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A Tutorial on Geometric Programming
"... A geometric program (GP) is a type of mathematical optimization problem characterized by objective and constraint functions that have a special form. Recently developed solution methods can solve even largescale GPs extremely efficiently and reliably; at the same time a number of practical problems ..."
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Cited by 123 (11 self)
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A geometric program (GP) is a type of mathematical optimization problem characterized by objective and constraint functions that have a special form. Recently developed solution methods can solve even largescale GPs extremely efficiently and reliably; at the same time a number of practical problems, particularly in circuit design, have been found to be equivalent to (or well approximated by) GPs. Putting these two together, we get effective solutions for the practical problems. The basic approach in GP modeling is to attempt to express a practical problem, such as an engineering analysis or design problem, in GP format. In the best case, this formulation is exact; when this isn’t possible, we settle for an approximate formulation. This tutorial paper collects together in one place the basic background material needed to do GP modeling. We start with the basic definitions and facts, and some methods used to transform problems into GP format. We show how to recognize functions and problems compatible with GP, and how to approximate functions or data in a form compatible with GP (when this is possible). We give some simple and representative examples, and also describe some common extensions of GP, along with methods for solving (or approximately solving) them.
Gate sizing to radiation harden combinational logic
, 2006
"... A gatelevel radiation hardening technique for cost– effective reduction of the soft error failure rate in combinational logic circuits is described. The key idea is to exploit the asymmetric logical masking probabilities of gates, hardening gates that have the lowest logical masking probability to ..."
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Cited by 50 (3 self)
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A gatelevel radiation hardening technique for cost– effective reduction of the soft error failure rate in combinational logic circuits is described. The key idea is to exploit the asymmetric logical masking probabilities of gates, hardening gates that have the lowest logical masking probability to achieve cost– effective tradeoffs between overhead and soft error failure rate reduction. The asymmetry in the logical masking probabilities at a gate is leveraged by decoupling the physical from the logical (Boolean) aspects of soft error susceptibility of the gate. Gates are hardened to singleevent upsets (SEUs) with specified worst case characteristics in increasing order of their logical masking probability, thereby maximizing the reduction in the soft error failure rate for specified overhead costs (area, power, and delay). Gate sizing for radiation hardening uses a novel gate (transistor) sizing technique that is both efficient and accurate. A full set of experimental results for process technologies ranging from 180 to 70 nm demonstrates the costeffective tradeoffs that can be achieved. On average, the proposed technique has a radiation hardening overhead of 38.3%, 27.1%, and 3.8 % in area, power, and delay for worst case SEUs across the four process technologies.
GLOBAL INTERCONNECT SIZING AND SPACING WITH CONSIDERATION OF COUPLING CAPACITANCE
, 1997
"... This paper presents an efficient approach to perform global interconnect sizing and spacing (GISS) for multiple nets to minimize interconnect delays with consideration of coupling capacitance, in addition to area and fringing capacitances. We introduce the formulation of symmetric and asymmetric wir ..."
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Cited by 47 (14 self)
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This paper presents an efficient approach to perform global interconnect sizing and spacing (GISS) for multiple nets to minimize interconnect delays with consideration of coupling capacitance, in addition to area and fringing capacitances. We introduce the formulation of symmetric and asymmetric wire sizing and spacing. We prove two important results on the symmetric and asymmetric effectivefringing properties which leadtoavery effective bound computation algorithm to compute the upper and lower bounds of the optimal wire sizing and spacing solution for all nets under consideration. Our experiments show that in most cases the upper and lower bounds meet quickly after a few iterations and we actually obtain the optimal solution. To our knowledge, this is the first indepth study of global wire sizing and spacing for multiple nets with consideration of coupling capacitance. Experimental results show that our GISS solutions lead to substantial delay reduction than existing single net wiresizing solutions without consideration of coupling capacitance.
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 41 (8 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.
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. Inac ..."
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Cited by 31 (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.
Closing the Gap Between ASIC and Custom: An ASIC Perspective
 DAC 2000
, 2000
"... We investigate the differences in speed between applicationspecific integrated circuits and custom integrated circuits when each are implemented in the same process technology, with some examples in 0.25 micron CMOS. We first attempt to account for the elements that make the performance different ..."
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Cited by 30 (0 self)
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We investigate the differences in speed between applicationspecific integrated circuits and custom integrated circuits when each are implemented in the same process technology, with some examples in 0.25 micron CMOS. We first attempt to account for the elements that make the performance different and then examine ways in which tools and methodologies may close the performance gap between applicationspecific integrated circuits and custom circuits.
A New Statistical Optimization Algorithm for Gate Sizing
"... In this paper, we approach the gate sizing problem in VLSI circuits in the context of increasing variability of process and circuit parameters as technology scales into the nanometer regime. We present a statistical sizing approach that takes into account randomness in gate delays by formulating a ..."
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Cited by 29 (2 self)
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In this paper, we approach the gate sizing problem in VLSI circuits in the context of increasing variability of process and circuit parameters as technology scales into the nanometer regime. We present a statistical sizing approach that takes into account randomness in gate delays by formulating a robust linear program that can be solved efficiently. We demonstrate the efficiency and computational tractability of the proposed algorithm on the various ISCAS’85 benchmark circuits. Across the benchmarks, compared to the deterministic approach, the power savings range from 23 − 30 % for the same timing target and the yield level, the average power saving being 28%. The runtime is reasonable, ranging from a few seconds to around 10 mins, and grows linearly.
CostEffective Radiation Hardening Technique for Combinational Logic
, 2004
"... A radiation hardening technique for combinational logic circuits is described. The key idea is to exploit the asymmetric logical masking probabilities of gates, hardening gates that have the lowest logical masking probability to achieve costeffective tradeoffs between overhead and soft error failu ..."
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Cited by 27 (0 self)
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A radiation hardening technique for combinational logic circuits is described. The key idea is to exploit the asymmetric logical masking probabilities of gates, hardening gates that have the lowest logical masking probability to achieve costeffective tradeoffs between overhead and soft error failure rate reduction. The technique, which decouples the physical from the logical aspects of soft error susceptibility of a gate, uses a novel gate (transistor) sizing technique that is both efficient and accurate (in comparison to SPICE). A full set of experimental results demonstrate the costeffective tradeoffs that can be achieved.
Gate Sizing Using Lagrangian Relaxation Combined with a Fast GradientBased PreProcessing Step
 PROC. ICCAD
, 2002
"... In this paper, we present Forge, an optimal algorithm for gate sizing using the Elmore delay model. The algorithm utilizes Lagrangian relaxation with a fast gradientbased preprocessing step that provides an effective set of initial Lagrange multipliers. Compared to the previous Lagrangianbased ..."
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Cited by 26 (1 self)
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In this paper, we present Forge, an optimal algorithm for gate sizing using the Elmore delay model. The algorithm utilizes Lagrangian relaxation with a fast gradientbased preprocessing step that provides an effective set of initial Lagrange multipliers. Compared to the previous Lagrangianbased approach, Forge is considerably faster and does not have the inefficiencies due to difficulttodetermine initial conditions and constant factors. We compared the two algorithms on 30 benchmark designs, on a Sun UltraSparc60 workstation. On average Forge is 200 times faster than the previously published algorithm. We then improved Forge by incorporating a slewratebased convex delay model, which handles distinct rise and fall gate delays. We show that Forge is 15 times faster, on average, than the AMPS transistorsizing tool from Synopsys, while achieving the same delay targets and using similar total transistor area.
Crosstalkdriven interconnect optimization by simultaneous gate and wire sizing
 IEEE Transactions on ComputerAided Design of Integrated Circuits and systems
, 2000
"... Abstract—Noise, as well as area, delay, and power, is one of the most important concerns in the design of deep submicrometer integrated circuits. Currently existing algorithms do not handle simultaneous switching conditions of signals for noise minimization. In this paper, we model not only physical ..."
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Cited by 25 (3 self)
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Abstract—Noise, as well as area, delay, and power, is one of the most important concerns in the design of deep submicrometer integrated circuits. Currently existing algorithms do not handle simultaneous switching conditions of signals for noise minimization. In this paper, we model not only physical coupling capacitance, but also simultaneous switching behavior for noise optimization. Based on Lagrangian relaxation, we present an algorithm which can optimally solve the simultaneous noise, area, delay, and power optimization problem by sizing circuit components. Our algorithm, with linear memory requirement and linear runtime, is very effective and efficient. For example, for a circuit of 6144 wires and 3512 gates, our algorithm solves the simultaneous optimization problem using only 2.1MB memory and 19.4min runtime to achieve the precision of within 1 % error on a SUN Sparc UltraI workstation. Index Terms—Deep submicrometer, gate sizing, interconnect, performance optimization, physical design, routing. I.