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Log Depth Circuits for Division and Related Problems

1986
*
SIAM journal on computing (Print)
*

We present optimal

doi:10.1137/0215070
fatcat:pntuddlzzvgzngh6ianmnmpid4
*depth*Boolean*circuits*(*depth*O(*log*n))*for*integer*division*, powering*and*multiple products We also show t h a t these three*problems*are of equivalent uniform*depth**and*space complexity ... In addition, we describe an algorithm*for*testing drvisibility t h a t is optimal*for*both*depth**and*space ...*Divisibility*Although the*DIVISION**problem*has P-uniform O(1ogn)*depth**circuits*, it is still unclear whether or not it has*log*-space uniform O ( 1 o g n )*depth**circuits*. ...##
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Log Depth Circuits For Division And Related Problems

*
25th Annual Symposium onFoundations of Computer Science, 1984.
*

We present optimal

doi:10.1109/sfcs.1984.715894
dblp:conf/focs/BeameCH84
fatcat:epcp5i6irjdhtjzhfm2zhxvdj4
*depth*Boolean*circuits*(*depth*O(*log*n))*for*integer*division*, powering*and*multiple products We also show t h a t these three*problems*are of equivalent uniform*depth**and*space complexity ... In addition, we describe an algorithm*for*testing drvisibility t h a t is optimal*for*both*depth**and*space ...*Divisibility*Although the*DIVISION**problem*has P-uniform O(1ogn)*depth**circuits*, it is still unclear whether or not it has*log*-space uniform O ( 1 o g n )*depth**circuits*. ...##
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Division Is In Uniform TC0
[chapter]

2001
*
Lecture Notes in Computer Science
*

Integer

doi:10.1007/3-540-48224-5_9
fatcat:vu6dhiadb5drjduq47va4sfobq
*division*has been known since 1986 [4, 13, 12] to be in slightly non-uniform TC 0 , i.e., computable by polynomial-size, constant*depth*threshold*circuits*. ... This has been perhaps the outstanding natural*problem*known to be in a standard*circuit*complexity class, but not known to be in its uniform version. We show that indeed*division*is in uniform TC 0 . ... In this paper, we construct uniform constant*depth**circuits**for**division**and*iterated multiplication. ...##
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Ranking and formal power series

1991
*
Theoretical Computer Science
*

As a consequence ranking

doi:10.1016/0304-3975(91)90144-q
fatcat:e6bcguyfqfejdpcv3vk6yqphia
*problems**for*regular languages are NC'-reducible to integer*division**and*hence computable by*log*-space uniform boolean*circuits*of polynomial size*and**depth*O(*log*n*log**log*n), ... or by p-uniform boolean*circuits*of polynomial size*and**depth*O(*log*n). ...*circuits*of polynomial size*and**depth*O(*log*' n) [8] . ...##
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Uniform constant-depth threshold circuits for division and iterated multiplication

2002
*
Journal of computer and system sciences (Print)
*

1 2 HESSE, ALLENDER

doi:10.1016/s0022-0000(02)00025-9
fatcat:vwqdpnv7njghzi2326rxqprtca
*AND*BARRINGTON It has been known since the mid-1980's [15, 46, 47] that integer*division*can be performed by poly-time uniform constant-*depth**circuits*of Majority gates; equivalently ... The essential idea in the fast parallel computation of*division**and**related**problems*is that of Chinese remainder representation (CRR) -storing a number in the form of its residues modulo many small primes ... We also thank Dieter van Melkebeek, Samir Datta, Michal Koucký, Rüdiger Reischuk,*and*Sambuddha Roy*for*helpful discussions. ...##
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Boolean circuits versus arithmetic circuits

1991
*
Information and Computation
*

Over Q

doi:10.1016/0890-5401(91)90078-g
fatcat:aydwnfiodfgy3bizzgauso2sei
*and*finite fields, Boolean*circuits*can simulate arithmetic*circuits*efficiently with respect to size. ... Over finite lields of small characteristic, the two models are equally powerful when size is considered, but Boolean*circuits*are exponentially more powerful than arithmetic*circuits*with respect to*depth*... ACKNOWLEDGMENTS We thank Martin Tompa*and*an anonymous referee*for*valuable suggestions. RECEIVED May 12, 1989; FINAL MANUSCRIPT RECEIVED October 27, 1989 ...##
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Recursively divisible problems
[chapter]

1996
*
Lecture Notes in Computer Science
*

We call

doi:10.1007/bfb0009494
fatcat:h7z2lslhivcljlyafhzrdtvl5e
*problems*that are recursively (n ; O(1))-*divisible*in a work-optimal way with 0 < < 1 ideally*divisible**and*give motivation drawn from parallel computing*for*the relevance of that concept. ...*For*example, ideally*divisible**problems*appear to be a proper subclass of the functional complexity class FP of sequentially feasible*problems*. ...*Relations*to Complexity Theory In this section we present a*circuit*construction*for*recursively (n ; d)-*divisible**problems*. ...##
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On Relating Time and Space to Size and Depth

1977
*
SIAM journal on computing (Print)
*

In doing so, we

doi:10.1137/0206054
fatcat:f45h6aubu5br3cgwhdz2t6nnp4
*relate*the power of nondeterminism*for*space bounded computations to the*depth*required*for*the transitive closure*problem*. ... Turing machine space complexity is*related*to*circuit**depth*complexity. ... Cook*for*a number of important suggestions,*and*to A. Meyer, L. Stockmeyer*and*the referees*for*their helpful comments. ...##
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Depth efficient neural networks for division and related problems

1993
*
IEEE Transactions on Information Theory
*

The

doi:10.1109/18.256501
fatcat:5ahnr5q4i5fwropv64l46qgo5q
*depth*of a*circuit*represents the number of unit delays or the time*for*parallel computation. The size of a*circuit*is the number of gates*and*measures the amount of hardware. ... It was known that traditional logic*circuits*consisting of only unbounded fanin*AND*, OR, NOT gates would require at least R(*log*nllog*log*n)*depth*to compute common arithmetic functions such as the product ... Theorem 8: Any polynomial size neural network*for*SORT-ING must have*depth*at least 3. Proof: We show that if we can sort 2n + 1 integers of length (*log*n + 3)-bits, then the Inner Product Modulo ...##
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Page 1057 of Mathematical Reviews Vol. , Issue 88b
[page]

1988
*
Mathematical Reviews
*

James (3-TRNT-C)

*Log**depth**circuits**for**division**and**related**problems*. SIAM J. Comput. 15 (1986), no. 4, 994-1003. ... The*divisibility**problem*, i.e., testing whether y divides x (x*and*y are n-bit numbers), is shown to be solvable by*log*-space uniform*circuits*of logarithmic*depth*. ...##
###
Page 1495 of Mathematical Reviews Vol. , Issue 87c
[page]

1987
*
Mathematical Reviews
*

“Furthermore, this paper describes Boolean

*circuits*of*depth*O(*log*n(loglogn)) which, given n-bit binary numbers, compute the product of n numbers*and*carry out integer*division*. ... n)-*depth*computation of maximal independent sets, which considerably improves the known O(*log*‘ n)-*depth*algorithm*for*a large class of graphs.” ...##
###
Quantum circuits for Toom-Cook multiplication

2018
*
Physical Review A
*

The asymptotic bounds

doi:10.1103/physreva.98.012311
fatcat:6nd6ki33wndwfdbbjd6e6rjgwy
*for*different performance metrics of the proposed quantum*circuit*are superior to the prior implementations of multiplier*circuits*using schoolbook*and*Karatsuba algorithms. ... In this paper, we report efficient quantum*circuits**for*integer multiplication using Toom-Cook algorithm. ... As quantum*division*is costlier in terms of Toffoli count*and*Toffoli*depth*than simple addition or shift operations, our overall*circuit*costs are decreased. ...##
###
A taxonomy of problems with fast parallel algorithms

1985
*
Information and Control
*

The notion of NC~-reducibility is introduced

doi:10.1016/s0019-9958(85)80041-3
fatcat:owvr44vfhvejlovtyk2x3jgjsu
*and*used throughout (*problem*R is NCl-reducible to*problem*S if R can be solved with uniform*log*-*depth**circuits*using oracles*for*S). ... 2 (solvable by uniform Boolean*circuits*of*depth*O(lof n)*and*polynomial size). ... The smallest*depth*known*for*a*log*-space uniform family of polynomial size*circuits**for**division*is O(*log*n*log**log*n) (Reif, 1983 ) (these*circuits*are probably also UE. uniform). ...##
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Probabilistic parallel prefix computation

1993
*
Computers and Mathematics with Applications
*

Hence, our

doi:10.1016/0898-1221(93)90089-e
fatcat:lz4vg24d2bf7xixr4xr2ifndci
*circuits*have O(*log**log*n)*depth**and*furthermore, have error probability which can be set to nYa*for*any constant (Y > 0. ... binary numbers by m, must have*depth*at least logf(('u*log*n)/*log*(2m) -2), if m is not*divisible*by two. ...##
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Quantum Circuits for GCD Computation with O(n n) Depth and O(n) Ancillae
[article]

2013
*
arXiv
*
pre-print

The proposed construction is based on the binary GCD algorithm

arXiv:1304.7516v1
fatcat:h2glr3dy4rdz3cly5e4flabus4
*and*it benefits from*log*-*depth**circuits**for*1-bit shift, comparison/subtraction,*and*managing ancillae. ... In this paper, we propose quantum*circuits**for*GCD computation with O(n n)*depth*with O(n) ancillae. Prior*circuit*construction needs O(n^2) running time with O(n) ancillae. ... Conditional Fred(A, B) can be implemented by*log*n*depth*with n ancillae -a*log*-*depth**circuit*to replicate the conditional on n ancillae*and*a*circuit*with*depth*1*for*Fred(A, B). ...
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