Legitimate Skew Clock Routing with Buffer Insertion

Buffer insertion is an efficient technique in interconnect optimization. This paper presents a graph based algorithm for optimal buffer insertion under accurate delay models. In our algorithm, a signal is accurately represented by a finite ramp which is characterized by two parameters, shift time and transition time. Any accurate delay model, such as delay models based on the transmission line model and SPICE simulations, can be incorporated into our algorithm. The algorithm determines the optimal number of buffers and their locations on a wire such that some optimization objective is satisfied. Two typical examples of such optimization objectives are minimizing the 50% threshold delay and minimizing the transition time. Both can be easily determined in our algorithm. We show that the buffer insertion problem can be reduced to a shortest path problem. The algorithm can be easily extended for simultaneous buffer insertion and wire-sizing, and complexity is still polynomial. The algorithm can also be extended to deal with problems such as buffer insertion subject to transition time constraints at any position along the wire.

The classic buffer insertion algorithm of van Ginneken has time and space complexity O(n/sup 2/), where n is the number of possible buffer positions. For more than a decade, van Ginneken's algorithm has been the foundation of buffer insertion. In this paper, we present a new algorithm that computes the same optimal buffer insertion, but runs much faster. For 2-pin nets, our time complexity is O(nlogn) and space complexity is O(n). For multipin nets, our time complexity is O(nlog/sup 2/n) and space complexity is O(nlogn). The speedup is achieved by four novel techniques: predictive pruning, candidate tree, fast redundancy check, and fast merging. On industrial test cases, the new algorithms is 2-80 times faster than van Ginneken's algorithm and uses 1/4-1/500 of the memory. Since van Ginneken's algorithm and its variations are used by most existing algorithms on buffer insertion and buffer sizing, our new algorithm significantly improves the performance of all these algorithms. The predictive pruning technique has been applied to buffer cost minimization (Shi et al., 2004), and significantly improved the running time.

Most existing buffer insertion algorithms, such as van Ginneken's algorithm, consider only individual nets. As a result, these algorithms tend to over buffer when applied to combinational circuits, since it is difficult to decide how many buffers to insert in each net. Recently, Sze, et al. [1] proposed a path-based algorithm for buffer insertion in combinational circuits. However their algorithm is inefficient for large circuits when there are many critical paths.In this paper, we present a new buffer insertion algorithm for combinational circuits such that the timing requirements are met and the buffer cost is minimized. Our algorithm iteratively inserts buffers in the circuit to improve the circuit delay. The core of this algorithm is simple but effective technique that guides the search for a good buffering solution. Experimental results on ISCAS85 circuits show that our new algorithm on average uses 36% less buffers and runs 3 times faster than Sze's algorithm.

Given a routing tree for a multi-pin net, two algorithms extending the van Ginneken algorithm (van Ginneken, 1990) for concurrent flip-flop and buffer insertion were presented in (Cocchini, 2002). One algorithm called MiLa targets at minimizing the latency, and the other algorithm called GiLa aims to find a feasible solution subject to given latency constraints imposed on sinks. However, they both do not consider the case where buffer/flip-flop blockages are present. In this paper, we enhance the MiLa algorithm and GiLa algorithm to consider blockage avoidance by finding alternative registered-buffered paths between each internal node inside a blockage and its parent node. The experimental results show that in comparison to the MiLa algorithm, our approach is able to find a solution with the same latency (for about half of the test cases) or even better latency (for the remaining test cases) and the same wirelength, while the buffer/flip-flop usage and CPU time are comparable or acceptable. In comparison to the GiLa algorithm, our approach is able to find a feasible solution for each test case while the Gila algorithm fails to do so for several test cases.

Most existing buffering algorithms neglect the impact of inductance on circuit performance, which causes large error in circuit analysis and optimization. Even for the approaches considering inductance effects, their delay models are too simplistic to catch the actual performance. As delay-length dependence is approaching linear with inductance effect (Ismail and Friedman, 1999), fewer buffers are needed to reduce RLC delay. This motivates this work to propose a new algorithm for RLC buffer insertion. In this paper, a new buffer insertion algorithm considering inductance for intermediate and global interconnect is proposed, based on downstream impedance instead of traditional downstream capacitance. A new pruning technique that provides tremendous speedup and a new frequency estimation method that is very accurate in delay computation are also proposed. Experiments on industrial netlists demonstrate that our new algorithm reduces the number of buffers up to 34.4% over the traditional van Ginneken's algorithm that ignores inductance. Our impedance delay estimation is very accurate compared to SPICE simulations, with only 10% error while the delay model used in the previous RLC algorithm has 20% error (Ismael et al., 2001). The accurate delay model not only reduces the number of buffers, but also brings high fidelity to the buffer solutions. Incorporating slew constraints, the algorithm is accelerated by about 4times with only slight degradation in solution quality

Buffer insertion is a fundamental technology for VLSI interconnect optimization. Several existing buffer insertion algorithms have evolved from van Ginneken's classic algorithm. In this work, we extend van Ginneken's algorithm to handle blockages in the layout. Given a Steiner tree containing a Steiner point that overlaps a blockage, a local adjustment is made to the tree topology that enables additional buffer insertion candidates to be considered. This adjustment is adaptive to the demand on buffer insertion and is incurred only when it facilitates the maximal slack solution. This approach can be combined with any performance-driven Steiner tree construction. The overall time complexity has linear dependence on the number of blockages and quadratic dependence on the number of potential buffer locations. Experiments on several large nets confirm that high-quality solutions can be obtained through this technique with little CPU cost.

With shrinking VLSI feature sizes and increasing overall chip areas, buffering has emerged as an effective solution to the problem of growing interconnect delays in modern designs. The problem of buffer insertion in a single net has been the focus of most previous researches. However, efficient algorithms for buffer insertion in whole circuits are generally needed. In this paper, we relate the timing constrained minimal buffer insertion problem to the min-cost flow dual problem, and propose two algorithms based on min-cost flow and min-cut techniques, respectively, to solve it in combinational circuits. We compare our approaches to a traditional approach based on Lagrangian relaxation. Experimental results demonstrate that our approaches are efficient and effective. On the average, our approaches achieve 45% and 39% reduction, respectively, on the number of buffers inserted in comparison to the traditional approach.

As gate delays decrease faster than wire delays for each technology generation, buffer insertion becomes a popular method to reduce the interconnect delay. Several modern buffer insertion algorithms (e.g., [7, 6, 15]) are based on van Ginneken's dynamic programming paradigm [14]. However, van Ginneken's original algorithm does not control buffering resources and tends to over-buffering, thereby wasting area and power. It has been a major open problem whether it is possible to optimize slack and at the same time minimize the buffer usage.This paper settles this open problem by showing that for arbitrary integer cost functions, the problem is NP-complete. We also extend the pre-buffer slack technique [12] to minimize the buffer cost. This technique can significantly reduce the running time and memory in buffer cost minimization problem. The experimental results show that our algorithm can speed up the running time up to 17 times and reduces the memory to 1/30 of traditional best know algorithm. Finally, we show how to efficiently deal with multiway merge in buffer insertion.

We study the problem of performing buffer insertion in the context of a given layout in a practical situation, there are restrictions on where buffers may be inserted while routing over such regions may be possible (e.g., due to the presence of macro cells). As a result, it is desirable to perform route planning and biffer insertion simultaneously. Further it is necessary that such an algorithm be aware of the tradeoff between cost (e.g. total capacitance) and delay. In this context we propose the Delay Reduction to Cost Ratio (DRCR) problem and present a fast algorithm for the same. Solutions identified by the algorithm are characterized with respect to the overall cost vs. performance tradeoff curve. Computational experiments demonstrate the viability of the approach.

In this paper, we investigate the problem of computing a lower bound on the number of buffers required when given a maximum clock slew rate (or rise time) constraint and a predefined clock tree. Using generalized properties of published CMOS timing models, we formulate a novel nonlinear buffer insertion problem. Next, we derive an algorithm that bounds the capacitance for each buffer stage without sacrificing the generality of the timing models. With this capacitance bound we formulate a second linear buffer insertion problem, which we solve optimally in O(n) time. The basic formulation and algorithm are extended to include a skew upper bound constraint. Using these algorithms we propose further algorithmic extensions that allow area and phase delay tradeoffs. Our results are verified using SPICE3e2 simulations with MCNC MOSIS 2.0 /spl mu/ models and parameters. Experiments with these test cases show that the buffer insertion algorithms proposed herein can be used effectively for designs with high clock speeds and small skews.

Buffer insertion is an increasingly critical optimization for achieving timing closure, and the number of buffers required increases significantly with technology migration. It is imperative for an automated buffer insertion algorithm to be able to efficiently optimize tens of thousands of nets. One must also be able to effectively navigate the existing layout, including handling large blockages, blockages with holes specifically for buffers, specially allocated buffer blocks, placement porosity, and routing congestion. The algorithm must also be flexible enough to know when to use and when not to use expensive layout resources. Although several previous works have addressed buffer insertion in the presence of blockages, this is the first to present a complete solution that can manage the physical layout environment.

As devices and lines shrink into the deep submicron range, the propagation delay of signals can be effectively improved by repowering the signals using intermediate buffers placed within the routing trees. Almost no existing timing driven floorplanning and placement approaches consider the option of buffer insertion. As such, they may exclude solutions, particularly early in the design process, with smaller overall area and better routability. In this paper, we propose a new methodology in which buffered trees are used to estimate wire delay during floorplanning. Instead of treating delay as one of the objectives, as done by the majority of previous work, we formulate the problem in terms of delay bounded buffered trees (DBB-tree) and propose an efficient algorithm to construct a DBB spanning tree for use during floorplanning. Experimental results show that the algorithm is very effective. Using buffer insertion at the floorplanning stage yields significantly better solutions in terms of both chip area and total wire length.

In this study a new approach for skew compensation in energy recovery clock distribution networks is introduced by manipulating the operating speed of the flip-flops. The STMicroelectronics 90 nm technology allows the use of devices with different threshold voltages, namely: high threshold voltage (HVT), standard threshold voltage (SVT) and low threshold voltage (LVT). Three types of flip-flops of equal input load: fast , standard and slow are used. Timing parameters of the flip-flops are adjusted by manipulating the switching threshold of the clock port of the flip-flops. A fast/slow flip-flop has a shorter/longer T DQ delay, compared with a standard flip-flop for the same setup time (T DCLK ). Distributing flip-flops according to their delay requirements would reduce the effect of the clock's skew on the outputs of sequentially adjacent flip-flops. Owing to the slow rise time of the sinusoidal clock signal used in energy recovery clock distribution networks compared to the conventional square-wave clock, the skew that can be compensated for in energy recovery clock distribution networks using this approach would be much higher than in square-wave clock distribution networks. This approach increases the skew bounds required by algorithms to balance the skew in the clock tree leading to reduced design complexity. Theoretical analysis and simulation results using STMicroelectronics 90 nm technology at a clock frequency of 500 MHz show that this approach is feasible and effective where a skew of up to 6.2% 1of the clock period can be compensated for in the example used. In addition, constructing clock trees using the skew slack provided in the proposed technique in a new modified differed merge embedding (DME) algorithm on five benchmarks have shown that the proposed technique enables an average reduction of 11.5% in total wire length and 53.2% reduction in the number of wire elongations. Balancing the skew in the clock tree using buffers was not considered here since inserting a buffer in the clock's path eliminates the energy recovery property. As an example of illustrating the proposed methodology, the authors have used the Elmore delay model with a selected energy recovery flip-flop to verify the practicality of the proposed scheme. Better results can be obtained by using different flip-flops. The method can generally be applied to energy recovery or square-wave clocking if different flip-flops of various speeds are used.

Historical index structure for reducing insertion and search cost in LBS

Clock Skew Scheduling has become a common practice in state-of-the-art FPGAs with the introduction of delay chains on the clock path in the hardware of both Xilinx and Intel <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">®</sup> FPGAs, as well as clock skew scheduling algorithms in the CAD tools. Ideally, globally optimal solutions are sought to find the best solution across the entire design. However, using Mixed-Integer Linear Programming (MILP) to find such optimal solutions has a large and sometimes unrealistic runtime cost, especially for bigger designs. Besides, a high runtime does not necessarily correlate to improved performance. We present, in this paper, techniques to reduce the runtime of mixed-integer linear programming approaches to clock skew scheduling, and provide alternatives that can achieve optimal or near-optimal performance gain with a fraction of the MILP runtime cost.