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Log Depth Circuits for Division and Related Problems

Paul W. Beame, Stephen A. Cook, H. James Hoover
1986 SIAM journal on computing (Print)  
We present optimal 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.  ... 
doi:10.1137/0215070 fatcat:pntuddlzzvgzngh6ianmnmpid4

Log Depth Circuits For Division And Related Problems

P.W. Beame, S.A. Cook, H.J. Hoover
25th Annual Symposium onFoundations of Computer Science, 1984.  
We present optimal 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.  ... 
doi:10.1109/sfcs.1984.715894 dblp:conf/focs/BeameCH84 fatcat:epcp5i6irjdhtjzhfm2zhxvdj4

Division Is In Uniform TC0 [chapter]

William Hesse
2001 Lecture Notes in Computer Science  
Integer 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.  ... 
doi:10.1007/3-540-48224-5_9 fatcat:vu6dhiadb5drjduq47va4sfobq

Ranking and formal power series

Alberto Bertoni, Danilo Bruschi, Massimiliano Goldwurm
1991 Theoretical Computer Science  
As a consequence ranking 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] .  ... 
doi:10.1016/0304-3975(91)90144-q fatcat:e6bcguyfqfejdpcv3vk6yqphia

Uniform constant-depth threshold circuits for division and iterated multiplication

William Hesse, Eric Allender, David A. Mix Barrington
2002 Journal of computer and system sciences (Print)  
1 2 HESSE, ALLENDER 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.  ... 
doi:10.1016/s0022-0000(02)00025-9 fatcat:vwqdpnv7njghzi2326rxqprtca

Boolean circuits versus arithmetic circuits

Joachim von zur Gathen, Gadiel Seroussi
1991 Information and Computation  
Over Q 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  ... 
doi:10.1016/0890-5401(91)90078-g fatcat:aydwnfiodfgy3bizzgauso2sei

Recursively divisible problems [chapter]

Rolf Niedermeier
1996 Lecture Notes in Computer Science  
We call 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.  ... 
doi:10.1007/bfb0009494 fatcat:h7z2lslhivcljlyafhzrdtvl5e

On Relating Time and Space to Size and Depth

Allan Borodin
1977 SIAM journal on computing (Print)  
In doing so, we 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.  ... 
doi:10.1137/0206054 fatcat:f45h6aubu5br3cgwhdz2t6nnp4

Depth efficient neural networks for division and related problems

K.-Y. Siu, J. Bruck, T. Kailath, T. Hofmeister
1993 IEEE Transactions on Information Theory  
The 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  ... 
doi:10.1109/18.256501 fatcat:5ahnr5q4i5fwropv64l46qgo5q

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

Srijit Dutta, Debjyoti Bhattacharjee, Anupam Chattopadhyay
2018 Physical Review A  
The asymptotic bounds 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.  ... 
doi:10.1103/physreva.98.012311 fatcat:6nd6ki33wndwfdbbjd6e6rjgwy

A taxonomy of problems with fast parallel algorithms

Stephen A. Cook
1985 Information and Control  
The notion of NC~-reducibility is introduced 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).  ... 
doi:10.1016/s0019-9958(85)80041-3 fatcat:owvr44vfhvejlovtyk2x3jgjsu

Probabilistic parallel prefix computation

J.H. Reif
1993 Computers and Mathematics with Applications  
Hence, our 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.  ... 
doi:10.1016/0898-1221(93)90089-e fatcat:lz4vg24d2bf7xixr4xr2ifndci

Quantum Circuits for GCD Computation with O(n n) Depth and O(n) Ancillae [article]

Mehdi Saeedi, Igor L. Markov
2013 arXiv   pre-print
The proposed construction is based on the binary GCD algorithm 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).  ... 
arXiv:1304.7516v1 fatcat:h2glr3dy4rdz3cly5e4flabus4
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