Minimal buffer insertion in clock trees with skew and slew rate constraints

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 μ 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