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Optimal size integer division circuits

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J. H. ReifAuthors Info & Claims

STOC '89: Proceedings of the twenty-first annual ACM symposium on Theory of computing

Pages 264 - 273

https://doi.org/10.1145/73007.73032

Published: 01 February 1989 Publication History

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Abstract

Division is a fundamental problem for arithmetic and algebraic computation. This paper describes Boolean circuits (of bounded fan-in) for integer division (finding reciprocals) that have size Ο(M(n)) and depth Ο(lognlog logn), where M(n) is the size complexity of Ο(logn) depth integer multiplication circuits. Currently, M(n) is known to be Ο(nlogn log,n), but any improvement in this bound that preserves circuit depth will be reflected by a similar improvement in the size complexity of our division algorithm. Previously, no one has been able to derive a division circuit with size Ο(n log^c n) for any c, and simultaneous depth less than Ω(log² n). Our circuits are logspace uniform; that is, they can be constructed by a deterministic Turing machine in space Ο(log n).

Our results match the best known depth bounds for logspace uniform circuits, and are optimal in size.

The general method of high order iterative formulas is of independent interest as a way of efficiently using parallel processors to solve algebraic problems. In particular, our algorithm implies that any rational function can be evaluated in these complexity bounds.

As an introduction to high order iterative methods we also present a circuit for finding polynomial reciprocals (where the coefficients come from an arbitrary ring, and ring operations are unit cost in the circuit) in size Ο(PM (n)) and depth Ο(log n log log n), where PM(n) is the size complexity of optimal depth polynomial multiplication.

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Cited By

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Mogensen T(2018)Garbage-Free Reversible Multiplication and DivisionReversible Computation10.1007/978-3-319-99498-7_18(253-268)Online publication date: 22-Aug-2018
https://doi.org/10.1007/978-3-319-99498-7_18
Meyer SSorenson J(2005)Efficient algorithms for computing the Jacobi symbolAlgorithmic Number Theory10.1007/3-540-61581-4_58(225-239)Online publication date: 2-Jun-2005
https://doi.org/10.1007/3-540-61581-4_58
Waack S(2005)On the parallel complexity of iterated multiplication in rings of algebraic integersParallel and Distributed Computing Theory and Practice10.1007/3-540-58078-6_4(35-44)Online publication date: 1-Jun-2005
https://doi.org/10.1007/3-540-58078-6_4
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Optimal size integer division circuits

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Published In

STOC '89: Proceedings of the twenty-first annual ACM symposium on Theory of computing

February 1989

600 pages

ISBN:0897913078

DOI:10.1145/73007

Editor:
D. S. Johnson

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 February 1989

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STOC89

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SIGACT

STOC89: 21st Annual ACM Symposium on the Theory of Computing

May 14 - 17, 1989

Washington, Seattle, USA

Acceptance Rates

STOC '89 Paper Acceptance Rate 56 of 196 submissions, 29%;

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

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Cited By

View all

Mogensen T(2018)Garbage-Free Reversible Multiplication and DivisionReversible Computation10.1007/978-3-319-99498-7_18(253-268)Online publication date: 22-Aug-2018
https://doi.org/10.1007/978-3-319-99498-7_18
Meyer SSorenson J(2005)Efficient algorithms for computing the Jacobi symbolAlgorithmic Number Theory10.1007/3-540-61581-4_58(225-239)Online publication date: 2-Jun-2005
https://doi.org/10.1007/3-540-61581-4_58
Waack S(2005)On the parallel complexity of iterated multiplication in rings of algebraic integersParallel and Distributed Computing Theory and Practice10.1007/3-540-58078-6_4(35-44)Online publication date: 1-Jun-2005
https://doi.org/10.1007/3-540-58078-6_4
(1992)BibliographyIntroduction to Parallel Algorithms and Architectures10.1016/B978-1-4832-0772-8.50008-X(785-801)Online publication date: 1992
https://doi.org/10.1016/B978-1-4832-0772-8.50008-X
Bini DPan VAggarwal A(1991)Parallel complexity of tridiagonal symmetric Eigenvalue problemProceedings of the second annual ACM-SIAM symposium on Discrete algorithms10.5555/127787.127855(384-393)Online publication date: 1-Mar-1991
https://dl.acm.org/doi/10.5555/127787.127855
Zeugmann T(1990)Computing large polynomial powers very fast in parallelMathematical Foundations of Computer Science 199010.1007/BFb0029653(538-544)Online publication date: 1990
https://doi.org/10.1007/BFb0029653

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