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k-Block parallel addition versus 1-block parallel addition in non-standard numeration systems

机译：非标准计算系统中的k块并行加法与1块并行加法

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Parallel addition in integer base is used for speeding up multiplication and division algorithms, k-block parallel addition has been introduced by Kornerup in [14]: instead of manipulating single digits, one works with blocks of fixed length k. The aim of this paper is to investigate how such notion influences the relationship between the base and the cardinality of the alphabet allowing block parallel addition. In this paper, we mainly focus on a certain class of real bases the so-called Parry numbers. We give lower bounds on the cardinality of alphabets of non-negative integer digits allowing block parallel addition. By considering quadratic Pisot bases, we are able to show that these bounds cannot be improved in general and we give explicit parallel algorithms for addition in these cases. We also consider the d-bonacci base, which satisfies the equation X-d = Xd-1 + Xd-2 + ... + X + 1. If in a base being a d-bonacci number I-block parallel addition is possible on an alphabet A, then #A >= d + 1; on the other hand, there exists a k is an element of N such that k-block parallel addition in this base is possible on the alphabet {0, 1, 2}, which cannot be reduced. In particular, addition in the Tribonacci base is 14-block parallel on alphabet {0, 1, 2}. (C) 2014 Elsevier B.V. All rights reserved.
机译：整数基数中的并行加法用于加速乘法和除法算法，Kornerup在[14]中引入了k块并行加法：而不是处理单个数字，而是处理固定长度k的块。本文的目的是研究这样的概念如何影响允许块并行加法的字母的基数和基数之间的关系。在本文中，我们主要关注某一类实数基，即所谓的Parry数。我们对非负整数的字母基数给出了下限，从而允许块并行加法。通过考虑二次Pisot基数，我们能够证明这些界限通常无法改善，并且在这些情况下我们给出了明确的并行算法进行加法。我们还考虑满足方程Xd = Xd-1 + Xd-2 + ... + X + 1的d-bonacci底数。如果以d-bonacci数为底数，则I-block可以并行加法。字母A，然后#A> = d +1;另一方面，k是N的元素，因此可以基于字母{0、1、2}在该基数上进行k块并行加法，而不能减少。特别是，在Tribonacci基数中的加法是在字母{0，1，2}上并行14块。 （C）2014 Elsevier B.V.保留所有权利。

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