Construction of Algorithms for Parallel Addition
Jan Legersk\'y, Milena Svobodov\'a

TL;DR
This paper introduces the Extending Window Method (EWM), an algorithm for constructing minimal alphabet parallel addition algorithms for algebraic number bases, including complex bases, enabling constant-time addition.
Contribution
The paper develops the EWM for designing minimal alphabet parallel addition algorithms for algebraic number bases, including complex and expanding algebraic integers.
Findings
EWM reproduces known algorithms for integer bases.
EWM finds new minimal alphabet algorithms for complex bases.
EWM aids in block parallel addition design.
Abstract
An algebraic number with no conjugate of modulus 1 can serve as the base of a numeration system with parallel addition, i.e., the sum of two operands represented in base with digits from 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 . The complexity of parallel addition algorithm depends heavily on the size 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…
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