Simultaneous interconnect delay and crosstalk noise optimization through gate sizing using game theory

The continuous scaling trends of interconnect wires in deep submicron (DSM) circuits result in increased interconnect delay and crosstalk noise. In this work, we develop a new postlayout gate sizing algorithm for simultaneous optimization of interconnect delay and crosstalk noise. The problem of postlayout gate sizing is modeled as a normal form game and solved using Nash equilibrium. The crosstalk noise induced on a net depends on the size of its driver gate and the size of the gates driving its coupled nets. Increasing the gate size of the driver increases the noise induced by the net on its coupled nets, whereas increasing the size of the drivers of coupled nets increases the noise induced on the net itself, resulting in a cyclic order dependency leading to a conflicting situation. It is pointed out that solving the postroute gate sizing problem for crosstalk noise optimization is difficult due to its conflicting nature. Game theory provides a natural framework for handling such conflicting situations and allows optimization of multiple parameters. By utilizing this property of game theory, the cyclic dependency of crosstalk noise on its gate sizes can be solved as well as the problem of gate sizing for simultaneous optimization of interconnect delay and crosstalk noise can be effectively modeled, whose objective function is again conflicting in nature. We have implemented two different strategies in which games are ordered according to 1) the noise criticality and 2) delay criticality of nets. The time and space complexities of the proposed gate sizing algorithm are linear in terms of the number of gates in the design. Experimental results for a noise critically ordered game theoretic approach on several medium and large open core designs indicate average improvements of 15.48 percent and 18.56 percent with respect to Cadence place and route tools in terms of interconnect delay and crosstalk noise, respectively, without any area overhead or the need for rerouting. Further, the algorithm performs significantly better than simulated annealing and genetic search as established through experimental results. A mathematical proof of existence for the Nash equilibrium solution for the proposed gate sizing formulation is also provided