{"title":"Asymptotically sharp estimates for the area of multiplexers in the cellular circuit model","authors":"S. A. Lozhkin, V. S. Zizov","doi":"10.1515/dma-2024-0009","DOIUrl":null,"url":null,"abstract":"\n A general cellular circuit of functional and switching elements (CCFSE) is a mathematical model of integral circuits (ICs), which takes into account peculiarities of their physical synthesis. A principal feature of this model distinguishing it from the well-known classes of circuits of gates (CGs) is the presence of additional requirements on the geometry of the circuit which ensure the accounting of the necessary routing resources for IC creation. The complexity of implementation of a multiplexer function of Boolean algebra (FBA) in different classes of circuits has been extensively studied. In the present paper, we give asymptotically sharp upper and lower estimates for the area of a CCFSE implementing a multiplexer FBA of order n. We construct a family of circuit multiplexers of order n of area equal to the halved upper estimate, and provide a method of delivering the corresponding lower estimate.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics and Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/dma-2024-0009","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
A general cellular circuit of functional and switching elements (CCFSE) is a mathematical model of integral circuits (ICs), which takes into account peculiarities of their physical synthesis. A principal feature of this model distinguishing it from the well-known classes of circuits of gates (CGs) is the presence of additional requirements on the geometry of the circuit which ensure the accounting of the necessary routing resources for IC creation. The complexity of implementation of a multiplexer function of Boolean algebra (FBA) in different classes of circuits has been extensively studied. In the present paper, we give asymptotically sharp upper and lower estimates for the area of a CCFSE implementing a multiplexer FBA of order n. We construct a family of circuit multiplexers of order n of area equal to the halved upper estimate, and provide a method of delivering the corresponding lower estimate.
功能和开关元件通用单元电路（CCFSE）是一种积分电路（IC）数学模型，它考虑到了积分电路物理合成的特殊性。该模型有别于众所周知的门电路（CG）的一个主要特点是，它对电路的几何形状有额外的要求，以确保为集成电路的创建提供必要的路由资源。关于在不同类别电路中实现布尔代数多路复用器函数（FBA）的复杂性，已经进行了广泛的研究。在本文中，我们给出了实现 n 阶多路复用器 FBA 的 CCFSE 面积的近似尖锐的上限和下限估计值。我们构建了一个面积等于上限估计值减半的 n 阶电路多路复用器系列，并提供了提供相应下限估计值的方法。
期刊介绍：
The aim of this journal is to provide the latest information on the development of discrete mathematics in the former USSR to a world-wide readership. The journal will contain papers from the Russian-language journal Diskretnaya Matematika, the only journal of the Russian Academy of Sciences devoted to this field of mathematics. Discrete Mathematics and Applications will cover various subjects in the fields such as combinatorial analysis, graph theory, functional systems theory, cryptology, coding, probabilistic problems of discrete mathematics, algorithms and their complexity, combinatorial and computational problems of number theory and of algebra.
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