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Unformatted text preview: On the Construction of ZeroDeficiency Parallel Prefix Circuits with Minimum Depth HAIKUN ZHU, CHUNGKUAN CHENG, and RONALD GRAHAM University of California, San Diego A parallel prefix circuit has n inputs x 1 , x 2 , ... , x n , and computes the n outputs y i = x i x i 1 x 1 , 1 i n , in parallel, where is an arbitrary binary associative operator. Snir proved that the depth t and size s of any parallel prefix circuit satisfy the inequality t + s 2 n 2. Hence, a parallel prefix circuit is said to be of zerodeficiency if equality holds. In this article, we provide a different proof for Snirs theorem by capturing the structural information of zerodeficiency prefix circuits. Following our proof, we propose a new kind of zerodeficiency prefix circuit Z ( d ) by constructing a prefix circuit as wide as possible for a given depth d . It is proved that the Z ( d ) circuit has the minimal depth among all possible zerodeficiency prefix circuits. Categories and Subject Descriptors: B.6.1 [ Logic Design ]: Design Styles Parallel circuits General Terms: Algorithms, Design, Theory Additional Key Words and Phrases: Zerodeficiency, parallel prefix circuits, depthsize tradeoff 1. INTRODUCTION 1.1 Problem Definition The prefix problem, which mostly gains research attention with the emergence of parallel computing, is actually the abstraction of many practical applications such as binary addition, radix sort, linear recurrences solving, polynomial eval uation, etc. [Lakshmivarahan and Dhall 1994]. Formally, the prefix problem is defined as follows: Definition 1.1 . [Prefix Problem] Given n inputs x 1 , x 2 , ... , x n and an ar bitrary binary associative operator , compute the prefix results Y i = x i x i 1 x 1 for 1 i n . This article is based on work previously published as Constructing ZeroDeficiency Parallel Prefix Adder of Minimum Depth, In Proceedings of the 2005 Asia and South Pacific Design Automation Conference ( ASPDAC 2005 ) . c 2005 IEEE. This work was supported in part under grants from National Science Foundation (NSF) project number MIP9987678, the California MICRO program, and SRC support. Authors address: Computer Science and Engineering, University of California, San Diego, La Jolla, CA 920930404; email: { hazhu,kuan,rgraham } @cs.ucsd.edu . Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or direct commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specific permission and/or a fee. Permissions may be requested from Publications Dept., ACM, Inc., 1515permission and/or a fee....
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 Spring '07
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