  Given a non-deterministic finite automaton (NFA) A with m states, and a
natural number n (presented in unary), the #NFA problem asks to determine the
size of the set L(A_n) of words of length n accepted by A. While the
corresponding decision problem of checking the emptiness of L(A_n) is solvable
in polynomial time, the #NFA problem is known to be #P-hard. Recently, the
long-standing open question -- whether there is an FPRAS (fully polynomial time
randomized approximation scheme) for #NFA -- was resolved in \cite{ACJR19}. The
FPRAS due to \cite{ACJR19} relies on the interreducibility of counting and
sampling, and computes, for each pair of state q and natural number i <= n, a
set of O(\frac{m^7 n^7}{epsilon^7}) many uniformly chosen samples from the set
of words of length i that have a run ending at q (\epsilon is the error
tolerance parameter of the FPRAS). This informative measure -- the number of
samples maintained per state and length -- also affects the overall time
complexity with a quadratic dependence.
  Given the prohibitively high time complexity, in terms of each of the input
parameters, of the FPRAS due to \cite{ACJR19}, and considering the widespread
application of approximate counting (and sampling) in various tasks in Computer
Science, a natural question arises: Is there a faster FPRAS for #NFA that can
pave the way for the practical implementation of approximate #NFA tools? In
this work, we demonstrate that significant improvements in time complexity are
achievable. Specifically, we have reduced the number of samples required for
each state to be independent of m, with significantly less dependence on $n$
and $\epsilon$, maintaining only \widetilde{O}(\frac{n^4}{epsilon^2}) samples
per state.
