  In this paper, we present a novel algorithm for the maximum a posteriori
decoding (MAPD) of time-homogeneous Hidden Markov Models (HMM), improving the
worst-case running time of the classical Viterbi algorithm by a logarithmic
factor. In our approach, we interpret the Viterbi algorithm as a repeated
computation of matrix-vector $(\max, +)$-multiplications. On time-homogeneous
HMMs, this computation is online: a matrix, known in advance, has to be
multiplied with several vectors revealed one at a time. Our main contribution
is an algorithm solving this version of matrix-vector $(\max,+)$-multiplication
in subquadratic time, by performing a polynomial preprocessing of the matrix.
Employing this fast multiplication algorithm, we solve the MAPD problem in
$O(mn^2/ \log n)$ time for any time-homogeneous HMM of size $n$ and observation
sequence of length $m$, with an extra polynomial preprocessing cost negligible
for $m > n$. To the best of our knowledge, this is the first algorithm for the
MAPD problem requiring subquadratic time per observation, under the only
assumption -- usually verified in practice -- that the transition probability
matrix does not change with time.
