A. Chiu. Complexity of parallel arithmetic using the Chinese Remainder representation. Master's thesis, U. Wisconsin-Milwaukee, 1995. G. Davida, supervisor. Home/Search Document Details and Download
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Uniform Constant-Depth Threshold Circuits for Division and.. - Hesse, Allender (2002) (1 citation) (Correct)
....compute Divisionand Powering; three layers suffice for IteratedMultiplication[44] It remained unknown whether Division could be computed in logarithmic space, until Chiu, Davida, and Litow [21] presented an improved algorithm that can be implemented in L uniform TC . Chiu s Master s thesis [20] also shows that division lies in fully uniform NC . In the next section, we present a simplified division algorithm that was inspired by [21] Our presentation is in terms of descriptive complexity, but can equally well be thought of explicitly in terms of circuits. Each step is a description ....
A. Chiu. Complexity of parallel arithmetic using the Chinese Remainder representation. Master's thesis, U. Wisconsin-Milwaukee, 1995. G. Davida, supervisor.
Uniform Circuits for Division: Consequences and Problems - Allender, Barrington, Hesse (2000) (2 citations) (Correct)
....with MAJORITY quantifiers and a single extra numerical predicate. This predicate calculates powers modulo a prime of O(log n) bits. We then consider various algorithms for powering modulo a small prime, and their consequences for the uniformity of division circuits. Following an argument of Chiu [10], powering modulo a small prime can be achieved by 1 [7] claimed only P uniform NC 1 , but it was observed later by Reif [23] that their algorithm is implementable in TC 0 . fully uniform circuits of logarithmic depth and fan in two ( Ruzzo uniform NC 1 ) and hence division is itself in ....
....A natural approach to determining the complexity of all these problems, then, is to consider a variety of algorithmic attacks on POW. It is easy to see that POW is in L, and hence that FOM POW is contained in L uniform TC 0 . A clever application of the result of [11] due to Chiu [10], gets us further: Proposition 4.3 The problem POW is in Ruzzo uniform NC 1 [24] Hence FOM POW is contained in both uniform NC 1 and NC 1 uniform TC 0 . Proof. of Proposition 4.3) Since POWERING and DI VISION are each in L uniform NC 1 by [11] for any k we can raise a k bit ....
A. Chiu. Complexity of parallel arithmetic using the Chinese Remainder representation. Master's thesis, U. WisconsinMilwaukee, 1995. G. Davida, supervisor.
The Division Breakthroughs - Allender (2001) (Correct)
....to compute this in polynomial time) A new approach was needed. 4 Breaking the Logspace Barrier Andrew Y. Chiu received his MS degree from the University of Wisconsin at Milwaukee in August, 1995. A mathematical prodigy, he subsequently left computer science to enter law school. His MS thesis [12] remained unknown to most of the community for several years. No paper summarizing its contributions was presented at any of the conferences where researchers usually announce their latest theorems. No technical re port was published on ECCC or on any of the other repositories for such material. ....
....theorems. No technical re port was published on ECCC or on any of the other repositories for such material. We all owe a great debt to Chiu s advisor, George Davida, and to his collaborator Bruce Litow, for preparing a paper building on his work for journal submission [13] Chiu s MS thesis [12] shows that division and ITERATED MULTIPLICATION lie in uniform NC 1 . In this survey, I ll sketch for now only a proof that these problems lie in Luniform TC 0 . As observed in the previous section, it is sufficient to show that one can convert from CRR to binary in L uniform TC 0 . ....
A. Chiu. Complexity of parallel arithmetic using the Chinese Remainder representation. Master's thesis, U. Wisconsin-Milwaukee, 1995. G. Davida, supervisor.
NC1 Division - Chiu, Davida, Litow (2000) (1 citation) Self-citation (Chiu) (Correct)
....1 Proof : Integer division is in NC1 because each of the following steps is in NC1. ffl Conversion from binary notation to CRR. See Theorem 7. ffl CRR division. See Theorem 11. ffl Conversion from CRR to binary notation. See Theorem 12. 2 4 Remarks It appears likely, based on material in [4] that integer division can be computed by the apparently more restrictive UE uniform O(log n) depth circuits. See [15] for a definition 15 of this kind of uniformity. However, we do not pursue this here because it seems to us that the most important consequence of an NC1 division algorithm is ....
Andrew Chiu. Complexity of parallel arithmetic using the chinese remainder representation. Master's thesis, U. Wisconsin-Milwaukee, 1995. G. Davida, supervisor.
Uniform Constant-Depth Threshold Circuits for Division and .. - Hesse, Allender, al. (2002) (1 citation) (Correct)
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A. Chiu. Complexity of parallel arithmetic using the Chinese Remainder representation. Master's thesis, U. Wisconsin-Milwaukee, 1995. G. Davida, supervisor.
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