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A Fixed-outline Floorplanning MethodBased on 2.5D

Sheqin Dong Qi Xie

Department of Computer Science and Technology

Tsinghua University, Beijing, 100084, P.R. China

Abstract

In this paper, a fixed-outline floorplan-ning algorithm based on 2.5D is proposed.By using constraints of area and number of pins to divide the modules into 4 layers,it confines the variations of the widths of the floorplans to a small region throughcommon subsequence of sequence pair representation. The goal of minimizing

the area is consistent with that of satisfy-ing the given outline. A set of bench-marks is used for test. Experiments showthat the proposed algorithm can achievehigh successful probabilities in short runtime, even with tight outlines and largeaspect ratios are imposed.

Keywords : Fixed-outline floorplan,Floorplanning based on 2.5D , CommonSubsequence

1. Intruduction

Floorplanning is a fundamental prob-lem in the design process of complexchips and has a profound impact on thearea, delay, power, and many other de-sign parameters. Many representations for floorplans had been proposed in the lasttwo decades, such as sequence pair [3],O-tree [4], CBL [5], B*tree [6] and BSG[7] etc.

Classical floorplanning is formulatedas a free-die problem. It roughly deter-mines the layout of a given set of mod-ules subject to various objectives, such asarea and estimated total wirelength. Inreality, the use of floorplanning duringthe chip synthesis process almost alwayscomes after the die size and package have

been chosen. As pointed in [1], modernfloorplanning formulation should be cast

as a fixed-outline problem, and the pack-ing must simultaneously achieve zerowhitespace and zero overlap for the givenoutline.

It is also showed in [2] that, empiri-cally, fixed-outline floorplan instancesare significantly harder than those with-out fixed outline. Several works have

been done addressed to it. [2] proposedan algorithm based on simulated anneal-ing and sequence pair. It tries to achievefixed-outline floorplanning by better localsearch, but the successful probabilities

are quite low. [8] presented a genetic al-gorithm, which can achieve high success-ful probabilities with loose outlines. Butno results of tighter outlines were re-

ported.On the other hand, interconnect opti-

mization has become a major concern inmodern floorplanning. Interconnect delayhas dominated over gate delay and timingconstraints have become more and moredifficult to be met with this huge number of interconnects involved. 2.5D chip is afeasible solution to these problems. It has

been shown that interconnect lengths can be greatly reduced in 2.5D ICs. A 2.5D

*This work is supported by NSFC 90307005

and NSFC 60473126

Proceedings of the 11th Joint Conference on Information Sciences (2008)Published by Atlantis Press

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chip is actually a chip with more than onesilicon layers to place modules.In this paper, a fixed-outlinefloorplanning algorithm based on 2.5D is

proposed. During simulated annealing,the 2.5D algorithm divides the inputmodules into 4 layers by adding the

constraints of area and number of pins.Then the fixed-outline algorithm is calledin each layer. It uses commonsubsequence and penalty function to bindthe variations of the widths of thefloorplans to a narrow range. Thus, asimple cost function which has no itemsabout the heights of the floorplans is

presented. Experiments show that the proposed algorithm can achieve highsuccessful probabilities in short run time,even with tight outlines and large aspectratios imposed. Also the proposed

algorithm performs well both on 2.5Dand fixed-outline.

2. Priliminaries

2.1. Sequence Pair and Common Sub-sequence

The concept of sequence pair was firstintroduced in [3]. A sequence pair is twosequences of all the module names.Horizontal and vertical constraint graphscan be derived from a given sequence

pair by the following two rules:1) (ab, ab) => a is placedleft to b;

2) (ba, ab) => a is placed below to b;

Fig.1 gives an example of sequence pair.

(a) (b) (c)

Figure 1. The (a) horizontal and (b) vertical constraint graphs of ( ecadfb,fcbead ), and (c) of

its floorplan .

Definition 1. Given two sequences X andY , a sequence Z is a common subse-quence of X and Y if Z is a subsequenceof both X and Y .

For example, assume a sequence pair P=(ecadfb,fcbead), cad is a commonsubsequence of P.

2.2. Sequence Pair and Common Sub-sequence

Definition 2. Given a modules set B and an outline with width of W d , let

B p={m p1 , m p2m pk } be a subset of B. If itholds that

1

k pi d

iw W , then B p is a pri-

mary modules set (PMS for short) of B.Let

1

k pi

iw be the length of B p and denote

it as L(B p ). Let | B p| be the number of modules in B p.

Definition 3. Given a sequence pair and a PMS B p, if there exists acommon subsequence of + and -, whichhas all the modules in B p, we call that < +, -> covers B p.

For example, assume B p={b, d, e } is aPMS of { a, b, c, d , e}. Since (bdeac , abdce )contains a common subsequence ,which includes all the modules in B p, wecall that it covers B p.

Given above definitions, the followinglemma can be drawn:Lemma 1 : If a sequence pair covers a

primary modules set B p, the width of itsfloorplan is no less than L(B p ).

3. The Fixed-outline Algorithm

The fixed-outline algorithm firstrandomly creates a sequence pair as theinitial solution, and then selects a PMS B p from the given sequence pair. Thenduring the simulated annealing process,adaptations are performed after

perturbations to make all the solutionsobtained also cover B p. After the

adaptations and perturbations, an


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algorithm is called trying to find a better PMS if the fixed-outline constraint hasnot been satisfied yet. Following aredetails of the fixed-outline algorithm.

Problem Definition: Each module mi isrectangular and defined by a ( wi, hi),where wi and h i are the width and height

of the module respectively. Given a set of modules B={m1 ,m2 ,mn} and their inter-connections, fixed-outline mode floor-

planning determines the layout of eachmodule subject to various objectives(such as estimate wirelength, etc.) whilesatisfying that no modules overlap and itholds that W F W d and H F H d , where W d and H d are the width and height of givenoutline, and W F and H F are those of thefloorplan.

For a given set of modules with totalarea AT and given maximum percent of dead-space , we construct a fixed outlinewith aspect ratio by following equations:

(1 )d T W A (1 ) /d T H A

Cost Function: The cost functionadopted in the fixed-outline algorithm is:

Cost =W1*area/totalArea+ W2*wdiff Where area is the area of the floorplan,

and totalArea is the sum of all the mod-ules area. While wdiff is defined like thisas the penalty function:

wdiff=(width-CurrentCPLength)/ Cur-rentCPLength

Here width is the width of the floorplan,and CurrenCPLength is the length of cur-rent common subseqence.

Unlike the cost functions adopted inother algorithms, neither the height of thefloorplan nor the excess of the heightappear in (1). And the excess of width isonly introduced as a penalty. The reasonlies in that: since all the solutionsobtained cover B p, their width must be noless than L( B p) (from Lemma 1); byadding penalty function, the widths of solutions are driven exactly to L(Bp). Thecriteria of selecting B p (in Section 3.3),

guarantees that L(B p ) is very close to W d .Thus we regard the width of the solutionsas constant, the goal of minimizing thearea is consistent with that of minimizingthe height. Therefore, it is reasonable touse this simple cost function, which maymake the optimization easier.

The Selection of PMS: Firstly, we canget the set S of all the modules which lieon the right edge of the floorplan usingthe property of the sequence pair. Thenwe can find a set of modules from theright edge of the floorplan to the left edgefor each module in S. This is a path in thehorizontal constraint graphs, also a com-mon sequence of the sequence pair. Aswe have all the common sequences of thefloorplan, its easy to choose the best one.We only need to compare the length of the common sequences and be sure thatits shorter than the width of the outline.After the fixed-outline algorithmrandomly creates a sequence pair as theinitial solution, the algorithm above will

be called firstly to generate the initialPMS. If the algorithm fails, we willrandomly choose only one module whoselength is less than the width of the outlineas PMS. Absolutely, such a floorplanwont satisfy the fixed-outline constraint.As pointed before, after the adaptationsand perturbations, this algorithm is calledtrying to find a better PMS if the fixed-

outline constraint has not been satisfiedyet. Thus, time and time again, we willget the perfect PMS whose length isexactly close to the width of the outline.As we have the reasonable cost function,the fixed-outline constraint will surely besatisfied at last.

Perturbations and Adaptations: The perturbations adopted in the fixed-outlinealgorithm are the most general ones usedin sequence pair representation, which arerotating a module, swapping two modulesin the first sequence or in both sequences.

Note that in order to maintain the length


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of B p, the modules in B p are not allowedto be rotated in our algorithm.

In the following cases, the solution ob-tained might not cover B p any longer: 1)to swap two modules in the first sequenceand only one of them belongs to B p; 2) toswap two modules in the first sequence

and both of them belong to B p; 3) to swaptwo modules in both sequences and onlyone of them belongs to B p.

To make the solution covers B p, adap-tations should be taken after perturbationsof the above cases. These adaptations can

be done in O(n)-time. In the following,we will introduce the adaptation for thefirst case in detail. The other two kinds of adaptations are quite similar and omittedhere for the sake of space.

Let < +, -> be the sequence pair which already have two modules, m1 and

m2 (m1 B p, m2 B p), swapped in the firstsequence. Let (i) denote the i-th modulein the sequence .

1) Set sequence BF = , scan - fromleft to right, and insert to BF themodules which belong to B p. Let x denote the position of m1 in BF and

p1 denote the position of m1 in -;

2) Set BF = , scan + from left to right,and insert to BF the modules which

belong to B p. Let y denote the posi-tion of m1 in BF and p2 denote the

position of m1 in +;3) If x= y, already covers B p,

so the procedure terminates. Other-wise, if p1< p2, goto 4; else goto 5;

4) For i= p1 to p2-1, set -(i)= -(i+1 ).Set -( p2)= m1 and stop;

For i= p2+1 to p1, set -(i)= -(i-1). Set -( p2)= m1 and stop;

4. The Algorithm Based on 2.5D

Before using the fixed-outline algorithm,we need an algorithm to group modules.

We set the numbers of layers to 4. Thedetails will be discussed next.

Constraint: We choose twoconstraints for our 2.5D algorithm. Theyare constraints of total area and totalnumber of pins. As in the fixed-outlinealgorithm, we construct a fixed outline by

using the total area of modules. So to besure that each layer has the same fixed-outline, the total area of each layer should

be more or less the same. We also choosethe total numbers of pins. The 2.5Dalgorithm is firstly designed to decreasethe interconnect lengths by addinginterconnects of layers. In theory, theinterconnects of layers will be bigger if the number of pins in each layer is moreapproximate.

Cost Function: We set our cost func-tion as this:

Cost=|ADiff(0)|++|ADiff(3)|+|PDiff(0)|++|Diff(3)|

Where ADiff(i) means the diff betweentotal area of layer i and quarter of totalarea of all the modules in 4 layers.Similarly, PDiff(i) covers the diff of number of pins. And we treat the areaconstraint and the pins constraint equally.The cost function is simple but effective.

Perturbations: We generate the initialsolution randomly. We choose the twolayers whose total area diff most. Thenrandomly choose two modules for swap.If after the swap, the diff between twolayers can be reduced by 10 percents atleast, we will accept it. Otherwise, wehave to choose another two.

5. Experimental Results

We implemented the proposed algo-rithm in C++ and run test data createdfrom MCNC benchmarks on a PC withAMD Athlon 64 Processor 3000+1.81GHZ and 1.00GB RAM. The follow-ing experimental results are the averages


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of 100 runs. Note that the modules han-dled in our experiments are all hard mod-ules.

Table 1 shows that the proposedalgorithm are quite promising in bothsuccessful probabilities and run time,even when the aspect ratio is as large as 6.

Another advantage of the proposedalgorithm is that high successful probabilities can be achieved with tightoutline imposed. When =7% and 6%, thesuccessful probabilities is 100%. When =5%, the successful probabilities rangefrom 70% to 90%, which are stillacceptable. Fig.2, 3 and 4 shows severalresultant floorplans of 198(created fromami33) with =7% and aspect ratios of 1,2 and 6 respectively. In the aboveexperiments, the algorithm terminatesonce a feasible floorplan. Otherwise, we

regard the algorithm failed.

6. Conclusions

In this paper, a fixed-outlinefloorplanning algorithm based on 2.5D is

proposed. The fixed-outline floorplanningalgorithm uses common subsequence of sequence pair. It tries to confine thevariations of the widths of the floorplansto a small region through commonsubsequence. Thus the goal of minimizing the area is consistent withthat of satisfying the given outline. Whilethe 2.5D algorithm uses the concept of layers. SA is used by adding constraintsof area and number of pins to divide themodules into 4 layers. A lot of standard

benchmark files are used for testing. Theresults are promising.

7. References

[1] A.B. Kahng, Classical FloorplanningHarmful? Proceedings of ISPD , SanDiego, 2000, pp. 207-213.

[2] Saurabh N. Adya, Igor L. Markov,Fixed-outline FloorplanningThrough Better Local Search, Pro-ceedings of ICCD , Austin, 2001, pp.328-334.

[3] H. Murata, K. Fujiyoshi, S. Nakatakeet.al, VLSI module placement basedon rectangle-packing by the sequence

pair, IEEE Trans. on CAD , 1996, vol.15(12), pp. 1518-1524.

[4] Y. Pang, C.-K. Cheng and T. Yoshi-mura, An Enhanced Perturbing Al-gorithm for Floorplan Design Usingthe O-tree Representation, Proceed-ings of ISPD , 2000, pp. 168-173.

[5] Xianlong Hong et al., Corner Block

List: An Effective and EfficientTopological Representation of on-Slicing Floorplan, Proceedings of

ICCAD , 2000, pp. 8-13.[6] Y.-C. Chang, Y.-W. Chang, G.-M.

Wu and S.-W. Wu, B*-Trees: A New Representation for Non-SlicingFloorplans, Proceedings of DAC ,2000, pp. 458-463.

[7] S. Nakatake, K. Fujiyoushi, H. Mu-rata and Y. Kajitani, Module Place-ment on BSG-structure and IC LayoutApplications, Proceedings of IC-

CAD, 2000, pp. 8-13.[8] Chang-Tzu Lin, De-Sheng Chen et al,Robust Fixed-outline FloorplanningThrough Evolutionary Search, Pro-ceedings of ASP-DAC , 2004, pp 42-44.

Table 1. The results of 198 with =7%

Aspect ratio =1 =2 =3 =4 =5 =6Run time 4.9s 3.1s 3.7s 3.3s 4.1s 4.9sSuccessful 100% 100% 100% 100% 100% 100%


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Figure 3. The resultant floorplans of 198 in 4 layers with =7% and aspect ratios of 2 .

Figure 2. The resultant floorplans of 198 in 4 layers with =7% and aspect ratios of 1.

Figure 4. The resultant floorplans of 198 in 4 layers with =7% and aspect ratios of 6 .


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