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Dirk Stroobandt Ghent University Electronics and Information Systems Department

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. Outline. Current status of wire length prediction models Multi-terminal net model

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Dirk Stroobandt Ghent University Electronics and Information Systems Department

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  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

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