1. Interconnect Estimation without Packing via ACG Floorplans Jia Wang and Hai Zhou Electrical & Computer Engineering Northwestern University U.S.A.

2. Jan. 2005NuCAD, Northwestern University U.S.A. 2 Floorplan and Constraint Graph Horizontal and vertical constraint graphs –At least one relation between any pair of modules. (1) There is redundancy in constraint graphs. –Transitive edges. (2) –Over-specification: more than one relation between two modules. (3) (1) (2) (3)

3. Jan. 2005NuCAD, Northwestern University U.S.A. 3 Reduce the Redundancy Remove all the transitive edges and allow exactly one relation between any pair of modules. –Remaining edges form a total order on vertices. Quadratic number of edges are still possible… –The basic structure could be identified as crosses in the graph.

4. Jan. 2005NuCAD, Northwestern University U.S.A. 4 Adjacent Constraint Graph (ACG) Definitions –Exactly one relation between every pair of modules. –No transitive edges. –No cross. Properties –Symmetry Still an ACG when edge directions reversed or edge types exchanged (H vs. V). –Constraint graph Packing is as simple as longest path computations. –Number of edges Conjecture: O(n). –Preserve geometrical adjacency information. Close to adjacency graph.

5. Jan. 2005NuCAD, Northwestern University U.S.A. 5 Edge Classification Every edge belongs to one of the following four classes. 1.The first V (H) edge. 2.The following V (H) edges before the first H (V ) edge. 3.The first H (V ) edge. 4.The edges follows the first H (V ) edge.

6. Jan. 2005NuCAD, Northwestern University U.S.A. 6 Reduced ACG Observation –Class 4 edges can be implied from other edges. –If we reverse all the edges … Class 1 or 4 edges remain unchanged. Class 2 and 3 edges exchanges. Reduced ACG –Simplify ACG by removing all the class 4 edges. –Total order remains. –Symmetry still holds. –No longer a constraint graph. Need to build a corresponding ACG for packing. –Number of edges is at most 3n.

7. Jan. 2005NuCAD, Northwestern University U.S.A. 7 Interconnect Estimation Common practice. 1.Obtain module positions by packing. 2.Compute wire lengths by module positions. Possible problems. –Area optimal packing is not necessarily interconnect optimal. –Interconnect optimization is more important in current floorplan problems. Estimate interconnect wire length directly on Reduced ACG. –Most edges in a Reduced ACG represent the closeness of the modules.

8. Jan. 2005NuCAD, Northwestern University U.S.A. 8 Estimation Algorithms The shortest path length on Reduced ACG is the estimation. –Such path exists because of the class 1 edges. –Large dead space degrades the estimation. However, floorplans with large dead space, e.g., larger than 50%, are not common in either area or interconnect optimizations. General estimation. –Ignore module sizes. –Path length is the number of the edges, i.e., edge length is 1. –Breadth-first-search computes the path. Accurate estimation. –Consider module sizes. –Edge length is half of the sum of the two module sizes. –Dynamic programming computes the path.

9. Jan. 2005NuCAD, Northwestern University U.S.A. 9 Experiments Three GSRC benchmarks: n100, n200, n300. Use simulated annealing to reach a floorplan of around 50% dead space for each benchmark. Compare the physical wire length and estimations for every pair of modules. –Physical wire length is computed as the Manhattan distances between the two modules after packing. General Estimation for n100 Accurate Estimation for n100

10. Jan. 2005NuCAD, Northwestern University U.S.A. 10 Experiments (Cont.) General Estimation for n300 Accurate Estimation for n300 General Estimation for n200 Accurate Estimation for n200