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 resistor-capacitor (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.

In this paper, we present an algorithm for the minimization of total power consumption via multiple V DD assignment, multiple V TH assignment, device sizing and stack forcing, while maintaining performance requirements. These four power reduction techniques are properly encoded in genetic algorithm and evaluated simultaneously. The overhead imposed by the insertion of level converters is also taken into account. The effectiveness of each power reduction mechanism is verified, as are the combinations of different approaches. Experimental results are given for a number of 65 nm benchmark circuits that span typical circuit topologies, including inverter chains, SRAM decoders, multiplier and a 32bit carry adders. From the experimental results, we show that the combination of four low power techniques is the effective way to achieve low power budget.

Most circuit sizing tools calculate the tradeoff between each gate’s delay and power or area, and then greedily change the gate with the best tradeoff. We show this is suboptimal. Instead we use a linear program to minimize circuit power. The linear program provides a fast and simultaneous analysis of how each gate affects gates it has a path to. Our approach reduces power by up to 30 % compared to commercial software, with a 0.13um library. The runtime for posing and solving the linear program scales linearly with circuit size.

Abstract — This paper revisits and extends a general linear programming (LP) formulation to exploit multiple knobs such as multi-Lgate footprint-compatible libraries and post-layout Lgatebiasing to minimize total leakage power under timing constraints. We minimize positive timing slack for each cell according to its leakage vs. delay sensitivity, so that unnecessary leakage power consumption is saved without degrading circuit performance. A key difference between our work and previous works is that we pre-process timing libraries to estimate the linear relation – in every slew-load condition – between the gate delay and gate length by linear fitting; we then optimize total leakage power by estimating the optimal gate length for each gate using fast linear programming. With a 65GP industry testbed, and directly comparing with commercial tools, we show the QOR and runtime advantages of our method for the multi-Lgate and Lgate-biasing knobs. We also show a promising application to circuit timing legalization, a problem which frequently arises when implementation and signoff timers differ. Overall, our results show strong viability of LP based estimation and optimization: compared with the commercial tools, we: (1) shift the achievable delay-leakage tradeoff curve in a positive way, and (2) more accurately maintain prescribed timing constraints.

Modern digital circuits consist of logic gates implemented in the complementary metal oxide semiconductor (CMOS) technology. The time taken for a logic gate output to change after one or more inputs have changed is called the delay of the gate. A conventional CMOS gate is designed to have the same input to output delay irrespective of which input caused the output to change. We propose a new gate design that has different delays along various input to output paths within the gate. This is accomplished by inserting selectively sized “permanently on ” series transistors at the inputs of the logic gate. We demonstrate the use of the variable input delay CMOS gates for a totally glitch-free minimum dynamic power implementation of a digital circuit. Applying a previously described linear programming method to the c7552 benchmark circuit, we obtained a power saving of 58 % over an unoptimized design. This power consumption was 18% lower than that for an alternative low power design using conventional CMOS gates. All circuits had the same overall delay. Since the overall delay was not allowed to increase, the glitch elimination with conventional gates required insertion of delay buffers on non-critical paths. The use of the variable input delay gates drastically reduced the required number of delay buffers. 1

Leakage power has become one of the most critical design concerns for the system-level chip designer. Multi-threshold techniques have been used to reduce runtime leakage power without sacrificing performance. In this paper, we present an effective and scalable transistor-level Vth assignment approach and show leakage reduction over standard cell-level Vth assignment. The main disadvantage of transistor-level Vth assignment is increased cell library size and characterization effort. In comparison to previous approaches, our approach yields better solution quality, requires smaller cell library, is more accurate in considering the impact of Vth assignment on propagation delay, slew (transition delay) and capacitance, and is significantly faster. 1.

Abstract-This paper presents a novel approach towards the simultaneous Vt-assignment and gatesizing problem. This inherently discrete problem is formulated as a continuous problem, allowing it to be solved using any of several widely available and highly efficient non-linear optimizers. We prove that, under our formulation, the optimal solution has discrete Vts assigned to almost every gate, thus eliminating the need for a sophisticated snapping heuristic. We show that this technique performs dual-Vt assignment and gate sizing in a very efficient manner. Compared to a sensitivity based method, we achieve average leakage savings of 31 % and average total power savings of 7.4 % with very efficient runtimes. 1.

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Abstract – Power consumption and power-related issues have become a first-order concern for most designs and loom as fundamental barriers for many others. And, while the primary method used to date for reducing power has been supply voltage reduction, this technique begins to lose its effectiveness as voltages drop to sub-one volt range and further reductions in the supply voltage begin to create more problems than are solved. Under these circumstances, the process of design and the automation tools required to support that process become the critical success factors. In the last decade, huge effort has been invested to come up with a wide range of design solutions that help solve the power dissipation problem for different types of electronic devices, components and systems. These techniques range from multiple voltage assignment and dynamic voltage scaling, to RTL power management and power-aware sequential logic synthesis, to leakage power reduction techniques. This tutorial paper explains a number of representative low power design techniques from this large set. More precisely, we will describe basic techniques, applicable at RT-level and below, that have proven to hold good potential for power optimization in practical design environments. 1

In this paper, we consider an optimization problem on directed acyclic graphs which is motivated by a standard task in low power VLSI design. With each vertex v of a directed acyclic graph, we associate two delay values d0(v) ≤ d1(v) and two leakage values c0(v) ≥ c1(v). The objective is to choose one of the indices 0 or 1 for each vertex, such that in first instance the corresponding total delay along any directed path is minimal, and in second instance the total leakage is minimized. We prove that a very restricted special case of this problem already is NP-hard, and we present four different heuristic approaches to the problem. Further, we test our algorithms on ISCAS85 benchmark circuits.