"Fast estimation of the partitioning Rent characteristic" Fast estimation of the partitioning Rent characteristic using a recursive partitioning model.

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Presentation transcript:

"Fast estimation of the partitioning Rent characteristic" Fast estimation of the partitioning Rent characteristic using a recursive partitioning model J. Dambre, D. Stroobandt and J. Van Campenhout SLIP’03 - April 5-6, 2003

"Fast estimation of the partitioning Rent characteristic" Presentation outline Rent’s rule vs. the Rent characteristicRent’s rule vs. the Rent characteristic Recursive (bi-)partitioning equationsRecursive (bi-)partitioning equations Cut probability modelsCut probability models Validation and application for fast estimation of the Rent characteristicValidation and application for fast estimation of the Rent characteristic Conclusions & future workConclusions & future work

"Fast estimation of the partitioning Rent characteristic" N c connections between gates (internal nets)N c connections between gates (internal nets) T ext Connections to circuit’s exterior (external nets or pins)T ext Connections to circuit’s exterior (external nets or pins) The circuit graph Circuit netlist, consists of: G c gatesG c gates Only two-terminal connections considered !

"Fast estimation of the partitioning Rent characteristic" Donath-based prediction of placement wire lengths circuitarchitecture Perform recursive partitioning of circuit and architecture

"Fast estimation of the partitioning Rent characteristic" Donath-based prediction of placement wire lengths circuit At each level: calculate number of cut nets...

"Fast estimation of the partitioning Rent characteristic" Donath-based prediction of placement wire lengths architecture... and their length distribution from the architecture site functions

"Fast estimation of the partitioning Rent characteristic" Recursive circuit partitioning... modeled by Rent’s rule?? For a particular circuit netlist: p: Rent exponent t: Rent coefficient Region III Region II Region I Perform complete recursive circuit partitioningPerform complete recursive circuit partitioning Find average data points for T vs. G (=Rent characteristic)Find average data points for T vs. G (=Rent characteristic) Fit power law to region I:Fit power law to region I:

"Fast estimation of the partitioning Rent characteristic" Recursive circuit partitioning... modeled by Rent’s rule?? T = G 0.45 ? T = 7.45 G 0.78 ? Which Rent parameters to use ??

"Fast estimation of the partitioning Rent characteristic" Rent’s rule vs. the Rent characteristic Rent’s rule is not a very accurate model, so use the Rent characteristic instead of Rent’s rule!

"Fast estimation of the partitioning Rent characteristic" Last year’s results... Prediction of average wire lengths, based on Donath’s technique, but... using the Rent characteristic instead of Rent’s rule Overestimation, but very good correlation !

"Fast estimation of the partitioning Rent characteristic"... becomes very inaccurate if too few levels... doesn’t help much if many levels 5 levels of partitioning Rent’s rule vs. the Rent characteristic Problem: full recursive partitioning takes a lot of time Solution I*: perform only limited number of partitioning levels and interpolate... * suggested in, e.g. Hagen et al., “On the intrinsic Rent parameter and spectra-based partitioning methodologies”, 1994

"Fast estimation of the partitioning Rent characteristic" Rent’s rule vs. the Rent characteristic Problem: recursive partitioning takes a lot of time Solution II: find a better model than Rent’s rule ??

"Fast estimation of the partitioning Rent characteristic" Presentation outline Rent’s rule vs. the Rent characteristicRent’s rule vs. the Rent characteristic Recursive (bi-)partitioning equationsRecursive (bi-)partitioning equations Cut probability modelsCut probability models Validation and application for fast estimation of the Rent characteristicValidation and application for fast estimation of the Rent characteristic Conclusions & future workConclusions & future work

"Fast estimation of the partitioning Rent characteristic" Verplaetse’s recursive (bi-)partitioning equations Level k-1 G k-1 = 16 T k-1 = 6 N k-1 = 30 Hierarchy levels k = 0...H, with: level 0 = entire circuit, level H = single gate

"Fast estimation of the partitioning Rent characteristic" Verplaetse’s recursive (bi-)partitioning equations Level k Partition in two equal parts (balanced)

"Fast estimation of the partitioning Rent characteristic" Define:  k-1 : fraction of N k-1 that is cut (=cut probability) Verplaetse’s recursive bipartitioning model Level k Recursive partitioning equations:

"Fast estimation of the partitioning Rent characteristic" Verplaetse’s recursive bipartitioning model Level k New problem: find expression for  k-1 instead of expression for T k-1 ?? Define:  k-1 : fraction of N k-1 that is cut (=cut probability)

"Fast estimation of the partitioning Rent characteristic" Presentation outline Rent’s rule vs. the Rent characteristicRent’s rule vs. the Rent characteristic Recursive (bi-)partitioning equationsRecursive (bi-)partitioning equations Cut probability modelsCut probability models Validation and application for fast estimation of the Rent characteristicValidation and application for fast estimation of the Rent characteristic Conclusions & future workConclusions & future work

"Fast estimation of the partitioning Rent characteristic" module 1 module 2 Cut probabilities for random partitioning Random bipartitioning of a two- terminal net = randomly assigning both connected gates to two modules P cut = (1/2) G k /(G k -1) =  k, rand 4 possible assignments, net is cut for two of them:

"Fast estimation of the partitioning Rent characteristic" Cut probabilities for random partitioning 0.5  k, rand = (1/2) G k /(G k -1)

"Fast estimation of the partitioning Rent characteristic" Cut probabilities for optimal circuit partitioning Measured values of  k ?? Must perform some manipulations on Rent characteristic because: circuit size not equal to 2 Hcircuit size not equal to 2 H partitionings not perfectly balancedpartitionings not perfectly balanced

"Fast estimation of the partitioning Rent characteristic" Cut probabilities for optimal circuit partitioning random ibm17 Always = 1 (boundary condition) Always = 1 (boundary condition)

"Fast estimation of the partitioning Rent characteristic" Cut probabilities for optimal circuit partitioning Verplaetse’s model for  k : With  and  parameters to be fitted

"Fast estimation of the partitioning Rent characteristic" Cut probabilities for optimal circuit partitioning random ibm17 Verplaetse

"Fast estimation of the partitioning Rent characteristic" Cut probabilities for optimal circuit partitioning New model for  k : based on observed relationship between  k and  k,rand +

"Fast estimation of the partitioning Rent characteristic" Cut probabilities for optimal circuit partitioning random ibm17 Verplaetse New model

"Fast estimation of the partitioning Rent characteristic" Cut probabilities for optimal circuit partitioning E+06 Module size G k Module terminals  k ibm17 Verplaetse New model

"Fast estimation of the partitioning Rent characteristic" Presentation outline Rent’s rule vs. the Rent characteristicRent’s rule vs. the Rent characteristic Recursive (bi-)partitioning equationsRecursive (bi-)partitioning equations Cut probability modelsCut probability models Validation and application for fast estimation of the Rent characteristicValidation and application for fast estimation of the Rent characteristic Conclusions & future workConclusions & future work

"Fast estimation of the partitioning Rent characteristic" Cut probabilities for optimal circuit partitioning Both models have comparable fitting quality New model only has a single fitting parameter

"Fast estimation of the partitioning Rent characteristic" Correlation coefficients: Verplaetse vs. Rc: 0.988Verplaetse vs. Rc: New model vs. Rc: 0.995New model vs. Rc: Update of last year’s results... Prediction of average wire lengths, based on Donath’s technique, but... using the new partitioning model instead of the Rent characteristic (Rc)

"Fast estimation of the partitioning Rent characteristic" Can models for  k be applied to obtain accurate estimations of the entire Rent characteristic based on a few partitioning levels only?? Fast estimation of the Rent characteristic

"Fast estimation of the partitioning Rent characteristic" Fast estimation of the Rent characteristic all levels New model converges faster: for most benchmarks, two levels is enough use 3 or 4 levels to be sure

"Fast estimation of the partitioning Rent characteristic" Update of last year’s results... Prediction of average wire lengths, based on Donath’s technique, but... using the new partitioning model, fit for 3 levels of partitioning Correlation coefficients: Verplaetse vs. Rc: 0.902Verplaetse vs. Rc: New model vs. Rc: 0.824New model vs. Rc: 0.824

"Fast estimation of the partitioning Rent characteristic" Presentation outline Rent’s rule vs. the Rent characteristicRent’s rule vs. the Rent characteristic Recursive (bi-)partitioning equationsRecursive (bi-)partitioning equations Cut probability modelsCut probability models Validation and application for fast estimation of the Rent characteristicValidation and application for fast estimation of the Rent characteristic Conclusions & future workConclusions & future work

"Fast estimation of the partitioning Rent characteristic" Conclusions Adaptation of Verplaetse’s recursive partitioning equations and cut probability modelAdaptation of Verplaetse’s recursive partitioning equations and cut probability model Application of this model for fast estimation of entire partitioning Rent characteristicApplication of this model for fast estimation of entire partitioning Rent characteristic

"Fast estimation of the partitioning Rent characteristic" Future work Extension to multi-terminal netsExtension to multi-terminal nets Try to find theoretical foundations for cut probability modelTry to find theoretical foundations for cut probability model Is it possible to estimate cut probabilities from circuit graph without performing partitioning??Is it possible to estimate cut probabilities from circuit graph without performing partitioning??