  An algebraic number $\beta \in \mathbb{C}$ with no conjugate of modulus 1 can
serve as the base of a numeration system $(\beta, \mathcal{A})$ with parallel
addition, i.e., the sum of two operands represented in base $\beta$ with digits
from $\mathcal{A}$ is calculated in constant time, irrespective of the length
of the operands. In order to allow parallel addition, sufficient level of
redundancy must be given to the alphabet $\mathcal{A}$. The complexity of
parallel addition algorithm depends heavily on the size $\#\mathcal{A}$ of the
alphabet: the bigger alphabet is considered, the lower complexity of the
parallel addition algorithm may be reached, and vice versa. Here we aim to find
parallel addition algorithms on alphabets of the minimal possible size, for a
given base. As the complexity of these algorithms becomes quite huge in
general, we introduce a so-called Extending Window Method (EWM) - in fact an
algorithm to construct parallel addition algorithms. This method can be applied
on bases $\beta$ which are expanding algebraic integers, i.e., $\beta$ whose
all conjugates are greater than 1 in modulus. Convergence of the EWM is not
guaranteed, nevertheless, we have developed tools for revealing
non-convergence, and there is a number of successful applications. Firstly, the
EWM provides the same parallel addition algorithms as were previously
introduced by A. Avizienis, C.Y. Chow & J.E. Robertson or B. Parhami for
integer bases and alphabets. Then, by applying the EWM on selected complex
bases $\beta$ with non-integer alphabets $\mathcal{A} \subset
\mathbb{Z}[\beta]$, we obtain new results - parallel addition algorithms on
alphabets of minimal possible size, which could not be found so far (manually).
The EWM is helpful also in the case of block parallel addition.
