Circuit-wise Buffer Insertion Circuit-wise Buffer Insertion and Gate Sizing Algorithm and Gate Sizing Algorithm

with Scalabilitywith Scalability

Zhanyuan JiangZhanyuan Jiang and Weiping Shiand Weiping Shi

DAC 2008, June 8–13, 2008, Anaheim, California, USA.

OutlineOutline

IntroductionIntroduction Problem FormulationProblem Formulation AlgorithmAlgorithm

Post-buffering Timing Estimation Linear Modeling of Non-linear Delay vs. Cost

Tradeoff Dynamic Critical Sink Selection Linear Programming Circuit Partition

Experimental ResultsExperimental Results ConclusionConclusion

OutlineOutline

IntroductionIntroduction Problem FormulationProblem Formulation AlgorithmAlgorithm

Post-buffering Timing Estimation Linear Modeling of Non-linear Delay vs. Cost

Tradeoff Dynamic Critical Sink Selection Linear Programming Circuit Partition

Experimental ResultsExperimental Results ConclusionConclusion

IntroductionIntroduction

As VLSI technology enters the nanoscale regime, a great amount of efforts have been made for timing optimization.

Among them, buffer insertion stands out as an effective technique to reduce interconnect delay.

Due to technology shrinking, more and more gates are placed on a chip, and algorithms without scalability can not fit into future physical synthesis flow.

OutlineOutline

IntroductionIntroduction Problem FormulationProblem Formulation AlgorithmAlgorithm

Post-buffering Timing Estimation Linear Modeling of Non-linear Delay vs. Cost

Tradeoff Dynamic Critical Sink Selection Linear Programming Circuit Partition

Experimental ResultsExperimental Results ConclusionConclusion

Problem FormulationProblem Formulation

We represent a combinational circuit as a Directed Acyclic Graph (DAG) G = (V,E).

Problem FormulationProblem Formulation

The paper abstract the routing tree of the circuit and ignore all the details (i.e., Steiner node and interconnect tree structure, etc.) within the routing tree.

The vertices only represent PI/PO of the circuit and input/output pins of modules while edges only for input-to-output paths within a module.

Problem FormulationProblem Formulation

OutlineOutline

IntroductionIntroduction Problem FormulationProblem Formulation AlgorithmAlgorithm

Post-buffering Timing Estimation Linear Modeling of Non-linear Delay vs. Cost

Tradeoff Dynamic Critical Sink Selection Linear Programming Circuit Partition

Experimental ResultsExperimental Results ConclusionConclusion

Post-buffering Timing Estimation

A post-buffering timing estimation technique is proposed in [12], which derives delay equations along a buffered wire segment and applies the equations for the delay estimation upon multiple-sink nets.

Linear Modeling of Non-linear Linear Modeling of Non-linear Delay vs. Cost TradeoffDelay vs. Cost Tradeoff

Table 1 shows that the impact of varying downstream sink size from 1X to 4X is negligible at the driver.

Linear Modeling of Non-linear Linear Modeling of Non-linear Delay vs. Cost TradeoffDelay vs. Cost Tradeoff

Linear Modeling of Non-linear Linear Modeling of Non-linear Delay vs. Cost TradeoffDelay vs. Cost Tradeoff

A curve fitting method is adopted to approximate each tradeoff as several linear segments.

In this paper, the number of segments is set as 2, which gives good accuracy.( ( If the number of segments is 3, the final circuit Elmore delay improves less than 0.1% while the linear programming solver time increases more than 50%. ))

Linear Modeling of Non-linear Linear Modeling of Non-linear Delay vs. Cost TradeoffDelay vs. Cost Tradeoff

QQrootroot + c + c11XXcc + c + c33 ≤ Q ≤ Qsinksink, , (1)(1)

QQrootroot + c + c22XXcc + c + c44 ≤ Q ≤ Qsinksink, , (2)(2)

LLcc ≤ X ≤ Xcc ≤ U ≤ Ucc, , (3)(3)

QQroot root :RAT values at root:RAT values at root

QQsink sink :RAT values at sink:RAT values at sink

XXc :c :the number of buffers at this netthe number of buffers at this net

CCi :i :the curve fitting coefficientthe curve fitting coefficient

LLc, c, UUc :c :lower bound and upper bound of the number of bufferslower bound and upper bound of the number of buffers

Linear Modeling of Non-linear Linear Modeling of Non-linear Delay vs. Cost TradeoffDelay vs. Cost Tradeoff

Dynamic Critical Sink Selection

A multiple-sink net contains sink S1, S2, · · · , Sn, and each sink has corresponding RAT Q1, Q2, · · · , Qn. It is hard to know which is critical sink before the stage of buffer insertion.

At the root, for a specific buffer number, we select the solution that minimizes the maximum delay among all sinks. Thus, only one delay cost tradeoff curve is returned.

Dynamic Critical Sink Selection

Dynamic Critical Sink Selection

Solution set 1Solution set 2

Dynamic Critical Sink Selection

Qroot + c1Xc + c5 ≤ Qsinkone, (1)Qroot + c2Xc + c6 ≤ Qsinkone, (2)Qroot + c3Xc + c7 ≤ Qsinktwo, (3)Qroot + c4Xc + c8 ≤ Qsinktwo, (4)Lc ≤ Xc ≤ Uc, (5)

QQroot root :RAT values at root:RAT values at rootQQsink sink :RAT values at sink:RAT values at sinkXXc :c :the number of buffers at this netthe number of buffers at this netCCi :i :the curve fitting coefficientthe curve fitting coefficient

LLc, c, UUc :c :lower bound and upper bound of the number lower bound and upper bound of the number of buffers of buffers

Linear Programming

CCi :i :the curve fitting coefficientthe curve fitting coefficient

Xi :Xi :the cost of routing tree RT(i)

Lc, Uc :lower bound and upper bound

Circuit Partition

This paper This paper adopt the divide-and-conquer scheme to speed up the algorithm.

The key components of the circuit partition technique are how to decide partition boundaries and how to set up side inputs/outputs in the sub-circuits.

Circuit Partition

In order to minimize partition error, the technique avoids partitioning the critical paths into different sub-circuits, which means that partition boundaries never cut through the most critical path.

If there is an overlap between different downstream cones, the overlap part belongs to the cone with the most critical primary input.

Circuit Partition

Figure 5: The circuit is partitioned into three subcircuits based on the downstream cones of primary inputs. The input a is the most critical primary input in the circuit.

OutlineOutline

IntroductionIntroduction Problem FormulationProblem Formulation AlgorithmAlgorithm

Post-buffering Timing Estimation Linear Modeling of Non-linear Delay vs.

Cost Tradeoff Dynamic Critical Sink Selection Linear Programming Circuit Partition

Experimental ResultsExperimental Results ConclusionConclusion

Experimental ResultsExperimental Results

Experimental ResultsExperimental Results

Experimental ResultsExperimental Results

OutlineOutline

IntroductionIntroduction Problem FormulationProblem Formulation AlgorithmAlgorithm

Post-buffering Timing Estimation Linear Modeling of Non-linear Delay vs.

Cost Tradeoff Dynamic Critical Sink Selection Linear Programming Circuit Partition

Experimental ResultsExperimental Results ConclusionConclusion

ConclusionConclusion

Experiments demonstrate that the circuit-wise algorithm achieves on average 17.4X speedup compared with the path based algorithm.

The whole circuit is partitioned into n downstream cones plus the remaining circuit.

The circuit is partitioned into n + m or n + k sub-circuits depending on whether the remaining circuit is disjointed or not.

The problem defined as follows: Given a DAG which represents a placed and

routed combinational circuit, possible candidate buffer locations, a buffer library and a gate library, find a buffering and gate sizing solution such that the total cost of buffers and gates are minimized, and the required arrival time at each primary input is less than a given constant constraint.

Elmore delayD(e) =R(e)[C(e)/2 + C(vj)]D(vD(vjj) = K(b) + R(b) ) = K(b) + R(b) ．． C(vC(vjj))

e: edge (vi, vj)R(e): resistance of eC(e): capacitance of eC(vj): downstream capacitance at vj

K(b): intrinsic delay of buffer bR(b): R(b): driving resistance of buffer b