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76
Random walks in a supply network
 Proceedings of the ACM/IEEE Design Automation Conference
, 2003
"... This paper presents a power grid analyzer based on a random walk technique. A lineartime algorithm is first demonstrated for DC analysis, and is then extended to perform transient analysis. The method has the desirable property of localizing computation, so that it shows massive benefits over conve ..."
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Cited by 49 (12 self)
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This paper presents a power grid analyzer based on a random walk technique. A lineartime algorithm is first demonstrated for DC analysis, and is then extended to perform transient analysis. The method has the desirable property of localizing computation, so that it shows massive benefits over conventional methods when only a small part of the grid is to be analyzed (for example, when the effects of small changes to the grid are to be examined). Even for the full analysis of the grid, experimental results show that the method is faster than existing approaches and has an acceptable error margin. This method has been applied to test circuits of up to 2.3M nodes. For example, for a circuit with 70K nodes, the solution time for a single node was 0.42 sec and the complete solution was obtained in 17.6 sec.
Power grid analysis using random walks
 IEEE Transactions on ComputerAided Design of Integrated Circuits and Systems
, 2005
"... This paper presents a lineartime algorithm for the DC analysis of a power grid, based on a random walk technique. Experimental results show that the proposed method is faster than existing approaches and has an acceptable error margin. It also has a desirable property of localizing computation, and ..."
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Cited by 30 (3 self)
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This paper presents a lineartime algorithm for the DC analysis of a power grid, based on a random walk technique. Experimental results show that the proposed method is faster than existing approaches and has an acceptable error margin. It also has a desirable property of localizing computation, and can be extended to RCnetwork transient analysis. This method has been applied to circuits of up to 70K nodes, for which the solution time for a single node was 0.42 sec and the complete solution was obtained in 17.6 sec. 1.
The Power Grid Transient Simulation in Linear Time Based on 3D AlternatingDirectionImplicit Method
 In ICCAD
, 2003
"... The rising power consumption and clock frequency of very large scale integration technology demand robust and stable power delivery. Extensive transient simulations on largescale power delivery structures are required to analyze power delivery fluctuation caused by dynamic IR drop and Ldi/dt drop a ..."
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Cited by 16 (0 self)
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The rising power consumption and clock frequency of very large scale integration technology demand robust and stable power delivery. Extensive transient simulations on largescale power delivery structures are required to analyze power delivery fluctuation caused by dynamic IR drop and Ldi/dt drop as well as package and onchip resonance. In this paper, we develop a novel and efficient transient simulation algorithm for the power distribution networks. Our algorithm, threedimensional (3D) transmissionlinemodeling alternatingdirectionimplicit (TLMADI) method, first models the power delivery structure as 3D transmission line shuntnode structure and transfer those equations to the telegraph equation. Finally, we solve it by the alternating direction implicit method. The 3D TLMADI method, with linear runtime and memory requirement, is also unconditionally stable, which ensures that the time steps are not limited by any stability requirement. Extensive numerical simulation results show that the proposed algorithm is not only over 300 000 times faster than SPICE but also extremely memory saving and accurate.
A fast decoupling capacitor budgeting algorithm for robust onchip power delivery
 in Proc. Asia South Pacific Design Automation Conf. (ASPDAC
, 2004
"... AbstractIn this paper, we present an efficient method to budget onchip decoupling capacitors (decaps) to optimize power delivery networks in an area efficient way. Our algorithm is based on an efficient gradientbased nonlinear programming method for searching the solution. Our contributions are ..."
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Cited by 14 (7 self)
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AbstractIn this paper, we present an efficient method to budget onchip decoupling capacitors (decaps) to optimize power delivery networks in an area efficient way. Our algorithm is based on an efficient gradientbased nonlinear programming method for searching the solution. Our contributions are an efficient gradient computation method (timedomain merged adjoint network) and a novel equivalent circuit modeling technique to speed up the optimization process. Experimental results demonstrate that the algorithm is capable of efficiently optimizing very large scale P/G networks. I
Hierarchical randomwalk algorithms for power grid analysis
 In Proceedings of the Asia/South Pacific Design Automation Conference
, 2004
"... Abstract — This paper presents a power grid analyzer that combines a divideandconquer strategy with a randomwalk engine. A singlelevel hierarchical method is first described and then extended to multilevel and “virtuallayer ” hierarchy. Experimental results show that these algorithms not only ..."
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Cited by 13 (4 self)
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Abstract — This paper presents a power grid analyzer that combines a divideandconquer strategy with a randomwalk engine. A singlelevel hierarchical method is first described and then extended to multilevel and “virtuallayer ” hierarchy. Experimental results show that these algorithms not only achieve speedups over generic randomwalk method, but also are more robust in solving various types of industrial circuits. For example, a 71Knode circuit is solved in 4.16 seconds, showing a more than 4 times speedup over the generic method; a 348Knode wirebond power grid, for which the performance of the generic method degrades, is solved in 75.88 seconds. I.
Floorplan and power/ground network cosynthesis for fast design convergence
 In ISPD ’06: Proceedings of the 2006 international symposium on Physical design
, 2006
"... As technology advances, the metal width decreases while the global wire length increases. This trend makes the resistance of the power wire increase substantially. Further, the threshold voltage scales nonlinearly, raising the ratio of the threshold voltage to the supply voltage and making the volta ..."
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Cited by 8 (2 self)
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As technology advances, the metal width decreases while the global wire length increases. This trend makes the resistance of the power wire increase substantially. Further, the threshold voltage scales nonlinearly, raising the ratio of the threshold voltage to the supply voltage and making the voltage (IR) drop in the power/ground (P/G) network a serious problem in modern IC design. Traditional P/G network analysis methods are often very computationally expensive, and it is thus not feasible to cosynthesize P/G network with floorplan. To make the cosynthesis feasible, we need not only an efficient, effective, and flexible floorplanning algorithm, but also a very efficient, yet sufficiently accurate P/G network analysis method. In this paper, we present a method for floorplan and P/G network cosynthesis based on an efficient P/G network analysis scheme and the B*tree floorplan representation. We integrate the cosynthesis into a commercial design flow to develop an effective power integrity (IRdrop) driven design methodology. Experimental results based on a realworld circuit design and the MCNC benchmarks show that our design methodology successfully fixes the IRdrop errors earlier at the floorplanning stage and thus enables the singlepass design convergence.
An algebraic multigrid solver for analytical placement with layout based clustering
 In Proceedings of the IEEE/ACM Design Automation Conference, ACM
, 2003
"... An efficient matrix solver is critical to the analytical placement. As the size of the matrix becomes huge, the multilevel methods turn out to be more efficient and more scalable. Algebraic Multigrid (AMG) is a multilevel technique to speedup the iterative matrix solver [10]. We apply the algebraic ..."
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Cited by 7 (0 self)
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An efficient matrix solver is critical to the analytical placement. As the size of the matrix becomes huge, the multilevel methods turn out to be more efficient and more scalable. Algebraic Multigrid (AMG) is a multilevel technique to speedup the iterative matrix solver [10]. We apply the algebraic multigrid method to solve the linear equations that arise from the analytical placement. A layout based clustering scheme is put forward to generate coarsening levels for the multigrid method. The experimental results show that the algebraic multigrid solver is promising for analytical placement.
Partial Random Walk for Large Linear Network Analysis
 in Proc. International Symposium on Circuits and Systems (ISCAS
, 2004
"... This paper proposes a new simulation algorithm for analyzing large power distribution networks, modeled as linear RLC circuits, based on a novel partial random walk concept. The random walk simulation method has been shown to be an efficient way to solve for a small number of nodes in a lager power ..."
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Cited by 6 (1 self)
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This paper proposes a new simulation algorithm for analyzing large power distribution networks, modeled as linear RLC circuits, based on a novel partial random walk concept. The random walk simulation method has been shown to be an efficient way to solve for a small number of nodes in a lager power distribution network [6], but the algorithm becomes expensive to solve for nodes that are more than a few. In this paper, we combine direct methods like LU factorization with the random walk concept to solve power distribution networks when a significant number of node waveforms is required. We also apply an equivalent circuit modeling method to speed up the direct simulation of subcircuits. Experimental results show that the resulting algorithm, called partial random walk (PRW), has significant advantages over the pure random walk method especially when the VDD/GND nodes are sparse and accuracy requirement is high. 1.
Statistical Analysis of OnChip Power Delivery Networks Considering Lognormal Leakage Current Variations with Spatial Correlation
, 2007
"... As the technology scales into 90nm and below, processinduced variations become more pronounced. In this paper, we propose an efficient stochastic method for analyzing the voltage drop variations of onchip power grid networks, considering lognormal leakage current variations with spatial correlat ..."
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Cited by 4 (1 self)
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As the technology scales into 90nm and below, processinduced variations become more pronounced. In this paper, we propose an efficient stochastic method for analyzing the voltage drop variations of onchip power grid networks, considering lognormal leakage current variations with spatial correlation. The new analysis is based on the Hermite polynomial chaos (PC) representation of random processes. Different from the existing Hermite PC based method for power grid analysis [10], which models all the random variations as Gaussian processes without considering spatial correlation, the new method consider both wire variations and subthreshold leakage current variations, which are modeled as lognormal distribution random variables, on the power grid voltage variations. To consider the spatial correlation, we apply orthogonal decomposition to map the correlated random variables into independent variables. Our experiment results show that the new method is more accurate than the Gaussianonly Hermite PC method using the Taylor expansion method for analyzing leakage current variations. It is two orders of magnitude faster than the Monte Carlo method with small variance errors. We also show that the spatial correlation may lead to large errors if not being considered in the statistical analysis.