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FlowMap: An Optimal Technology Mapping Algorithm for Delay Optimization in LookupTable Based FPGA Designs
 IEEE TRANS. CAD
, 1994
"... The field programmable gatearray (FPGA) has become an important technology in VLSI ASIC designs. In the past a few years, a number of heuristic algorithms have been proposed for technology mapping in lookuptable (LUT) based FPGA designs, but none of them guarantees optimal solutions for general Bo ..."
Abstract

Cited by 317 (41 self)
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The field programmable gatearray (FPGA) has become an important technology in VLSI ASIC designs. In the past a few years, a number of heuristic algorithms have been proposed for technology mapping in lookuptable (LUT) based FPGA designs, but none of them guarantees optimal solutions for general Boolean networks and little is known about how far their solutions are away from the optimal ones. This paper presents a theoretical breakthrough which shows that the LUTbased FPGA technology mapping problem for depth minimization can be solved optimally in polynomial time. A key step in our algorithm is to compute a minimum height Kfeasible cut in a network, which is solved optimally in polynomial time based on network flow computation. Our algorithm also effectively minimizes the number of LUTs by maximizing the volume of each cut and by several postprocessing operations. Based on these results, we have implemented an LUTbased FPGA mapping package called FlowMap. We have tested FlowMap on a large set of benchmark examples and compared it with other LUTbased FPGA mapping algorithms for delay optimization, including Chortled, MISpgadelay, and DAGMap. FlowMap reduces the LUT network depth by up to 7% and reduces the number of LUTs by up to 50% compared to the three previous methods.
Almost Optimal Lower Bounds for Small Depth Circuits
, 1986
"... We give improved lower bounds for the size of small depth circuits computing several functions. In particular we prove almost optimal lower bounds for the size of parity circuits. Further we show that there are functions computable in polynomial size and depth k but requires exponential size when ..."
Abstract

Cited by 274 (8 self)
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We give improved lower bounds for the size of small depth circuits computing several functions. In particular we prove almost optimal lower bounds for the size of parity circuits. Further we show that there are functions computable in polynomial size and depth k but requires exponential size
Pseudorandom generators without the XOR Lemma (Extended Abstract)
, 1998
"... Impagliazzo and Wigderson [IW97] have recently shown that if there exists a decision problem solvable in time 2 O(n) and having circuit complexity 2 n) (for all but finitely many n) then P = BPP. This result is a culmination of a series of works showing connections between the existence of har ..."
Abstract

Cited by 135 (23 self)
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of hard predicates and the existence of good pseudorandom generators. The construction of Impagliazzo and Wigderson goes through three phases of "hardness amplification" (a multivariate polynomial encoding, a first derandomized XOR Lemma, and a second derandomized XOR Lemma) that are composed
A RiskFactor Model Foundation for RatingsBased Bank Capital Rules
 Journal of Financial Intermediation
, 2003
"... When economic capital is calculated using a portfolio model of credit valueatrisk, the marginal capital requirement for an instrument depends, in general, on the properties of the portfolio in which it is held. By contrast, ratingsbased capital rules, including both the current Basel Accord and i ..."
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Cited by 283 (1 self)
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When economic capital is calculated using a portfolio model of credit valueatrisk, the marginal capital requirement for an instrument depends, in general, on the properties of the portfolio in which it is held. By contrast, ratingsbased capital rules, including both the current Basel Accord
On the Hopf lemma
, 2007
"... Abstract. The Hopf Lemma for second order elliptic operators is proved to hold in domains with C 1,α, and even less regular, boundaries. It need not hold for C 1 boundaries. Corresponding results are proved for second order parabolic operators. 1. The Hopf Lemma, a purely local result, is a basic to ..."
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Cited by 3 (2 self)
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Abstract. The Hopf Lemma for second order elliptic operators is proved to hold in domains with C 1,α, and even less regular, boundaries. It need not hold for C 1 boundaries. Corresponding results are proved for second order parabolic operators. 1. The Hopf Lemma, a purely local result, is a basic
On the ring lemma
, 2008
"... The sharp ring lemma states that if n >= 3 cyclically tangent discs with pairwise disjoint interiors are externally tangent to and surround the unit disc, then no disc has a radius below cn = (F 2 n−1 +F2 n−2 −1)−1 – where Fk denotes the k th Fibonacci number – and that the lower bound is attaine ..."
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Cited by 1 (0 self)
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The sharp ring lemma states that if n >= 3 cyclically tangent discs with pairwise disjoint interiors are externally tangent to and surround the unit disc, then no disc has a radius below cn = (F 2 n−1 +F2 n−2 −1)−1 – where Fk denotes the k th Fibonacci number – and that the lower bound
The expression lemma ⋆
"... Abstract. Algebraic data types and catamorphisms (folds) play a central role in functional programming as they allow programmers to define recursive data structures and operations on them uniformly by structural recursion. Likewise, in objectoriented (OO) programming, recursive hierarchies of objec ..."
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Cited by 1 (1 self)
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in deriving refactorings that turn sufficiently disciplined functional programs into OO programs of a designated shape and vice versa. Key words: expression lemma, expression problem, functional object, catamorphism, fold, the composite design pattern, program calculation, distributive law, free monad, cofree
Lemma: Let
, 1992
"... Diophantus probably knew, and Lagrange[L] proved, that every positive integer can be written as a sum of four perfect squares. Jacobi[J] proved the stronger result that the number of ways in which a positive integer can be so written 3 equals 8 times the sum of its divisors that are not multiples of ..."
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of many great mathematicians, and we encourage the reader to look up Grosswald’s[G] erudite masterpiece on this subject. The crucial part of our proof is played by two simple identities, that we state as one Lemma.
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