cell objects in the cell library (mirroring ignored) 395 395 # mapped cell objects (mirroring ignored, one direction) 74 111 # mapped cell objects (mirroring considered, two directions) 128 191 # mapped cell objects (mirroring considered, four directions) 211 303

The EB shots under the five cases were sought out by solving mathematical problem instances and are shown in Table 5. In the table, the parenthesized values show the numbers of cell objects. The areas and delay times of Circuit 2 are shown in Table 6. In our experiment, areas and delay times of Circuit 1 were not examined because the benchmark

Circuit 1 15268 (74) 17300 16915 (128) 30925 29733 (211) Circuit 2 51055 (111) 69760 69589 (191) 93774 91187 (303)

> Area [݉݉ଶ] 768,685.5 654,885.0 654,885.0 Delay time [ns] 7.15 6.97 7.09

According to Table 5, as the number of cell objects increases, in other words, the number of cell directions increases, the number of EB shots increases. This is because the reduction of cell directions enables more cell functions to be on a CP aperture mask and to be projected with the CP. The area of Circuit 2 under Case 1 was largest among the five cases as shown in Table 6 because only a single direction, that is a basic direction, was adopted for place-androute. This was because the gaps between cell areas came to arise and each cell area got to own its own power and ground lines. Theoretically speaking, the design with the four cell directions should be best among the five cases with regard to area and delay time. Similarly, the design with the two cell directions should be intermediate. The experimentally obtained values of areas do not reflect this supposition. The delay time of Circuit 2 under Cases 2 and 3 was found best. This is because the CAD tool returned approximate solutions of layout and happened to result against the supposition. Note that the values shown in Table 1 and Table 6 are nothing more than the ones the CAD tool reported. If a design obtained with the two cell directions is better than a design obtained with the four cell directions, the design of

Comparing the number of EB shots of Circuit 1 under Case 4 with that under Case 5, 3.85% reduction of the number of EB shots was achieved. The difference in the numbers of EB shots was caused by solving the problem instances exactly or approximately. The optimal sets of cells was selected exactly under Cases 3 and 5 while sets of cells was selected

two-cell direction may be adopted as a design of four-cell direction.

Table 4. Benchmark circuits

circuit was not logic-synthesizable.

Table 5. EB shots and cell objects

Table 6. Area and delay time of Circuit 2

Feature size [m] 0.25 0.35
