UTACO: a unified timing and congestion optimization algorithm for standard cell global routing

Timing performance and routability are two main goals of global routing. These two targets are mutually conflicting if we view and handle their effects independently. In this paper, we adopt a shadow price mechanism to incorporate the two issues into one unified objective function. We formulate global routing as a multicommodity flow problem. The objective function is the slack of congestion with the clock period as the delay limit from registers and inputs to registers and outputs. The multicommodity flow is expressed by a linear-programming formulation as a primal problem. We then convert the primal problem into a dual formulation using the shadow price as the variables. The shadow price of a net is the sum of its congestion price and timing price. The primal and dual formulation offers theoretical upper and lower bounds of the routing solution. Throughout the optimization process, the difference of the two bounds reduces, which provides the user's insight into the quality of the solutions. Based on the new formulation, this paper presents the UTACO algorithm for standard cell global routing.