TPL-Aware Displacement-driven Detailed Placement Refinement with Coloring Constraints

To minimize the effect of process variation for a design in triple patterning lithography (TPL), it is beneficial for all standard cells of the same type to share a single coloring solution. In this paper, we investigate the TPL-aware detailed placement refinement problem under these coloring constraints. Given an initial detailed placement, the positions of standard cells are perturbed and a TPL solution complying with the coloring constraints is derived while minimizing cell displacement, lithography conflicts and stitches. We prove that this problem is NP-complete and show that it can be formulated as a mixed integer linear program. Since mixed integer linear programming is very time consuming, we propose an effective heuristic algorithm. In our approach, important adjacent pairs of standard cells are recognized firstly, since they have significant impact on cell displacement. Then a tree-based heuristic is applied to generate a good initial solution for our linear programming-based refinement. Experimental results show that compared with mixed integer linear programming, our heuristic approach is comparable in solution quality while using very short CPU runtime.