1. Dirk Stroobandt Ghent University Electronics and Information Systems Department Multi-terminal Nets do Change Conventional Wire Length Distribution Models Talk at SLIP 2001 March 31, 2001

2. Dirk Stroobandt, SLIP 2001 2 Current status of wire length prediction models Multi-terminal net model Wire length prediction for multi-terminal nets Discussion and results Outline

3. March 31, 2001Dirk Stroobandt, SLIP 2001 3 Current status of wire length prediction models Multi-terminal net model Wire length prediction for multi-terminal nets Discussion and results Outline

4. March 31, 2001Dirk Stroobandt, SLIP 2001 4 Net Terminal / pin Conventional Wire Length Models Circuit model Placement and routing model Model for the architecture Pad Channel Manhattan grid using Manhattan metric Cell Logic block T = t B p Rent’s rule Only for two-terminal nets!

5. March 31, 2001Dirk Stroobandt, SLIP 2001 5 Previous Work on Multi-terminal Nets Stroobandt & Kurdahi* Hierarchical model with recursive net degree distributions Depend on Rent exponent and several circuit properties Modelled average net degree is exact Zarkesh-Ha et al.** Closed form expression for net degree distributions Depend on Rent exponent and circuit size only Modelled average net degree is not exact * D. Stroobandt and F.J. Kurdahi. “On the characterization of multi-point nets in electronic designs.” Proc. 8 th Great Lakes Symposium on VLSI, pp. 344-350, February 1998. ** P. Zarkesh-Ha, J.A. Davis, W. Loh and J.D. Meindl. “Stochastic interconnect network fan- out distribution using Rent’s rule.” Proc. IEEE IITC, pp. 184-186, June 1998. ** P. Zarkesh-Ha, J.A. Davis, W. Loh and J.D. Meindl. “Prediction of interconnect fan-out distribution using Rent’s rule.” Proc. SLIP, pp. 107-112, April 2000.

6. March 31, 2001Dirk Stroobandt, SLIP 2001 6 Current status of wire length prediction models Multi-terminal net model Wire length prediction for multi-terminal nets Discussion and results Outline

7. March 31, 2001Dirk Stroobandt, SLIP 2001 7 Multi-terminal Nets: Stroobandt’s Model Number of new terminals T k in the cut calculated from Rent’s rule Relation new terminals – nets cut: Introduction of new parameter  Cut at level k Terminal at both levels New terminal at level k Module at level k Module at level k +1 Internal net (two new terminals; number of them = S i,k ) External net (one new terminal; number of them = S e,k ) Model based on hierarchical partitioning Pseudoconnection (no new terminals)

8. March 31, 2001Dirk Stroobandt, SLIP 2001 8 Assume: internal and external net degree distributions known at level k: W n (k) (normalized). Recursive equations are found: Cut at level k Terminal at both levels New terminal at level k Module at level k Module at level k +1 Internal net (two new terminals) External net (one new terminal) Multi-terminal Net Degree Distribution Pseudoconnection (no new terminals)

9. March 31, 2001Dirk Stroobandt, SLIP 2001 9 Numerical Evaluation and Power Law Approximation Resulting net degree distribution converges toward power law for large designs Analytical power law approximation based on value for 2- and 3-terminal nets. Net degree distribution depends on two parameters: 1)Rent exponent p 2)New parameter  and increases with increasing p and also with increasing  # Internal nets (normalized) Net degree 0.0001 0.001 0.01 0.1 1 110 theory, k=5 theory, k=10 theory, k=15 theory, k=20 theory, k=25 theory, k=30 approximation

10. March 31, 2001Dirk Stroobandt, SLIP 2001 10 Average net degrees for external and internal nets at each hierarchical level equal 2 if  =1/2 Average net degrees for all nets at each hierarchical level equal 2 if  =1/2 Average net degree in entire circuit exactly equals number of terminals over number of nets Both the internal and overall net degree are independent of the Rent exponent p for very large circuits. Average Net Degree

11. March 31, 2001Dirk Stroobandt, SLIP 2001 11 Multi-terminal nets: Zarkesh-Ha’s model No. of terminals for internal connections (per gate): Number of terminals shared through an i-point net: Average value does not correspond to actual value Model based on (recursive) terminal conservation 2 problems Overestimating T int

12. March 31, 2001Dirk Stroobandt, SLIP 2001 12 Experimental Validation Theoretical and measured distribution fit well for Stroobandt’s model. Zarkesh-Ha’s power law function deviates a lot for small net degrees. Scaled version nears Stroobandt’s power law approximation. No good fit for large net degrees but such nets are rare and there are a lot of net degrees that do not occur. ISCAS89 benchmark s953Benchmark industry3 0.1 1 10 100 1000 110100 Net degree Number of nets Measurement Average Stroobandt Zarkesh-Ha Zarkesh-Ha (scaled) 0.1 1 10 100 1000 10000 100000 110100 Net degree Number of nets Measurement Average Stroobandt Zarkesh-Ha Zarkesh-Ha (scaled)

13. March 31, 2001Dirk Stroobandt, SLIP 2001 13 Experimental Validation Zoming in on small net degrees… ISCAS89 benchmark s953Benchmark industry3 0 50 100 150 200 250 300 12345 Average Stroobandt Zarkesh-Ha Zarkesh-Ha (scaled) Measurement 0 2000 4000 6000 8000 10000 12000 14000 16000 12345 Measurement Average Stroobandt Zarkesh-Ha Zarkesh-Ha (scaled)

14. March 31, 2001Dirk Stroobandt, SLIP 2001 14 Current status of wire length prediction models Multi-terminal net model Wire length prediction for multi-terminal nets Discussion and results Outline

15. March 31, 2001Dirk Stroobandt, SLIP 2001 15 1. Partition the circuit into 4 modules of equal size such that Rent’s rule applies (minimal number of pins). 2. Partition the Manhattan grid in 4 subgrids of equal size in a symmetrical way. Donath’s* Hierarchical Placement Model * W. E. Donath. Placement and Average Interconnection Lengths of Computer Logic. IEEE Trans. on Circuits & Syst., vol. CAS-26, pp. 272-277, 1979.

16. March 31, 2001Dirk Stroobandt, SLIP 2001 16 3. Each subcircuit (module) is mapped to a subgrid. 4. Repeat recursively until all logic blocks are assigned to exactly one grid cell in the Manhattan grid. Donath’s Hierarchical Placement Model mapping

17. March 31, 2001Dirk Stroobandt, SLIP 2001 17 Length Estimation Model Donath’s assumption of uniformly distributed connections......or using the occupation probability* as a placement optimization model favouring shorter interconnections Adjacent ( A -) combination Diagonal ( D -) combination * D. Stroobandt and J. Van Campenhout. Accurate Interconnection Length Estimations for Pre- dictions Early in the Design Cycle. VLSI Design, Spec. Iss. on PD in DSM, 10 (1): 1-20, 1999.

18. March 31, 2001Dirk Stroobandt, SLIP 2001 18 Different Applications for Multi-terminal Net Models Delay-related applications - delays - power due to interconnect Routing-related applications - prediction of number of wiring layers - prediction of routing area needed - prediction of routing channel densities Source-sink lengthsSteiner lengths

19. March 31, 2001Dirk Stroobandt, SLIP 2001 19 Number of multi-terminal net connections at each hierarchical level Difference between delay- related and routing-related applications: - Source-sink pairs Assume A is source A-B at level k A-C and A-D at level k+1 Count as three connections - Entire Steiner tree lengths Segments A-B, C-D and E-F A-B and C-D at level k E-F at level k+1 Add lengths to one net length A B C D F E Level k +1 Level k Net terminal Steiner point Assumption: multi-terminal nets are split over only two partitions at every hierarchical level

20. March 31, 2001Dirk Stroobandt, SLIP 2001 20 Current status of wire length prediction models Multi-terminal net model Wire length prediction for multi-terminal nets Discussion and results Outline

21. March 31, 2001Dirk Stroobandt, SLIP 2001 21 Resulting Wire Length Distributions Source-sink pair lengthsSteiner tree lengths `Ideal' behaviour for point-to-point nets 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 110100100010000 F r a c t i o n o f w i r e s Interconnection length Stroobandt Source-sink pairs 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 110100100010000 F r a c t i o n o f w i r e s Steiner length distribution Stroobandt Interconnection length Source-sink length distribution

22. March 31, 2001Dirk Stroobandt, SLIP 2001 22 Scaling behaviour of average wire length Net segment length (previous model) Source-sink length (new model) Steiner tree length (new model) Resulting Wire Length Distributions 1 10 100 2481632641282565121024 A v e r a g e l e n g t h Two-terminal nets only Circuit size  =0.1  =0.2  =0.3  =0.4  =0.5

23. March 31, 2001Dirk Stroobandt, SLIP 2001 23 Experimental Verification Steiner tree lengths More accurate Steiner length estimates SA-based placement Steiner lengths measured by Geosteiner  New model better fits measured data (average lengths within 25%)

24. March 31, 2001Dirk Stroobandt, SLIP 2001 24 Experimental Verification Source-sink pair lengths are generally underestimated Steiner tree lengths are really close to measured ones 0 2 4 6 8 10 0.20.30.40.50.60.70.8 A v e r a g e l e n g t h Rent exponent Measured source-sink length distribution Stroobandt's length distribution New source-sink length distribution

25. March 31, 2001Dirk Stroobandt, SLIP 2001 25 Conventional wire length estimation models do not properly take multi-terminal nets into account. Fundamental difference between internal and external multi-terminal nets in a hierarchical placement model. Leads to multi-terminal net degree distribution model. Length distribution for multi-terminal nets found for delay-related and routing-related applications. Source-sink distributions are close to old net segment distributions but have a different scaling behaviour. Steiner length estimates are much more accurate than before. Conclusions