D. Minimum Cost Minimum Diameter A-tree Heuristic Iii. Minimum Diameter A-tree Algorithm for Pd-msr Problem A. Review of A-tree Algorithm Performance Driven Routing with Multiple Sources

Experimental results are shown in Tables 1 and 2. We compare tree length (TL), the sum total path length for all p i ! p j pairs (TPL), and tree diameter D. Also included are the maximum delay between any source-sink pair (MD) and the average of maximum delays for each source (AMD), averaged over all runs. Delays were measured from the input transition to the output reaching 90% of its nal value. Delays are shown in nanoseconds, while tree and path lengths are shown in centimeters. Table 3 shows the solutions by 1-Steiner, MD A-tree, and MC MD A-tree when only a (randomly chosen) subset of pairs are critical. This table assumes the CMOS IC parameters, and 8 points per test set. We report the total weighted path length TWPL, the required weighted path length RWPL (a lower bound), and the maximum and average maximum delays for the critical pairs. Compared with the best known Steiner tree heuristic, minimum cost minimum diameter A-trees ooer maximum delay improvements of 1% to 11% and 4% to 13% for 0:5 CMOS ICs and MCMs. Average maximum delays were improved by 0% to 4% and 1% to 12%. When only a subset of point pairs are critical, maximum delay improvements for 0:5 CMOS IC routings were from 3% to 7%. Table 3: Comparison of routing results for test cases where only a limited number of point pairs have W(p i ; p j) = 1. Figure 1: The smallest tilted rectangle containing the points, and a smallest tilted square which contains the rectangle. In fact, it is not always necessary to place the root of the A-tree at the center of an STS. It is easy to see that as long as the root satisses the constraint d(p i ; r) + d(r; p j) D for all p i and p j , the tree will have minimum diameter. In the Euclidean plane, the region which satisses the constraint is simply the intersection of all ellipses formed by pairs p i , p j. We deene an octilinear segment to be a segment that is either horizontal, vertical, or have slope 1. In the Manhattan plane, the set of points satisfying d(p i ; r) + d(r; p j) D is an octilinear ellipse (OE), bounded by no more than eight octilinear segments. If d(p i ; p j) = D 0 …