2003-4-8 IC-SOC'2003, Taiwan 1

On the Global Routing Algorithms ConsideringOn the Global Routing Algorithms ConsideringRoutabilityRoutability and Timing Performanceand Timing Performance

Tong Tong JingJing, , XianXian--Long HongLong HongJingJing--YuYu XuXu, , YiYi--CiCi CaiCai

EDA Lab., Dept . of Computer Science and Technology Tsinghua Univ., Beijing 100084, P. R. China

2003-4-8 IC-SOC'2003, Taiwan 2

OutlineOutline1. Single Net----Steiner Tree Algorithms

Topology optimization considering minimal wire length / timing performance

2. Routability----Congestion Reduction

3. Unified Timing and Congestion Optimization

4. Future Research

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IntroductionIntroduction

Minimizing congestion is still a problem

Fabrication technology → VDSM device size and giga-hertz clock frequencies

Interconnect delay → chip timing

Need efficient timing and congestion optimization global routing algorithms

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1.1. Single NetSingle Net--------SteinerSteiner Tree AlgorithmsTree AlgorithmsPrevious ResearchMinimum Wire Length Steiner Tree AlgorithmHierarchical Steiner Tree Algorithm Based on Dreyfus-Wagner Algorithm

H. Y. Bao, X. L. Hong, Y. C. Cai Microelectronics & Computer, 1998

Timing-Driven Steiner Tree AlgorithmsIDW: Iterative Dreyfus-Wanger BasedCFD: Constructed Force Directed Approach

X. L. Hong. ACM/IEEE DAC, 1993X. L. Hong. Chinese J. Computers, 1995 X. L. Hong. Chinese J. of Semiconductors, 1995

Timing-Driven Steiner Tree Algorithm Based on Sakurai ModelH. Y. Bao, X. L. Hong, Y. C. CaiChinese J. of Semiconductors, 1999

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Hierarchical Timing-Driven Steiner Tree AlgorithmJ. Y. Xu, X. L. Hong, T. Jing, Y. C. Cai, J. GuIEEE/ACM ASP-DAC, 2002

s

T1t

GRG

e

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Experimental Results--Comparison on run time

1

10

100

1000

10000

100000

1000000

6 7 8 9 10 11 12Number of Pins

CPU

Tim

e(m

s)Optimal SolutionCPU Time(ms)Hierarchical SolutionCPU Time(ms)

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ConclusionsBy decomposing the minimum time delay Steiner tree problem into hierarchy, the high-quality solutions are provided with a significant speed up.

Comparison on time delay show a total speed up of 1000x ~ 100000x with no degradation of wire length performance.

For nets with large number of pins, our approach also achieves satisfactory results in a very short time.

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Congestion Reduction Algorithms for Global RoutingCongestion Reduction Algorithms for Global Routing** Based on Search Space Traversing Technology (SSTT)* Random GR Independent of Net Ordering (RINO)

RINO Algorithm. H. Y. Bao, T. Jing, X. L. Hong, Y. C. Cai IEEE/ACM ASP-DAC, 1999 Chinese J. Computers, 2001

Parallel RINO Algorithm.J. Y. Xu, H. Y. Bao, X. L. Hong, Y. C. Cai, T. Jing Chinese J. of Semiconductors, 2002

SSTT Algorithm.T. Jing, X. L. Hong, H. Y. Bao, Y. C. Cai, J. Y. Xu

. IEEE ASICON, 2001Journal of Computer Science and Technology, 2003

2.2. Congestion ReductionCongestion ReductionStill a critical problem in global routing

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Local search method with search space smoothing (SSS, J. Gu, 1994)Manages to smooth the search space and “hides” some local

minimum points�Reduce the effect of local minimum pointsNeeds to find a suitable “trivial case” �The usage of this method is

limited.

T h e m in im u m so lu tion in o rig in al space

T he in itia l searchp o in t in o rig in a l space

T he sm o o thed so lu tion space 1

T h e sm oo th ed so lu tion space 2

… … …

T h e sm o o thed so lu tion space n

T he o rig in al so lu tion space

A n exam ple o f one d im ensiona l so lu tion space sm oo th ing : th e m in im u m so lu tiono f so lu tio n space i w ill be the in itia l s ta rting p o in t in th e so lu tion sp ace i+ 1

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Stochastic optimization strategyReroute a random subset of congested nets

simultaneously using the concurrent routing procedure R1 (search towards the optimum point in one direction)

Deterministic optimization strategyThe mid-solution got in R1 is then refined by

the sequential processing procedure R2 (search towards the optimum point in another direction)

There is a repeated iteration between R1 and R2

Local enumeration strategy

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Ncong = {n1, n2, . . . , ni}

1. Get the set of congestednets from GRG

Nran is a random subset of Ncong

2. Get the set of nets fromNcong to be rerouted

3. Free the GRG edge resourcesheld by nets in Nran

4. Rip up & reroutethe nets in Nran

GRG without nets in Nran

GRG after reroute

Rerouting done by R1

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|Nran||Ncong|

Iterations

0.2 Upper Bound

LowerBound

Get Mid-Solution Smid

Start R2Start R1

Get Sgood

Start R1

The repeating iterations between R1 and R2

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TheOriginal

Tree

The NewConstructed

Tree

The BestTree Circular

LinkedList

The best Steiner tree selecting for a congested net

done by R3

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Thus, it disturbs the environment of getting trapped in local minimum. As a result, it is able to transit from local minimum point and make a fast search in the whole search space.

an initial solution of one optimization sub-strategy

search trace

a local minimum point

a global minimum point

search space

search transition

Fig.3. The illustration of SSTT algorithm

a mid-solution

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Our router performs fast and get good routing solutions

Our router does well on both MCNC benchmarks and industrial circuits.

Since any arbitrary initial solution can be accepted, the initialization in our algorithm is greatly simplified.

Conclusions

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The main contribution of this workThe main contribution of this workDifferent from existing algorithms, the UTACO algorithm adopts a

shadow price mechanism to incorporate timing and congestion optimizing into one unified objective function. It can optimize timing and congestion simultaneously. The shadow price of a net is the sum of its congestion price and timing price.

The timing analysis strategy in UTACO is different from that in the above mentioned approaches. Based on the CC-net and its cut, we can reduce the delay in an overall survey instead of greedy trying.

UTACO AlgorithmUTACO AlgorithmT. Jing, X. L. Hong, H. Y. Bao, Y. C. Cai, J. Y. XuIEEE/ACM ASP-DAC, 2003

3.3. Unified timing and congestion optimizationUnified timing and congestion optimization

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Timing modelThe Sakurai-delay-based timing model is used here. Note that the

Sakurai-delay-based timing model is an extension of Elmore-delay-based timing model with adjusted weights on different terms.

Problem formulation

GRC1

GRGv1

e

v2

SteinerTree

Pin Chip

Global Routing Graph (GRG)

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The nets-based timing analysis strategy[J. Huang, X .L. Hong, C. K. Cheng et al, IEEE/ACM DAC, 1993]

The critical-path-based timing analysis strategy[X. L. Hong, T. X. Xue, J. Huang et al, IEEE TCAD, 1997]

The critical-network-based timing analysis strategy[J. Tong, X. L. Hong et al,

IEEE ISCAS, 2002Journal of Computer Science and Technology, 2003]

Prof. X. L. Hong introduced it in IC-SOC 2002, USA.

Previous ResearchTypical timing analysis strategies

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Ref. [Shragowitz’87, Carden IV’96, Albrecht’00] introduce the multi-commodity flow into global routing. The goal of these algorithms is only to minimize congestion. Timing optimization does not be considered and formulated.

Based on shadow price mechanism, we formulate global routing as a multi-commodity flow problem and incorporate timing and congestion optimizing into one unified objective function.

The objective function is the slack of congestion with the clock period as the delay limit that from registers and inputs to registers and outputs.

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In this paper, The multi-commodity flow is expressed by a linear programming formulation as a primal problem. We then convert the primal problem into a dual formulation using the shadow price as the variables.

In the dual formulation, the wiring congestion is reflected by the congestion price at each edge of the GRG. The congestion price of a net is defined to be the sum of the congestion prices on the edges passed by that net.

The signal delay is reflected by the timing price at each net. If we view each timing price as a timing flow on each net, the timing flow forms paths flowing from inputs and registers to registers and outputs. The amount of the timing flow on each path corresponds to the criticality of the timing constraint on that path.

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The shadow price of a net is the sum of its congestion price and timing price. The objective of the dual problem is to maximize the sum of shadow prices of all nets together with the clock period limit on the boundary of the circuit.

The primal and dual formulation offers theoretical upper and lower bounds of the routing solution. Throughout the optimization process, the difference of the two bounds reduces. When the difference approaches zero, we have an optimal solution.

However, the amount of routing flow is limited by discrete numbers, the difference always exists. The bounds thus provide the user’s insight into the quality of the solutions.

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Linear programming formulationMinimize sSubject to

re: ∀ e ∈ E (A)

λn: ∀ n ∈ N (B)

ωij: ∀ i ∈ ns, j ∈ nt, ∀ n ∈ N (C)

ui: ∀ node i (D)

ai ≥ 0, ≥0, s ≥ 0 ∀ node i, n ∈ N

0)(,

≥⋅+⋅− ∑∈

eNnf

fne

fn cstt ϕ

nf

fn bt ≥∑

01 ≥+⋅⋅−− ∑ jf

fn

fij

ni atdb

a

Pai −≥−

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Maximize (E)

Subject tos : (F)

: ∀ n ∈ N (G)

ai : ∀ node i (H)

λn ≥ 0, ωij ≥ 0 ∀ n ∈ N, i ∈ ns, j ∈ ntui ≥ 0, re ≥ 0 ∀ node i, e ∈ E

)( ∑∑ ⋅−⋅∈ i

iNn

nn Pubλ

1≤⋅∑∈Ee

ee rc

fnt

0≤−+− ∑∑ ij

jij

ij uωω

Dual linear programming formulation

01)(,

≤⋅⋅−+⋅− ∑∑∈∈∈

ijf

ijnjnin

nEe

fnee db

trts

ωλϕ

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By using the method of iterating the primal dual process, we will choose the nets with to reroute.)(max )1()(

,max+−=∆ k

tk

tnff

nf

nλλλ

)(max )1()(,max

+−=∆ kt

ktnf

fn

fn

λλλ

))1)((

)1)(((max

)1()1(

)()(

,

+

∈∈∈

+

∈∈∈

⋅⋅+⋅

−⋅⋅+⋅=

∑∑

∑∑

kij

fij

njninEe

fne

ke

kij

fij

njninEe

fne

kenf

db

tr

db

tr

ts

ts

ωϕ

ωϕ

∀ n ∈ N (O)

)1)(( ijf

ijnjninEe

fneet

db

trts

fn

ωϕλ ⋅⋅+⋅= ∑∑∈∈∈

Let . Then, )(min,

fntnfn

λλ =

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How to select the nets?----By creating the CC-net

Based on the information about congested edges and critical paths given by the primal iteration, we create a network NW.

NW= (Vcc, Ecc, , PI, PO), which consists of and only consists of all congested edges and critical paths. The network NW is called CC-net.

Where Vcc is the set of pins of NWEcc is the set of edges of NWPI is the source of NWPO is the sink of NW

λmax∆

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Partial CC-net in MCNC C2

PI

PO

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How can we get in CC-net?We select the min-cut of the CC-net. Then, decrease

the delay and congestion of all edges in the min-cut. As a result, we are able to get the nets with .

Based on the theory of maximum network flow, we can get the maximum flow of the CC-net. If the value of the maximum flow does not equal ∝, we will get its min-cut.

λmax∆

λmax∆

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Experimental results and conclusionsonclusions

The UTACO algorithm has been implemented in the C language on a Sun Ultra Enterprise 450.

We compare routing results between UTACO and SSTT algorithm on main optimization objectives. The experimental results are also compared with other typical algorithms.

The experimental results show that the UTACO algorithm is able to:

• Optimize both timing and congestion simultaneously and efficiently

• Reduce the delay in an overall survey

• Obtain good routing results on other optimizing objectives, such as wire length, overflow edges

• Take a very short running time

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Coupling considered GR algorithms• Coupling effects → longer delay • Coupling effects → noise (crosstalk)

Efficient timing models

Routing for special applications• Data-path routing inside SOC

Interconnect optimization• Topology optimization + buffer insertion / sizing

+ wire sizing

4. Future Research

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Thank you!Thank you!

Tong Tong JingJing ((����),), PhPh.D..D.Associate Prof.

Dept . of CST, Tsinghua Univ.Beijing 100084, P. R. ChinaTel.: +86-10-62785564Fax: +86-10-62781489E-mail: jingtong@tsinghua.edu.cn

jingtong_eda@hotmail.com