# NBTI and Power Reduction Using an Input Vector Control and Supply Voltage Assignment Method
## 3 Methodology
### Minimum NBTI Vector Selection Considering Power Effect
In order to solve this problem, the delay degradation is substituted for the postaging delay in the ILP formulation, and the ILP formulation for NBTI mitigation is modified as Equation (18):  
\[ \begin{split}\text{min}\ \Delta D\\ s.t.\ \ \ \ \Delta D\geq\left(D(p_{i})+\sum\limits_{g_{ij}\in p_{i}} \Delta D\big{(}g_{ij}\big{)}-D_{0}\right),\forall i\end{split}  \tag{18}\]  
where \(D_{0}\) is the maximum propagation delay on all paths in the initial state. Then, the co-optimization ILP formulation can be described as Equation (19):  
\[E=\Delta D/\Delta D_{min}+P_{\text{leak}}/P_{\text{leak},min} \tag{19}\]  
where \(\Delta D\) is the delay degradation during the circuit lifetime, and \(P_{\text{leak}}\) is the leakage power. \(\Delta D_{min}\) and \(P_{\text{leak},min}\) are the results obtained by ILP formulation in Equations (18) and (17), respectively. Because \(\Delta D\) is defined as difference between the largest original delay and the largest postaging delay in the entire path set, \(\Delta D/\Delta D_{min}\) and \(P_{\text{leak}}/P_{\text{leak},min}\) have the same order of magnitude. Then, we can obtain the global best input vector which can consider the NBTI effect and leakage power simultaneously, using the co-optimization ILP formulation in Equation (19). We name the ILP formulation for NBTI mitigation only, the ILP formulation for leakage reduction only, and our proposed co-optimization ILP formulation as IVC #1, IVC #2, and IVC #3, respectively, in the following section. In Figure 6, we compare the degradation and leakage reduction results of the above three IVC methods in ten years on c880 and c3540 circuits. In Figure 6, the x-axis and y-axis represent ten years of NBTI-induced delay degradation and leakage power, respectively, for each input vector when the circuit uses that input vector in standbymode. Point A (the rectangular point) and B (the triangle point) correspond to the result obtained by ILP formulation for NBTI mitigation and leakage reduction only, respectively. Point C (the asterisk point) represents the result of our proposed co-optimization method. The delay degradation and leakage power minimization results of the three ILP formulations are also compared with the results of Monte Carlo (MC) simulations. The iteration number for MC simulation is 100,000. We can see that compared with the MC simulation, the IVC #1 and IVC #2 methods can minimize the delay degradation and leakage power, respectively. However, the input vector obtained by the ILP formulation for a single effect cannot simultaneously reduce NBTI and leakage. When the degradation is minimized, the leakage is relatively high, and vice versa. In comparison, the input vector obtained by the proposed co-optimization ILP formulation can provide a balanced tradeoff between NBTI and leakage. Both specifications are near the potential optimal result, so it can then be combined with the SVA method to save more power during the circuit's lifetime. The effectiveness of our proposed IVC and SVA combination method will be discussed in the next section.