Interconnect Estimation without Packing via ACG Floorplans

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

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

7. Interconnect Estimation • Common practice. • Obtain module positions by packing. • 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. NuCAD, 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. NuCAD, 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 NuCAD, Northwestern University U.S.A.

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