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Prediction of Interconnect Net-Degree Distribution Based on Rent’s Rule Tao Wan and Malgorzata Chrzanowska- Jeske Department of Electrical and Computer.

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Presentation on theme: "Prediction of Interconnect Net-Degree Distribution Based on Rent’s Rule Tao Wan and Malgorzata Chrzanowska- Jeske Department of Electrical and Computer."— Presentation transcript:

1 Prediction of Interconnect Net-Degree Distribution Based on Rent’s Rule Tao Wan and Malgorzata Chrzanowska- Jeske Department of Electrical and Computer Engineering Portland State University This work has been partially supported by the NSF grant CCR 9988402

2 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske2 Outline Motivation Previous work Our models: - Model 1: Linear - Model 2: Exponential - Model 3: Weighted exponential Experimental results Behavior of internal fraction factor f Results Conclusions

3 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske3 A priori Interconnect Prediction Interconnect: importance of wires increases (they do not scale as devices do) - delay, power, area A priori: - before the actual layout is generated - very little information is known To narrow the solution search space To reduce the number of iterations To evaluate new architectures

4 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske4 Rent’s Rule Rent’s rule: underlying assumption for most of the interconnect prediction techniques. T = number of I/Os Circuit of B blocks T: number of terminals B: number of blocks k: Rent coefficient - average number of terminals per block p: Rent exponent - interconnect complexity - level of placement optimization

5 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske5 Rent’s Rule Region I: good approximation Region II : high B - technology constraints, limited pins - psychological constraints, designers Region III : low B - mismatch between local and global interconnect complexity, FPGAs Region I Region III Region II

6 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske6 What is a Net-Degree Distribution? A net-degree distribution is a collection of values, indicating, for each net degree i, how many nets have a net degree equal i. net degree12345678910 number of nets 056858652312 223 ISCAS89 s1423 net-degree distribution Internal 3-terminal net External 3-terminal net

7 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske7 Why Net-Degree Distribution ? For a priori interconnect prediction, the most important parameter to predict is wire length Most current estimation techniques on wire length distribution are based on two-terminal nets - not accurate, lots of wires in current design are multi-terminal nets - multi-terminal nets do change conventional wire length distribution models

8 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske8 Why Net-Degree Distribution ? To evaluate total wire length - we have to accurately estimate the net-degree distribution Current net-degree distribution models not accurate - Zarkesh-Ha et al.’s model underestimates the number of nets with small net degrees - Stroobandt’s model underestimates the number of nets with high net degrees The goal of this research was to derive more accurate net-degree distribution models

9 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske9 Previous Work 143 290 282

10 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske10 Zarkesh-Ha et al.’s Model A closed form expression for fan-out (net- degree) distribution Depends on Rent’s parameters k, p and circuit size B only Underestimates the distribution for low net degrees, finds inaccurate total number of nets and average net degree

11 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske11 Zarkesh-Ha et al.’s Model Zarkesh-Ha et al. made a few inaccurate assumptions Did not consider external nets Their original circuit model Their circuit model with external nets

12 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske12 Zarkesh-Ha et al.’s Model They assumed that adding one block to the boundary of m-1 blocks would only introduce m-terminal nets, which is not valid in most circumstances - new nets don’t have to be connected to all the blocks in the boundary boundary of 3 blocks boundary of 2 blocks

13 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske13 Zarkesh-Ha et al.’s Model They changed the definition of T Net (m), more specifically, they changed the boundary m blocks N g blocks They derived the expression of T Net (m) for the boundary of m blocks But when they applied it to the whole circuit consisting of N g blocks, they forgot to substitute N g for m

14 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske14 Stroobandt’s Model Hierarchical model with recursive net-degree distribution Depends on Rent’s parameters k, p, circuit size B, and several circuit parameters, such as internal fraction factor f or ratio of number of new input terminals to total number of new terminals  More accurate total number of nets and average net degree Underestimate the distribution for high net degrees - Maximum fan-out (maximum net degree -1)

15 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske15 Stroobandt’s Model Stroobandt’s derivation is based on the relationship between the number of new terminals and the number of nets cut His derivation did not consider the effect of Region II of Rent’s rule and did not take into account - technology constraint - psychological constraint Cut at level k Terminal at both levels New terminal at level k Module at level k Module at level k +1 Internal net External net Unchanged net (Source: Stroobandt’s paper)

16 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske16 Zarkesh-Ha et al. were interested in the number of terminals shared through an i-terminal net for each block. Stroobandt analyzed the number of new terminals generated by cutting nets. We directly look into the net-degree distribution - net-degree distribution - internal net-degree distribution - external net-degree distribution - their normalized expressions, and We record the change of and when circuit grows Our Models

17 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske17 Model 1: Linear Grow from m-1 blocks to m blocks: 1, 2, 3, 4… new external net new internal net unchanged external net internal net Outside terminals of m-1 blocks have equal probability to be selected for combining internal fraction factor f f = new internal net new internal + new external

18 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske18 Model 1: Linear Input: M, f, p, k Output: net-degree distribution N Algorithm net-degree-distribution(M) begin initialize vectors: nN i, nN e, and N; nN e (2)=k;(*nN e distribution for a single block*) for i=2 to M do update nN i and nN e ; end for for i=2 to M do N(i)=round(nN i (i)*M+nN e (i)*M); end for end. …M blocks A circuit of M blocks 12 34

19 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske19 Model 2: Exponential Grow from m blocks to 2m blocks: 1, 2, 4, 8… internal net unchanged external net new external net new internal net Outside terminals of m blocks have equal probability to be selected for combining internal fraction factor f f = new internal net new internal + new external

20 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske20 Model 2: Exponential … blocks A circuit of M blocks 12 34 Algorithm net-degree-distribution(M) Input: M, f, p, k Output: net-degree distribution N begin initialize vectors: nN i, nN e, and N; nN e (2)=k;(*nN e distribution for a single block*) N=ceil(log 2 M);(*Determine the number of iterations*) for h=0 to N-1 do m1=2 h ; m2=2 h+1 ; update nN i and nN e ; end for for i=2 to M do N(i)=round(nN i (i)*M+nN e (i)*M); end for end.

21 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske21 Model 1 and Model 2

22 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske22 Model 3: Weighted Exponential Technology constraint and psychological constraint will result in the fact that nets with higher degrees are generated with higher probability We defined a vector W(n) to associate the probability to the degree of newly generated nets

23 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske23 Model 3: Weighted Exponential internal net unchanged external net new external net: (1-f)P new internal net: fP internal net unchanged external net new external net: (1-f)PW(2) new internal net: fPW(2)

24 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske24 Experimental Results M: number of circuit blocks T: number of terminals N tot : total number of nets - found for ISCAS89 benchmarks k and p: Rent’s parameters - partitioning and Region I power-law fitting f: internal fraction factor Net-degree distribution - counted by a Matlab program for ISCAS89 benchmarks Region IRegion IIIRegion II k = 2.45 p = 0.40

25 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske25 Experimental Results Δ

26 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske26 The Least Squares Method: NameZarkesh-Ha et al.StroobandtLinearExponentialWeighted exponential s27370000 s208.1177611934618160 s29817815671191605302 s38661953393440132072126 s3447126525940640264 s3496961551994659280 s38221074671909721263 s4442481147601143133 s52635121998526525861345 s526n39302051516226021318 s5109754143611128421103 s420.1591592934611431555 s713465096381202710162 s9532103028917839011214 The distribution with smallest is the best approximation

27 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske27 Average Net Degree Average net degree and total number of nets are the most important parameters to represent interconnect complexity error = |prediction-measurement| measurement

28 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske28 Total Number of Nets error = |prediction-measurement| measurement

29 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske29 Maximum Fan-out error = |prediction-measurement| measurement Maximum fan-out is important to predict the length of the longest path

30 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske30 Behavior of Internal Fraction Factor f Internal fraction factor f, similarly to Rent’s rule, exhibits Region II and Region III We applied a best fitting procedure to get a refined value of f. With the refined value of f, our models and Stroobandt’s model can provide even more accurate estimation Region IIIRegion IRegion II f = 0.47 f = 0.51 f = new internal net new internal + new external

31 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske31 Comparison on Average Net Degree

32 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske32 Comparison on Total Number of Nets

33 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske33 Comparison on Maximum Fan-out

34 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske34 Conclusions We found an alternative way to derive net- degree distribution models by directly using normalized net-degree distribution - linear growth model - exponential growth model - weighted exponential model We accurately measured Rent’s parameters k and p, internal fraction factor f and net-degree distribution of ISCAS89 benchmarks, which can be good references for other researchers

35 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske35 Conclusions For the first time, we observed that internal fraction factor f, similarly to Rent’s rule, also exhibited Region II and Region III. With a refined value of f, our models and Stroobandt’s model can provide even more accurate estimation of net-degree distribution Weighted exponential growing model is more accurate than any other currently available net-degree distribution model Avg. net degreeTotal num. of netsMax. fan-out Zarkesh-Ha et al.56.65%87.15%-25.10% Stroobandt18.00%24.75%28.95% Accuracy enhancement of weighted exponential model avg. error of Str. – avg. error of weighted avg. error of Stroobandt accuracy enhancement =

36 02/15/04SLIP 2004 - Tao Wan & Malgorzata Chrzanowska-Jeske36 Conclusions Weighted exponential model is a general theoretical model - applicable to various levels of circuit descriptions and various physical architectures - system level, circuit level, physical level - copper interconnect, optical interconnect - lots of possible applications - one example: a benchmark generator for floorplanning. Our tool, Bgen, generates more realistic benchmark circuits using weighted exponential model than when using other models

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