Constraint-free Analog Placement with Topological Symmetry Structure

Transcription

1 Constraint-free Analog Plaement with Topologial Symmetry Struture Qing DONG Department of Information an Meia Sienes University of Kitakyushu Wakamatsu, Kitakyushu, Fukuoka, , Japan Shigetoshi NAKATAKE Department of Information an Meia Sienes University of Kitakyushu Wakamatsu, Kitakyushu, Fukuoka, , Japan Astrat In analog iruits, loks nee to e plae symmetrially to satisfy the evies mathing. Different from the existing onstraint-riven approahes, the propose topologial symmetry struture enales us to generate a symmetrial plaement without any onstraint. Simulate annealing is utilize as the framework of the optimization, an we propose new move operation to maintain the plaement s topologial symmetry. By inserting ummy loks, we present a physial skewe symmetry struture allowing non-symmetry partly, so that to enhane the plaement on area an wire length. Besies, we inorporate regularity into the evaluation of plaement. Experiments shows that our approah generate topologial omplete symmetry plaements without muh ompromise on hip area an wire length, ompare to the plaements with no symmetry. I. INTRODUCTION In analog iruits, mathing the partnere evies helps to avoi oth high offset voltage an egraation of power supply rejetion ratio [1], so partnere evies are often require to e plae symmetrially.constraint-riven approahes[3],[5], [7], [8] an yiel plaements satisfying the mathing requirement. Sine no effiient generation of onstraint has een estalishe yet, onstraint is generate manually, the onstraint-riven approahes still suffer from the time-onsuming onstraint generation proess. The struture plaement propose in [11] is a onstraintless approah to pursue the regularity of analog plaement. Following this onstraint-less onept, we introue another struture plaement fousing on topologial symmetry struture. Our ontriution is summarize as follows. Not speifying any lok s name, we an formulate the topologial omplete symmetry struture. Base on this struture, the plaement goes to symmetry naturally without any onstraint. Besies, this struture enales us to alulate the vertial oorinate in the linear time to the numer of eges of the onstraint graph. We introue the physial skewe symmetry struture for sarifiing the omplete symmetry against minimizing the area or wire length. In orer to improve the plaement, we utilize simulate annealing as a framework of the optimization proess an propose effiient moves to maintain the topologial symmetry at every step. Furthermore, we introue zero-size ummy loks to present Physial Skewe Symmetry Struture. As a result, we an release physially symmetry to get the plaement with less area or wire length. In experiments, we applie our struture plaement to inustrial instanes for analog lok esigns. In aition, we omine the evaluation of symmetry struture with the evaluation proeure introue in [11] for taking row an array strutures into onsieration. In the results, the plaements of the omplete symmetry struture oul e otaine y the ompromise of aout 5% on average with respet to the hip area an the wire length, ompare to the plaements with no symmetry. Furthermore, we teste the aility of physial skewe symmetry plaement. Those inserte ummy loks enue other loks more flexiility, an serve to reue the hip area an wire length as expete. The rest of this paper is organize as follows. Setion II esries preliminary onepts of our struture plaement. Setion III introues the topologial property of a topologial omplete symmetry. Setion IV esries the framework of the optimization of plaement with new move, an introues the physial skewe symmetry struture. Setion V emonstrates the experimental results. Setion VI onlues ontriution an future work. II. SEQUENCE-PAIR &SINGLE-SEQUENCE A sequene-pair [2] is an orere pair of Γ + an Γ to represent a plaement. Eah of Γ + an Γ is a permutation of names of given n loks. If lok x is the i-th in Γ +,weenote Γ + (i) =x, aswellasγ 1 + (x) =i. Similar notation is also use for Γ. For every lok pair (a, ), a is the left of (equivalently, is the right of a) ifγ 1 + (a) < Γ 1 + () an Γ 1 (a) < Γ 1 (). Analogously, a is elow (equivalently, is aove a)ifγ 1 + (a) > Γ 1 + () an Γ 1 (a) < Γ 1 (). The single-sequene [4], [6], [9], [10] an represent a plaement s topology without speifying lok s name. It is efine as S(k) =Γ 1 + (Γ (k)),thatis,s is the same as Γ when eah lok is rename as Γ + =(1, 2,...,n). III. TOPOLOGICAL SYMMETRY STRUCTURE Symmetry strutures are require in analog plaement to math evie pairs. Relate works in [3], [7] introue onstraint-riven approahes to realize a plaement satisfying the symmetry requirement.however, the speifiation of the /08/$ IEEE 186

2 axis vertial line horizontal line Fig. 2. An example of generation of symmetry struture. isshownintaleianfig. 2. Γ (k) an Δ + (k) (or Γ + (k) an Δ (k)) orrespon to a symmetry-pair. Note that if they orrespon to the same lok, it means a self-symmetry. 1. Make a hek-list of (1, 2,...,n). Set Γ + as (1, 2,...,n), Δ + as (n, n 1,...,1), Γ an Δ as empty, an k =1. Fig. 1. An example of sequene-pair with horizontal symmetry topology. onstraint is a tough task, an no effiient generation of appropriate onstraint has een estalishe yet. Therefore, we introue the generation of symmetry strutures without speifying any pair of evies or name of any lok. It is possile that a plaement naturally goes to a symmetry struture. Let a sequene-pair e SP =(Γ +, Γ ), an let the reverse sequenes of Γ + an Γ e Δ + an Δ, respetively. Originally, a sequene-pair is efine y the arrangement of loks from the left-sie to the right-sie over the hip. We all suh a sequene-pair LR-SP. On the other han, if a sequene-pair is efine from the right-sie to the left-sie, the sequene-pair is alle RL-SP. We notie that RL-SP is (Δ, Δ + ). Γ + (k) an Δ + (n k +1)orrespon to the same lok as well as Γ (k) an Δ (n k +1)o to the same one. If the LR-SP an RL-SP inue the same single-sequene, that is, Γ 1 + (Γ (k)) = Δ 1 (Δ +(k)), (1) then we say that the sequene-pair has a horizontal symmetry topology. Given an n n gri,assigning n loks into the gri aoring to (Γ +, Γ ) will result in the orresponing plaement. Denote k as j, enote Γ 1 + (Γ (k)) an Δ 1 (Δ + (k)) as i. The equation (1) means that if a lok was assigne to the joint of i-th horizontal line an the j-th vertial line, another lok(or itself, if it is on the axis) will appear on the symmetri position whih is the joint of reverse i-th vertial line an the reverse j-th horizontal line. The symmetry axis is the iagonal of the gri. Fig. 1 shows an example, the gri has een rotate 45 egree. The LR-SP(Γ +, Γ )=(ae, ae) an the RL-SP(Δ, Δ + )=(ea, ea) introue the same single-sequene whih is (13425), so the sequene-pair has a horizontal symmetry topology In the following, we introue the generation of a singlesequene with an aritrary horizontal symmetry topology. A horizontal symmetry topology has a single vertial axis. It an e also extene to a horizontal axis or plural axes, ut the extension is omitte here for the spae limitation. An example 2. Stop the proeure if all numers have een assigne to Γ, that is, the hek-list is empty. 3. Assign an aritrary numer x in the hek-list to Γ (k), an remove x from the hek-list. 4. Assign n k +1to oth Δ (x) an Γ (n x +1),an remove n k +1from the hek-list. 5. Inrement k until Γ (k) has not een assigne to, an return to step 2). Furthermore, in the tale, x = 2 an x = 1 are hosen aritrarily (see step 3) aove). If we take other hoies, the resultant Γ woul e ifferent. In other wors, we an ontrol the generation of a single-sequene with a horizontal symmetry topology y hoosing these numers. TABLE I AN EXAMPLE OF SINGLE-SEQUENCE GENERATION WITH A HORIZONTAL SYMMETRY TOPOLOGY. k xn k +1 n x +1 Γ an Δ hek-list initial (-, -, -, -, -) (1,2,3,4,5) (2, -, -, -, -) (1,3,4,5) (2, -, -, 5, -) (1, 3, 4) (2, 1, -, 5, -) (3, 4) (2, 1, -, 5, 4) (3) (2, 1, 3, 5, 4) (-) Consier the realization of a symmetry plaement from a sequene-pair with a horizontal symmetry topology. Prior works [3], [7] introue a speial alulation to plae a pair of loks impose a symmetry-onstraint on so that the y-axis eomes the enter etween them an their y-oorinates are the same [3], [7]. But, they suffer from the time-onsuming alulation of y-oorinates, eause it often nees several iterations to align y-oorinates of a horizontal symmetry-pair. However, we an give a single path alulation of y-oorinates of symmetry-pairs as long as the sequene-pair has a horizontal symmetry topology. Theorem III.1 (Vertial Feasiility) For eah symmetry-pair, if their y-oorinate is the same oorinate in the plaement, 187

3 the plaement is alle vertial feasile. A vertial onstraint graph inue y a sequene-pair with a horizontal symmetry topology is given. It has no irete yle. Y-oorinate of every lok of a vertial feasile plaement an e alulate in the time omplexity that is linear to the numer of eges of the onstraint graph. IV. SYMMETRY-ORIENTED OPTIMIZATION Simulate annealing is aopte as the optimization framework. In the following, we will esrie (i) generation of an initial plaement, (ii)evaluation of symmetry, (iii) ost funtion, (iv) move operation, (v) physial skewe symmetry struture. A. Generation of Initial Plaement As we esrie aove, we an generate a single-sequene satisfying a topologial omplete symmetry. Let the singlesequene e S symm. We assign loks to S symm, then generate the initial plaement. Note that this assignment orrespons to the onfiguration of a sequene-pair. If two numer onsist one symmetry-pair, one numer alls the other partner. Inthis assignment, loks are lassifie aoring to their size so that loks of the similar size elong to the same group. Hene, every lok is assigne to S symm suh that the partner is hosen from the same group. S symm onsists of a set of symmetry-pair numers an selfsymmetry numers. We set a limitation on the amount of selfsymmetry numers in orer to avoi too muh self-symmetry loks. Although our onept is onstraint-less, we are still ale to omine the onstraint-riven approah to our plaement. For a partiular pair of loks shoul e symmetrial, i.e. they nee to satisfy one symmetry onstraint, we assign them into a symmetry-pair numers, make them as the partner of eah other. B. Evaluation of Symmetry Topologial struture value V top an Physial imension ost C phy introue in [11] are use to evaluate the quality of a struture plaement. Multi-rows an arrays are extrate so the two values involving some fators suh as loal ompatness an e alulate. Sine we an ontrol the generation of the initial symmetry plaement, the symmetry strutures an e extrate uring the generation, then those fators of symmetry strutures an e also alulate. Given a symmetry x, x enotes the numer of loks an x self enotes the numer of self-symmetry loks. The with an the height of a lok are enote y w(), h() respetively. The topology of a symmetry x is value as: V top (x) = x x self ; (2) an the loal symmetriity of a symmetry x is value as: C sym (x) = w() w( ) h() h( ). (3) (, x) Although in this paper we fous on the symmetry of a plaement, multi-rows an arrays are still preferre in an analog layout, so we omine the fators of symmetries into V top an C phy. Let sets of arrays, multi-rows an symmetries e A, R an X, respetively. The Topologial struture value is efine as: V top = α σ(r)+β σ(a)+γ self ), r R a A x X( x x (4) where the first two items in formula(4) are the same in [11], σ figures out the aspets of arrays an multi-rows. α, β anγ are oeffiients to alane a trae-off among struture values. The Physial imension ost is efine as: C phy = α a A (C mp(a)+c uni (a)) +β r R (C mp(r)+c uni (r)) (5) +γ x X C sym(x), where the first two items in formula(5) are the same in [11], C mp an C uni esrie the loal ompatness an uniformity respetively, α, β an γ are oeffiients to alane a trae-off among osts. V top ontriutes to esrie the topologial shape of a plaement an C phy oes to esrie the loal ompatness an uniformity. C. Cost Funtion Our ost funtion E for a plaement P is esigne as follows. E(P )= Area(P ) WLen(P ) g(v top ) g(c phy ), where Area(P ) an WLen(P ) are the hip area an the wire length of P, respetively. g is a onversion that maps an input value to another value within [1.0, 1.1), an it is efine as: xm ln(0.5) g(x) = exp( x+ɛ ),wherex m is an average value of {x} an ɛ is a small value enough to ignore. The meaning of the funtion g is to e likely to egrae the hip area or the wire length y 10% to otain etter topologial struture value or less physial imension ost. D. Moves Keeping Symmetry We propose new moves to maintain the topologial symmetry uring an annealing proess. For every move, we hoose two loks a an. Our moves are explaine as a omination of FullExhange(a, ) an HalfExhange(a, ) introue in [2]. For the ompleteness, we riefly esrie these moves. FullExhange(a, ): Pair interhange of names of two lok a an in oth Γ + an Γ. HalfExhange(a, ): Pair interhange of two loks names in either Γ + or Γ. In the following, we esrie all the moves use in a simulate annealing. The simulate annealing ontrols the seletion of moves aoring to ranom values. 188

4 A: FullExhangeOfSymm on lok a an a a' A: omplete topologial symmetry a a' B: non-omplete topologial symmetry * * * * * * a a' C: Put ummy loks to get skewe symmetry Fig. 4. Physial Skewe Symmetry Struture. E. Physial Skewe Symmetry Struture B: HalfExhangeOfSymm on lok a an Fig. 3. Moves Keeping Symmetry. 1. RotateBlok(a): Rotate the lok a y 90 egree. 2. FlipBlok(a): Flip the lok a horizontally. 3. FullExhangeOfSymm(a, ): Let partners of a an e a an, respetively. If a is equal to, apply FullExhange(a, ) introue in [2]. Otherwise, apply oth FullExhange(a, ) an FullExhange(a, ). An example is shown in Fig. 3(A). 4. HalfExhangeOfSymm(a, ): Let partners of a an e a an, respetively. If a an are a symmetry-pair or two self-symmetry loks, apply HalfExhange(a, ) introue in [2]. Note that in this exhange, two selfsymmetry loks will e turne into a symmetry-pair, while a symmetry-pair will e separate into two selfsymmetry loks. Otherwise, apply HalfExhange(a, ) on Γ + (or Γ ) an HalfExhange(a, )onγ (or Γ + ). An example is shown in Fig. 3(B). Beause of the limitation of the lok s size or shape, a omplete symmetry plaement may lea to too muh ompromise on hip area an wire length. Fig. 4 shows an example with 5loks:a, a,,, an. a an a havethesamesizean shape. A omplete symmetry-oriente plaer trens to get a plaement like Fig. 4(A) whih is omplete topologial symmetry, ut might not so goo at the area. But there oul e another solution like Fig. 4(B), whih an not e otaine y a omplete symmetry plaement, ut the plaement still looks symmetry eause the area of it s left sie an right sie are almost same. In orer to make improvement on those topologial omplete symmetry plaement, we insert some ummy loks into the single-sequene, an the size of eah ummy lok is 0. In a symmetry struture, if a lok s symmetry partner is a ummy lok, we say that they formulate a Physial Skewe Symmetry Struture. During the generation of the initial plaement an the optimization, the single-sequene keeps eing topologial symmetry. The ummy loks are ignore in the symmetry oorinate alulation, so it is possile to generate a physial skewe symmetry plaement inluing non-symmetry parts. As a result we may get a more ompat plaement. In Fig. 4(C), ummy loks, an are inserte into the single-sequene, an form a topologial omplete symmetry plaement, finally help to lea to the result like Fig. 4(B). 189

5 BLK[35 ] BLK[34 ] BLK[24 ] BLK[19 ] BLK[11 ] BLK[122 ] BLK[63 ] BLK[33 ] BLK[32 ] BLK[25 ] BLK[27 ] BLK[106 ] BLK[1 ] BLK[23 ] BLK[111 ] BLK[6 ] BLK[113] BLK[144] BLK[114] BLK[133] BLK[37 ] BLK[124] BLK[152] BLK[106] BLK[114] BLK[127] BLK[126] BLK[125] BLK[160] BLK[165] BLK[31 ] BLK[124] BLK[136] BLK[133] BLK[137] BLK[47 ] BLK[141] BLK[139] BLK[17 ] BLK[32 ] BLK[48 ] BLK[44 ] BLK[49 ] BLK[132] BLK[53 ] BLK[162] BLK[105] BLK[40 ] BLK[41 ] BLK[135] BLK[36 ] BLK[39 ] BLK[138] BLK[121] BLK[113] BLK[145] BLK[142] BLK[125] BLK[148] BLK[149] BLK[46 ] BLK[35 ] BLK[34 ] BLK[109] BLK[107 ] BLK[108] BLK[19 ] BLK[131] BLK[130] BLK[140] BLK[45 ] BLK[55 ] BLK[154] BLK[49 ] BLK[82 ] BLK[33 ] BLK[31 ] BLK[12 ] BLK[13 ] BLK[37 ] BLK[44 ] BLK[29 ] BLK[30 ] BLK[51 ] BLK[26 ] BLK[27 ] BLK[25 ] BLK[11 ] BLK[24 ] BLK[19 ] BLK[20 ] BLK[128 ] BLK[158] BLK[76 ] BLK[118] BLK[139] BLK[117] BLK[122] BLK[143] BLK[147] BLK[146] BLK[119] BLK[134] BLK[123] BLK[120] BLK[142] BLK[145] BLK[138] BLK[149] BLK[164] BLK[130] BLK[129] BLK[137] BLK[128] BLK[116 ] BLK[162] BLK[107] BLK[109] BLK[148] BLK[108] BLK[111] BLK[144 ] BLK[79 ] BLK[94 ] BLK[93 ] BLK[126] BLK[115] BLK[104] BLK[103] BLK[141 ] BLK[117] BLK[134] BLK[143] BLK[127] BLK[73 ] BLK[118] BLK[155] BLK[14 ] BLK[152] BLK[132] BLK[151] BLK[165] BLK[161] BLK[154] BLK[163 ] BLK[157] BLK[135 ] BLK[153] BLK[112] BLK[101] BLK[102] BLK[150] BLK[121] BLK[158] BLK[140] BLK[164 ] BLK[150] BLK[156] BLK[101 ] BLK[110] BLK[163] BLK[161] BLK[129] BLK[155] BLK[136] BLK[131] BLK[157] BLK[159] BLK[115] BLK[159] BLK[160] BLK[116] BLK[120] BLK[103] BLK[18 ] BLK[23 ] BLK[22 ] BLK[156] BLK[22 ] BLK[110] BLK[102] BLK[112] BLK[119] BLK[151] BLK[146] BLK[147 ] BLK[104] BLK[105] BLK[153] BLK[123] 2C-2 ata A: symmetry ata C: normal ata D: normal ata A: normal ata C: omplete symmetry ata B: normal ata D: omplete symmetry ata C: physial skewe symmetry ata D: physial skewe symmetry ata B: symmetry Fig. 5. Resultant plaements of ata A, B, where the primary ojetive is the prout of the hip area an the wire length: normal an symmetry are normal plaement an our symmetry plaement respetively. V. EXPERIMENTS We implemente our symmetry-oriente struture plaement, an performe it on 14 instanes of analog lok sets from inustries. A normal lok plaement ase on sequenepair was also implemente. A. Comparison with normal plaement First, we ompare our struture plaement with normal one. Fig. 5 shows two pairs of resultant plaements. The results of normal plaement are well ompate, ut not well organize in loal area, an lak symmetry. Our symmetry struture plaement is well ompate either an istinguishes for the perfet symmetry generate without any onstraint. Sine we omine the evaluation of row an array strutures, our plaement also shows goo quality on loal ompatness an regularity. Our plaement is stritly limite to e symmetrial, ut the hip area an the wire length are not neessarily inreasing heavily ue to this limitation. The numerial results are shown in Tale II. The normal plaement an our plaement are enote y normal an symm respetively. The tale inlues the results aout the hip area an the wire length y oth plaements. The results show, as we expete, our plaement with a omplete symmetry struture oul e otaine y the ompro- Fig. 6. Resultant plaements of ata C an D, where normal, omplete symmetry an physial skewe symmetry are normal plaement, omplete symmetry plaement an physial skewe symmetry plaement respetively. mise of at most aout 5 % on average, ompare to the normal plaement. B. Physial Skewe Symmetry Seonly, we emonstrate the effetiveness of physial skewe symmetry struture. For the same instanes, we generate the ummy loks as the 10%, 20%, an 30% of the numer of input loks, an teste eah ase. The numerial test result is shown in Tale III. In this tale, values aout hip area an wire length are ratios to the hip area an wire length y symm in Tale II. The olumns of Best show the est result among the those of ummy 0%, ummy 10%, ummy 20% an ummy 30%. We attaine the 6% an 14% reution on average with respet to hip area an wire length. Fig. 6 shows the result y normal, omplete symmetry an physial skewe symmetry. We are onvine that the insertion of the ummy loks expans the solution spae of the symmetry-oriente plaement, an helps to reue the hip area an wire length. VI. CONCLUSION This paper presente a struture plaement fousing on topologial symmetry struture. Unlike the existing onstraintriven approah, it oes not nee any onstraint. We formu- 190

6 TABLE II THE NUMERICAL RESULT OF ANALOG BLOCK DESIGNS, WHERE WLEN IS THE WIRE LENGTH, NORMAL AND SYMMETRY REFER TO NORMAL PLACEMENT AND OUR SYMMETRY PLACEMENT RESPECTIVELY. ata #loks #nets normal symm (symm-normal)/symm area(μm 2 ) wlen(μm) time(se) area(μm 2 ) wlen(μm) time(se) area(%) wlen(%) A ,660 9,356 1, ,940 10, B ,784 5, ,082 7, C ,068 16,520 1, ,420 14, D ,090 2, ,844 3, E ,077 3, ,760 3, F ,105 9, ,968 9, G ,622 5, ,543 6, H ,631 4, ,953 3, I ,683 2, ,008 2, J ,507 15,576 7,926 80,565 12,645 1, K ,860 38, ,491 36, L ,452 3,392 1,868 45,456 4,692 1, M ,753 8, ,151 10, N ,799 3, ,681 3, average: TABLE III THE NUMERICAL RESULT OF ANALOG BLOCK DESIGNS USING Physial Skewe Symmetry Struture. ata ummy 0% ummy 10% ummy 20% ummy 30% Best(0 30%) area wlen #ummy area(%) wlen(%) #ummy area(%) wlen(%) #ummy area(%) wlen(%) area(%) wlen(%) A B C D E F G H I J K L M N average late the property of a topologial omplete symmetry in terms of single-sequene. Besies, we propose an effiient move use in a simulate annealing to maintain the topologial symmetry of every plaement. Furthermore, we introue physial skewe symmetry struture to ompromise symmetry for less hip area or wire length. In experiments, we showe that our approah generate topologial omplete plaements without muh ompromise on hip area an wire length, ompare to the plaements with no symmetry. In future works, we are going on stuying the struture plaement with hierarhial regular strutures suh as array, row, an symmetry. REFERENCES [1] J.Cohn, D. Garro, R.Rutenar an L.Carley, Analog Devie-level Automation, Kluwer Aa. Puli., [2] H. Murata, K. Fujiyoshi, S. Nakatake, an Y. Kajitani, Retanglepaking-ase moule plaement, Pro. ICCAD 1995, pp , [3] B. Balasa an K. Lampaert, Symmetry within the sequene-pair representation in the ontext of plaement for analog esign, IEEE Trans. on CAD, Vol.19, No.7, pp , [4] C. Koama an K. Fujiyoshi, Selete Sequene-Pair: An effiient eoale paking representation in linear time using Sequene-Pair, Pro. ASP-DAC 2003, pp , [5] T. Nojima, X. Zhu, Y. Takashima, S. Nakatake, an Y. Kajitani, Multilevel plaement with iruit shema ase lustering, Pro. ASP-DAC 2004, pp , [6] X. Zhang an Y. Kajitani, Spae-planning: Plaement an moules with ontrolle empty area y Single-Sequene, Pro. ASP-DAC 2004, pp.25-30, [7] F. Balasa, S. C. Maruvaa, an K. Krishnamoorthy, On the exploration of the solution spae in analog plaement with symmetry onstraints, IEEE Trans. on CAD, Vol.23, No.2, pp , [8] T. Nojima, Y. Takashima, S. Nakatake, an Y. Kajitani, A evie-level plaement with multi-iretional onvex lustering, Pro. GLSVLSI 2004, pp , [9] X. Zhu, X. Zhang, an Y. Kajitani, A general paking algorithm ase on Single-Sequene, Pro. ICCCAS 2004, pp , [10] X. Zhang an Y. Kajitani, Thoery of T-juntion floorplans in terms of single-sequene, Pro. ISCAS 2004, pp , [11] S. Nakatake, Struture Plaement with Topologial Regularity Evaluation, Pro. ASP-DAC 2007, pp