Handling soft modules in general nonslicing floorplan using Lagrangian relaxation

In the early stage of floorplan design, many modules have large flexibilities in shape (soft modules). Handling soft modules in general nonslicing floorplan is a complicated problem. Many previous works have attempted to tackle this problem using heuristics or numerical methods, but none of them can solve it optimally and efficiently. In this paper, we show how this problem can be solved optimally by geometric programming using the Lagrangian relaxation technique. The resulting Lagrangian relaxation subproblem is so simple that the optimal size of each module can be computed in linear time. We implemented this method in a simulated annealing framework based on the sequence pair representation. The geometric program is invoked in every iteration of the annealing process to compute the optimal size of each module to give the best packing. The execution time is much faster (at least 15 times faster for data sets with more than 50 modules) than that of the most updated previous work by Murata and Kuh (1998). For a benchmark data with 49 modules, we take 3.7 h in total for the whole annealing process using a 600-MHz Pentium III processor while the convex programming approach described by Murata and Koh needs seven days using a 250-MHz DEC Alpha. Our technique will also be applicable to other floorplanning algorithms that use constraint graphs to find module positions in the final packing.