2] C =  D = 0 Input 1 ------- N = 2 Strings '00' and '11' are only strings on length 2 which are accepted. So, answer is 2. Input 2 ------- N = 1 String '0' is the only string. Answer is 1.