Optimal algorithms for planar over-the-cell routing problems

In this paper, we consider the two row maximum planar subset (TRMPS) problem in over-the-cell routing. The TRMPS problem requires selection of the maximum planar subset of nets, which can be routed between two rows of terminals in a cell row. This problem was first encountered by Gong, Liu, and Preas (1990). They stated the complexity of this problem to be unknown, and presented a min {1,k/d(S)} approximation algorithm, where k is the number of tracks available over the cell area and d(S) is the density of a solution S. We show that TRMPS problem can be solved optimally in polynomial time. We present a O(kn²) dynamic programming algorithm for the TRMPS problem, where n is the number of nets. We also present a parallel version of our algorithm, which has a complexity of O(kn). Our algorithm can also be extended to solve the TRMPS problem, in the presence of prerouted nets, a chosen subset of nets, as well as for planar channel routing. We also apply our technique to obtain a 0.5 approximation, for over the cell routing in middle terminal model, thus improving the best known existing algorithm. The weighted version of the TRMPS problem, as well as, all the extensions can also be solved in O(kn²) time