Dirk Stroobandt Ghent University Electronics and Information Systems Department

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

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

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

4. Model for the architecture Rent’s rule Cell p T = t B Pad Channel Manhattan grid using Manhattan metric Placement and routing model Conventional Wire Length Models Circuit model Logicblock Net Terminal / pin Only for two-terminal nets! Dirk 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. 8th 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. Dirk Stroobandt, SLIP 2001

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

7. Cut at level k Internal net (two new terminals; number of them = Si,k) External net (one new terminal; number of them = Se,k) Pseudoconnection (no new terminals) Multi-terminal Nets: Stroobandt’s Model Model based on hierarchical partitioning • Number of new terminals Tk in the cut calculated from Rent’s rule • Relation new terminals – nets cut: • Introduction of new parameter g Module at levelk +1 Module at level k Module at level k Terminal at both levels New terminal at level k Dirk Stroobandt, SLIP 2001

8. Cut at level k Internal net (two new terminals) External net (one new terminal) Pseudoconnection (no new terminals) Multi-terminal Net Degree Distribution • Assume: internal and external net degree distributions known at level k: Wn(k) (normalized). • Recursive equations are found: Module at levelk +1 Module at level k Module at level k Terminal at both levels New terminal at level k Dirk 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: • Rent exponent p • New parameter g • and increases with increasing p and also with increasing g # Internal nets (normalized) 1 theory, k=5 theory, k=10 theory, k=15 theory, k=20 0.1 theory, k=25 theory, k=30 approximation 0.01 0.001 0.0001 1 10 Net degree Dirk Stroobandt, SLIP 2001

10. Average Net Degree • Average net degrees for external and internal nets at each hierarchical level equal 2 if g=1/2 • Average net degrees for all nets at each hierarchical level equal 2 if g=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. Dirk Stroobandt, SLIP 2001

11. Multi-terminal nets: Zarkesh-Ha’s model Model based on (recursive) terminal conservation • 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 Overestimating Tint 2 problems Dirk Stroobandt, SLIP 2001

12. 1000 100000 Measurement Measurement Average Average 10000 Stroobandt Stroobandt 100 Zarkesh-Ha Zarkesh-Ha 1000 Zarkesh-Ha (scaled) Zarkesh-Ha (scaled) Number of nets 10 Number of nets 100 10 1 1 0.1 0.1 1 10 100 1 10 100 Net degree Net degree Experimental Validation ISCAS89 benchmark s953 Benchmark industry3 • 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. Dirk Stroobandt, SLIP 2001

13. Measurement Measurement 300 16000 Average Average 14000 Stroobandt 250 Stroobandt Zarkesh-Ha 12000 Zarkesh-Ha 200 Zarkesh-Ha (scaled) 10000 Zarkesh-Ha (scaled) 8000 150 6000 100 4000 50 2000 0 0 1 2 3 4 5 1 2 3 4 5 Experimental Validation • Zoming in on small net degrees… ISCAS89 benchmark s953 Benchmark industry3 Dirk Stroobandt, SLIP 2001

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

15. Donath’s* Hierarchical Placement Model 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. • * W. E. Donath. Placement and Average Interconnection Lengths of Computer Logic. IEEE Trans. on Circuits & Syst., vol. CAS-26, pp. 272-277, 1979. Dirk Stroobandt, SLIP 2001

16. mapping Donath’s Hierarchical Placement Model 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. Dirk Stroobandt, SLIP 2001

17.   Length Estimation Model Donath’s assumption of uniformly distributed connections... Adjacent (A-) combination Diagonal (D-) combination ...or using the occupation probability* as a placement optimization model favouring shorter interconnections • * 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. Dirk Stroobandt, SLIP 2001

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

19. Level k +1 C A E F B D Level k Net terminal Steiner point 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 Assumption: multi-terminal nets are split over only two partitions at every hierarchical level Dirk Stroobandt, SLIP 2001

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

21. 1 1 `Ideal' behaviour for point-to-point nets Source-sink length distribution Stroobandt 0.1 Stroobandt 0.1 Source-sink pairs Steiner length distribution 0.01 0.01 s s e e r r i i w w 0.001 0.001 f f o o n n o o i i t 0.0001 t 0.0001 c c a a r r F F 1e-05 1e-05 1e-06 1e-06 1e-07 1e-07 1 10 100 1000 10000 1 10 100 1000 10000 Interconnection length Interconnection length Resulting Wire Length Distributions Source-sink pair lengths Steiner tree lengths Dirk Stroobandt, SLIP 2001

22. 100 g =0.1 g =0.2 h g =0.3 t g n e l g =0.4 e 10 g a r e g v =0.5 A T wo-terminal net s only 1 2 4 8 16 32 64 128 256 512 1024 Circuit size Resulting Wire Length Distributions • Scaling behaviour of average wire length • Net segment length • (previous model) • Source-sink length (new model) • Steiner tree length (new model) Dirk Stroobandt, SLIP 2001

23. 1000 Theoretical Steiner length distribution Experimentally measured length distribution Previous distribution of Stroobandt 100 s e r i w f o 10 r e b m u N 1 0.1 1 10 100 Interconnection length 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%) Dirk Stroobandt, SLIP 2001

24. 10 12 Measured source-sink length distribution Measured Steiner length distribution Stroobandt's length distribution Stroobandt's length distribution New source-sink length distribution 10 New Steiner length distribution 8 8 h h t t g 6 g n n e e l l e e 6 g g a a r r e e 4 v v A A 4 2 2 0 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Rent exponent Rent exponent Experimental Verification Source-sink pair lengths are generally underestimated Steiner tree lengths are really close to measured ones Dirk Stroobandt, SLIP 2001

25. Conclusions • 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. Dirk Stroobandt, SLIP 2001