并行加法算法的构造
Construction of Algorithms for Parallel Addition
摘要：
没有模1共轭的代数数$β\in \mathbb{C}$可以用作具有并行加法的计数系统$(β, \mathcal{A})$的基数，即，在恒定时间内计算以具有来自$\mathcal{A}$的数字的基$β$表示的两个操作数的和，而与操作数的长度无关。为了允许并行加法，必须给字母表$\mathcal{A}$提供足够的冗余度。并行加法算法的复杂度很大程度上取决于字母表的大小$\#\mathcal{A}$：考虑的字母表越大，并行加法算法的复杂度可能越低，反之亦然。在这里，我们的目标是找到一个给定的基础上的最小可能大小的字母表并行加法算法。由于这些算法的复杂性变得相当巨大的一般情况下，我们介绍了一个所谓的扩展窗口方法（EWM）-实际上是一个算法来构造并行加法算法。该方法可以应用于扩展代数整数的基$β$上，即，$β$，其所有共轭物的模量大于1。EWM的收敛性并不能保证，尽管如此，我们已经开发出了揭示非收敛性的工具，并且有一些成功的应用。首先，EWM提供了与A. Avizienis，C.Y. Chow & J.E. Robertson或B.整数基和字母表的Parhami。然后，通过应用EWM在选定的复杂基地$β$与非整数字母$\mathcal{A} \subset \mathbb{Z}[β]$，我们得到了新的结果-并行加法算法的最小可能的大小，这是迄今为止（手动）找不到的字母。EWM在块并行加法的情况下也是有帮助的。
英文摘要：
An algebraic number $β\in \mathbb{C}$ with no conjugate of modulus 1 can serve as the base of a numeration system $(β, \mathcal{A})$ with parallel addition, i.e., the sum of two operands represented in base $β$ 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 $β$ which are expanding algebraic integers, i.e., $β$ 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 $β$ with non-integer alphabets $\mathcal{A} \subset \mathbb{Z}[β]$, 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.
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