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124
New Algorithms for Gate Sizing: A Comparative Study
 IN DAC
, 1996
"... Gate sizing consists of choosing for each node of a mapped network a gate implementation in the library so that some cost function is optimized under some constraints. It has a significant impact on the delay, power dissipation, and area of the final circuit. This paper compares five gate sizing alg ..."
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Cited by 29 (0 self)
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Gate sizing consists of choosing for each node of a mapped network a gate implementation in the library so that some cost function is optimized under some constraints. It has a significant impact on the delay, power dissipation, and area of the final circuit. This paper compares five gate sizing algorithms targeting discrete, nonlinear, nonunimodal, constrained optimization. The goal is to overcome the nonlinearity and nonunimodality of the delay and the power to achieve good quality results within a reasonable CPU time, e.g., handling a 10000 node network in 2 hours. We compare the five algorithms on constraint free delay optimization and delay constrained power optimization, and show that one method is superior to the others.
Optimal Wire and Transistor Sizing for Circuits with NonTree Topology
 in Proc. Int. Conf. on Computer Aided Design
, 1997
"... Conventional methods for optimal sizing of wires and transistors use linear RC circuit models and the Elmore delay as a measure of signal delay. If the RC circuit has a tree topology the sizing problem reduces to a convex optimization problem which can be solved using geometric programming. The tree ..."
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Cited by 28 (11 self)
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Conventional methods for optimal sizing of wires and transistors use linear RC circuit models and the Elmore delay as a measure of signal delay. If the RC circuit has a tree topology the sizing problem reduces to a convex optimization problem which can be solved using geometric programming. The tree topology restriction precludes the use of these methods in several sizing problems of significant importance to highperformance deep submicron design including, for example, circuits with loops of resistors, e.g., clock distribution meshes, and circuits with coupling capacitors, e.g., buses with crosstalk between the lines. The paper proposes a new optimization method which can be used to address these problems. The method uses the dominant time constant as a measure of signal propagation delay in an RC circuit, instead of Elmore delay. Using this measure, sizing of any RC circuit can be cast as a convex optimization problem which can be solved using the recently developed efficient interi...
Minimization of Dynamic and Static Power Through Joint Assignment of Threshold Voltages and Sizing Optimization
 in Proc. Int. Symp. on Low Power Electronics and Design
, 2003
"... We describe an optimization strategy for minimizing total power consumption using dual threshold voltage (Vth) technology. Significant power savings are possible by simultaneous assignment of Vth with gate sizing. We propose an efficient algorithm based on linear programming that jointly performs Vt ..."
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Cited by 28 (0 self)
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We describe an optimization strategy for minimizing total power consumption using dual threshold voltage (Vth) technology. Significant power savings are possible by simultaneous assignment of Vth with gate sizing. We propose an efficient algorithm based on linear programming that jointly performs Vth assignment and gate sizing to minimize total power under delay constraints. First, linear programming assigns the optimal amounts of slack to gates based on powerdelay sensitivity. Then, an optimal gate configuration, in terms of Vth and transistor sizes, is selected by an exhaustive local search. Benchmark results for the algorithm show 32 % reduction in power consumption on average, compared to sizing only power minimization. There is up to a 57 % reduction for some circuits. The flow can be extended to dual supply voltage libraries to yield further power savings.
Digital Circuit Optimization via Geometric Programming
 Operations Research
, 2005
"... informs ® doi 10.1287/opre.1050.0254 © 2005 INFORMS This paper concerns a method for digital circuit optimization based on formulating the problem as a geometric program (GP) or generalized geometric program (GGP), which can be transformed to a convex optimization problem and then very efficiently s ..."
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Cited by 27 (7 self)
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informs ® doi 10.1287/opre.1050.0254 © 2005 INFORMS This paper concerns a method for digital circuit optimization based on formulating the problem as a geometric program (GP) or generalized geometric program (GGP), which can be transformed to a convex optimization problem and then very efficiently solved. We start with a basic gate scaling problem, with delay modeled as a simple resistorcapacitor (RC) time constant, and then add various layers of complexity and modeling accuracy, such as accounting for differing signal fall and rise times, and the effects of signal transition times. We then consider more complex formulations such as robust design over corners, multimode design, statistical design, and problems in which threshold and power supply voltage are also variables to be chosen. Finally, we look at the detailed design of gates and interconnect wires, again using a formulation that is compatible with GP or GGP.
GradientBased Optimization of Custom Circuits Using a StaticTiming Formulation
, 1999
"... This paper describes a method of optimally sizing digital circuits on a statictiming basis. All paths through the logic are considered simultaneously and no input patterns need be specified by the user. The method is unique in that it is based on gradientbased, nonlinear optimization and can accom ..."
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Cited by 26 (4 self)
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This paper describes a method of optimally sizing digital circuits on a statictiming basis. All paths through the logic are considered simultaneously and no input patterns need be specified by the user. The method is unique in that it is based on gradientbased, nonlinear optimization and can accommodate transistorlevel schematics without the need for precharacterization. It employs efficient timedomain simulation and gradient computation for each channelconnected component. A largescale, generalpurpose, nonlinear optimization package is used to solve the tuning problem. A prototype tuner has been developed that accommodates combinational circuits consisting of parameterized library cells. Numerical results are presented.
Gate Sizing Using Incremental Parameterized Statistical Timing Analysis
 In ICCAD
, 2005
"... Abstract — As technology scales into the sub90nm domain, manufacturing variations become an increasingly significant portion of circuit delay. As a result, delays must be modeled as statistical distributions during both analysis and optimization. This paper uses incremental, parametric statistical ..."
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Cited by 25 (1 self)
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Abstract — As technology scales into the sub90nm domain, manufacturing variations become an increasingly significant portion of circuit delay. As a result, delays must be modeled as statistical distributions during both analysis and optimization. This paper uses incremental, parametric statistical static timing analysis (SSTA) to perform gate sizing with a required yield target. Both correlated and uncorrelated process parameters are considered by using a firstorder linear delay model with fitted process sensitivities. The fitted sensitivities are verified to be accurate with circuit simulations. Statistical information in the form of criticality probabilities are used to actively guide the optimization process which reduces runtime and improves area and performance. The gate sizing results show a significant improvement in worst slack at 99.86 % yield over deterministic optimization. I.
Methods for true power minimization
 in ICCAD
, 2002
"... This paper presents methods for efficient power minimization at circuit and microarchitectural levels. The potential energy savings are strongly related to the energy profile of a circuit. These savings are obtained by using gate sizing, supply voltage, and threshold voltage optimization, to minimi ..."
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Cited by 21 (6 self)
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This paper presents methods for efficient power minimization at circuit and microarchitectural levels. The potential energy savings are strongly related to the energy profile of a circuit. These savings are obtained by using gate sizing, supply voltage, and threshold voltage optimization, to minimize energy consumption subject to a delay constraint. The true power minimization is achieved when the energy reduction potentials of all tuning variables are balanced. We derive the sensitivity of energy to delay for each of the tuning variables connecting its energy saving potential to the physical properties of the circuit. This helps to develop understanding of optimization performance and identify the most efficient techniques for energy reduction. The optimizations are applied to some examples that span typical circuit topologies including inverter chains, SRAM decoders, and adders. At a delay of 20 % larger than the minimum, energy savings of 40 % to 70 % are possible, indicating that achieving peak performance is expensive in terms of energy. Energy savings of about 50 % can be achieved without delay penalty with the balancing of sizes, supplies, and thresholds. 1.
Statistical Optimization of Leakage Power Considering Process Variations using DualVth and Sizing
, 2004
"... Increasing levels of process variability in sub100nm CMOS design has become a critical concern for performance and power constraint designs. In this paper, we propose a new statistically aware DualVt and sizing optimization that considers both the variability in performance and leakage of a design ..."
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Cited by 21 (3 self)
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Increasing levels of process variability in sub100nm CMOS design has become a critical concern for performance and power constraint designs. In this paper, we propose a new statistically aware DualVt and sizing optimization that considers both the variability in performance and leakage of a design. While extensive work has been performed in the past on statistical analysis methods, circuit optimization is still largely performed using deterministic methods. We show in this paper that deterministic optimization quickly looses effectiveness for stringent performance and leakage constraints in designs with significant variability. We then propose a statistically aware dualVt and sizing algorithm where both delay constraints and sensitivity computations are performed in a statistical manner. We demonstrate that using this statistically aware optimization, leakage power can be reduced by 1535 % compared to traditional deterministic analysis. The improvements increase for strict delay constraints making statistical optimization especially important for high performance designs.
UncertaintyAware Circuit Optimization
 IN DAC
, 2002
"... Almost by definition, welltuned digital circuits have a large number of equally critical paths, which form a socalled "wall" in the slack histogram. However, by the time the design has been through manufacturing, many uncertainties cause these carefully aligned delays to spread out. Inaccuracies i ..."
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Cited by 19 (1 self)
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Almost by definition, welltuned digital circuits have a large number of equally critical paths, which form a socalled "wall" in the slack histogram. However, by the time the design has been through manufacturing, many uncertainties cause these carefully aligned delays to spread out. Inaccuracies in parasitic predictions, clock slew, modeltohardware correlation, static timing assumptions and manufacturing variations all cause the performance to vary from prediction. Simple statistical principles tell us that the variation of the limiting slack is larger when the height of the wall is greater. Although the wall may be the optimum solution if the static timing predictions were perfect, in the presence of uncertainty in timing and manufacturing, it may no longer be the best choice. The application of formal mathematical optimization in transistor sizing increases the height of the wall, thus exacerbating the problem. There is also a practical matter that schematic restructuring downstream in the design methodology is easier to conceive when there are fewer equally critical paths. This paper describes a method that gives formal mathematical optimizers the incentive to avoid the wall of equally critical paths, while giving up as little as possible in nominal performance. Surprisingly, such a formulation reduces the degeneracy of the optimization problem and can render the optimizer more effective. This "uncertaintyaware" mode has been implemented and applied to several highperformance microprocessor macros. Numerical results are included.
Tradeoffs between Gate Oxide Leakage and Delay for Dual Tox Circuits
 Proc. ACM/IEEE DAC
"... Gate oxide tunneling current (Igate) will become the dominant component of leakage in CMOS circuits as the physical oxide thickness (Tox) goes below 15˚A. Increasing the value of Tox reduces the leakage at the expense of an increase in delay, and a practical tradeoff between delay and leakage can be ..."
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Cited by 19 (0 self)
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Gate oxide tunneling current (Igate) will become the dominant component of leakage in CMOS circuits as the physical oxide thickness (Tox) goes below 15˚A. Increasing the value of Tox reduces the leakage at the expense of an increase in delay, and a practical tradeoff between delay and leakage can be achieved by assigning one of the two permissible Tox values to each transistor. In this paper, we propose an algorithm for dual Tox assignment to optimize the total leakage power under delay constraints, and generate a leakage/delay tradeoff curve. As compared to the case where all transistors are set to low Tox, our approach achieves an average leakage reduction of 83 % under 100nm models.