TPL-Aware Displacement-driven Detailed Placement Refinement with Coloring Constraints

Transcription

1 TPL-ware Dsplacement-drven Detaled Placement Refnement wth Colorng Constrants Tao Ln Iowa State Unversty Chrs Chu Iowa State Unversty BSTRCT To mnmze the effect of process varaton for a desgn n trple patternng lthography (TPL), t s benefcal for all standard cells of the same type to share a sngle colorng soluton. In ths paper, we nvestgate the TPL-aware detaled placement refnement problem under these colorng constrants. Gven an ntal detaled placement, the postons of standard cells are perturbed and a TPL soluton complyng wth the colorng constrants s derved whle mnmzng cell dsplacement, lthography conflcts and sttches. We prove that ths problem s NP-complete and show that t can be formulated as a mxed nteger lnear program. Snce mxed nteger lnear programmng s very tme consumng, we propose an effectve heurstc algorthm. In our approach, mportant adjacent pars of standard cells are recognzed frstly, snce they have sgnfcant mpact on cell dsplacement. Then a tree-based heurstc s appled to generate a good ntal soluton for our lnear programmng-based refnement. Expermental results show that compared wth mxed nteger lnear programmng, our heurstc approach s comparable n soluton qualty whle usng very short CPU runtme. 1. INTRODUCTION Wth the technology node scalng to sub-16nm, electron beam (E-beam), extreme ultravolet lthography (EUVL) and TPL are consdered the most promsng lthography technologes. In ths paper, we are focusng on TPL. There are many prevous works on TPL optmzaton. The fundamental problem of TPL s to elmnate lthography conflcts whle mnmzng sttch count. [1 8] are related to TPL layout decomposton. [1 4] focus on -Dmenson layout decomposton. [5, 6] focus on row-based 1-Dmenson layout decomposton. [9, 10] consder TPL durng detaled routng stage. Recently, [11] presents a TPL aware detaled placement approach n whch layout decomposton and placement are resolved smultaneously. The approach s effectve n resolv- Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. Copyrghts for components of ths work owned by others than CM must be honored. bstractng wth credt s permtted. To copy otherwse, or republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. Request permssons from permssons@acm.org. ISPD 15 March 9 prl 1, 015, Monterey, C, US. Copyrght c 015 CM ng lthography conflcts. However, the approach only consders the optmzaton of wrelength together wth lthography conflcts and sttch number. It s not clear how to ncorporate other placement objectves lke tmng and routablty. Besdes, [6] ponts out the advantage of assgnng the same lthography pattern for the same standard cell type durng TPL layout decomposton. Ths would mnmze the effect of process varaton and best guarantee that those standard cells of the same type eventually have smlar physcal and electrcal characterstcs. However, [6] only consders the decomposton of a fxed layout, and hence often cannot completely satsfy these constrants. In ths paper, we nvestgate the TPL-aware detaled placement refnement problem under the colorng constrants that all standard cells of the same type should share the same TPL colorng soluton. Gven an ntal detaled placement, the postons of standard cells are perturbed and a TPL soluton complyng wth the colorng constrants s derved whle mnmzng total cell dsplacement, lthography conflcts and sttches smultaneously. Dfferent from [11], our approach s appled to an optmzed detaled placement under any conventonal placement metrcs. By refnng t wth mnmal perturbaton, the qualty of the detaled placement can be preserved. In addton, we consder the colorng constrants. Compared wth [6], as placement perturbaton s allowed, the colorng constrants are always satsfed n our approach. We prove that ths problem s NP-complete and show that t can be formulated as a mxed nteger lnear program (MILP). Snce the MILP s tme consumng to solve, we propose an effectve heurstc algorthm to solve t. In our algorthm, mportant adjacent pars of standard cells are recognzed frstly, snce they have sgnfcant mpact on cell dsplacement. Then a treebased heurstc s appled to generate a good ntal soluton whch s then refned by a lnear programmng (LP)-based technque. Expermental results show that compared wth MILP soluton, the heurstc method s comparable n soluton qualty whle usng very lmted CPU runtme. The contrbutons of ths paper are summarzed as follows. We formulate a new TPL optmzaton problem consderng TPL colorng constrants for standard cells durng detaled placement. We prove that ths new problem s NP-complete. We propose a MILP formulaton for ths new problem. Snce MILP s very tme consumng to solve, we propose an effectve heurstc algorthm.

2 B B B C C C (a) Gven ntal detaled placement. (b) One soluton: try to optmze the dsplacement of the second row. (c) nother soluton: try to optmze the dsplacement of the frst row. Fgure 1: n nstance of problem: choosng dfferent colorng solutons for types, B and C plus cell shftng. The rest of paper s organzed as follows. In Secton, we gves the formal problem defnton and ts MILP formulaton. In Secton 3, we prove that ths problem s NPcomplete. In Secton 4, we llustrate the heurstc algorthm. In Secton 5, we present the expermental results. Fnally, we make our conclusons n Secton 6.. PROBLEM DEFINITION Gven a standard cell lbrary, all feasble colorng solutons for each cell type are found out frstly. Snce each cell contans only a small number of layout features, the enumeratve approach proposed n [11] works well. Besdes, ths step s performed once per lbrary. For the -th type of cell denoted by t, there are n feasble colorng solutons p 1, p,, p n. The correspondng sttch counts are s 1, s,..., s n. The wdth of t s w. There are k types of standard cells n the lbrary. Gven a detaled placement, whch has n rows. For the j-th row, the types of standard cells ordered from left to rght are c 1 j, c j,, c r j j, where rj s the number of cells n the j-th row. The TPL-aware dsplacement-drven detaled placement wth colorng constrants s defned as follows. Gven a standard cell lbrary wth a set of feasble colorng solutons for each standard cell type, and an ntal detaled placement, elmnate all lthography conflcts by choosng one colorng soluton for each type of standard cell and shftng the standard cells wthout changng the cell orderng n each row. The objectve s to mnmze the total cell dsplacement and the number of sttches. Fg. 1 gves an nstance of ths problem. By choosng colorng solutons for types, B and C and shftng cells, conflcts are elmnated. In Fg. 1(a), an ntal detaled placement wth two rows s gven. In Fg. 1(b), cell dsplacement of the second row s optmzed well whle that of the frst row s not. On the contrary, n Fg. 1(c), cell dsplacement of the frst row s optmzed well whle that of the second row s not. It shows that dfferent TPL solutons may lead to sgnfcantly dfferent cell dstrbuton n each row..1 MILP formulaton The above problem can be formulated as a MILP. We use a bnary varable b j to denote whether the colorng soluton p j s assgned to standard cell type t. In the -th row, the orgnal central x-coordnates of cells ordered from left to rght are o 1, o,, o r, ther new central x-coordnates are x 1, x,, x r, ther dsplacement are q1, q,, q r. For any two adjacent cells, the type of left one s t and ts colorng soluton s p u, the type of rght one s t j and ts colorng soluton s p v j. To avod lthography conflct, the mnmal dstance between these two cells s a constant denoted by d u,v,j. For any two adjacent cells n the row, let x j 1 and x j be ther central x-coordnates, ther actual dstance s denoted by z j. Besdes, the wdth W of placement regon s also gven. The problem can be formulated nto the followng mathematcal programmng. Note that n ths paper, for any par of adjacent cells, the dstance s from the center of the left one to the center of the rght one. Mnmze: α n Subject to: n j r b j j=1 r n c j b k c j =1 j=1 k=1 = 1, 1 k s k + β n c j r q j =1 j=1 x j xj 1 = z j, 1 n j r n j 1 n j c z j c b u b v d u,v, 1 n c j 1 c j c j 1 u=1 v=1,c j x j oj qj, 1 n 1 j r o j xj qj, 1 n 1 j r j c, 1 n 1 j r W j c, 1 n 1 j r = 0 or 1, 1 k 1 j n b j The objectve s a weghted sum of total cell dsplacement and sttch count. The frst constrant represents that standard cells of the same type should have the same colorng soluton. The second and thrd constrants represent that for any two adjacent cells, there s enough dstance to avod lthography conflct. The fourth and ffth constrants represent cell dsplacement. Fnally, the last two constrants mean that cells should be put nsde of placement regon. The product of two bnary varables n the thrd constrant can be transformed nto lnear constrants as follows: c = a b a + b c 1 a c 0 b c 0, where a, b, c are all bnary varables. Therefore, the problem can be formulated as a MILP. 3. COMPLEXITY OF PROBLEM To see the complexty of ths problem, let us look at a specal verson of ts decson problem frstly.

3 x 1 x x 3 x 4 (a) n nstance of 3-colorng problem. The three colors are RED, BLUE and GREEN. t 1 t t 0 t 1 t 3 t 0 t t 3 t 0 t 3 t 4 (b) n nstance of sngle-row verson. The wdths of cells are 1. The wdth of row s 11. For any type of standard cell t, t has three feasble colorng solutons ( ) p 1, p, p 3. p 1, p and p 3 are respectvely correspondng to RED, BLUE and GREEN. Fgure : The reducton from 3-colorng problem to sngle-row verson. Defnton 1 (Sngle-row verson). The gven ntal detaled placement has only one row. The problem s to decde whether there s a feasble soluton to accommodate all cells wthout conflcts. Theorem 1. The sngle-row verson s NP-complete. Proof. It s easy to see that the sngle-row verson s NP. We show that the 3-colorng problem can be reduced to sngle-row verson. Snce the 3-colorng s NP-complete [1], the sngle-row verson s NP-complete. Suppose n a 3-colorng problem nstance, there are n nodes denoted by x 1, x,, x n. There are m edges denoted by e 1, e,, e m. We can construct the followng sngle-row verson nstance. Each node x s correspondng to one type of standard cell t, whch has three feasble colorng solutons p 1, p, p 3. p 1, p and p 3 are correspondng to RED, BLUE and GREEN respectvely. There s a specal type of standard cell t 0. The wdth of standard cells are all 1. We defne the mnmal dstance between t and t j to elmnate conflct as follows. d u,v,j = 1 f u v and 0 and j 0 f u = v and 0 and j 0 1 f = 0 or j = 0 It means that for any par of adjacent cells, f the type of ether one s t 0, the mnmal dstance between these two cells to avod conflct s 1 no matter what the fnal colorng solutons are. Otherwse, f the left one s assgned the colorng soluton whch s correspondng to p k (1 k 3) and the rght one s assgned the colorng soluton whch s correspondng to p k j, the mnmal dstance between these two cells to avod lthography conflct s. Otherwse the mnmal dstance s 1. For any two nodes x and x j, suppose < j wthout loss of generalty. If there s an edge e = (x, x j), then we construct a par of adjacent cells (t, t j). Besdes, we add a standard cell of type t 0 between any two pars of constructed adjacent cells. nd the wdth of row s defned as the number of constructed standard cells,.e., 3m-1. Fg. (b) shows the correspondng sngle-row verson nstance of the 3-colorng problem nstance n Fg. (a). If the above 3-colorng problem nstance s true, then n the constructed sngle-row verson nstance, for any two adjacent cells t and t j ( < j), we can choose the colorng solutons so that the mnmal dstance between these two cells to avod lthography conflct s 1. Therefore, all the constructed standard cells can be put nsde of the row. Smlarly, f sngle-row verson nstance s true, then we can fnd a soluton that satsfes the correspondng 3-colorng problem nstance. The dsplacement-drven TPL-aware detaled placement wth orderng and colorng constrants s a generalzaton of the sngle-row verson, so t s also NP-complete [1]. 4. METHODOLOGY Snce the problem s NP-complete and MILP s very tme consumng, we propose an effectve heurstc algorthm to solve ths problem. In ths secton, we frstly show the motvaton of our approach. Next, we present ts overvew whch s composed of three stages. Fnally, we llustrate these three stages respectvely. 4.1 Motvaton Snce standard cells of the same type should have the same colorng soluton, we defne adjacent par as follows. Defnton. n adjacent par s a par of types of two adjacent standard cells. For example, f the type of left cell s t and the type of rght one s t j, the correspondng adjacent par s (t, t j). The mnmal dstances of adjacent pars to avod lthography conflcts have sgnfcant mpact on soluton qualty of ths problem. There are two reasons. Frstly, f these mnmal dstances are not optmzed well, then t would be dffcult to put all cells nsde of the row regon, as shown n Fg. 3(a). Secondly, dfferent adjacent pars have dfferent mpact on total cell dsplacement, as shown n Fg. 3(b). Therefore, our method tres to focus on the mnmal dstances of mportant adjacent pars. 4. Overvew Our approach s composed of three stages. In the frst stage, we propose a method to recognze the mportant adjacent pars. In the second stage, we try to optmze mnmal dstances of mportant adjacent pars and a tree-based

4 B C C B B B C C B B (a) The upper fgure represents that all the cells are put nsde of row f the mnmal dstances (to elmnate lthography conflcts) of adjacent pars are optmzed well. On the contrary, n the lower fgure, the rght most cell B s outsde of row f those mnmal dstances of adjacent pars are not optmzed well. B C C C C B C C C C (b) In the upper fgure whch represents the orgnal placement, the left-most adjacent par (cell B and cell C) s the most mportant one to optmze cell dsplacement. If the mnmal dstance of ths par to elmnate conflct s not optmzed well, all the other cells on the rght hand sde would be shfted rght as shown n lower fgure. Fgure 3: The two examples reveal the motvaton of our heurstc approach. heurstc s appled to get a good ntal soluton. In the last stage, we apply LP-based method to refne the soluton. The overvew s presented n Fg. 4. Standard cell lb Detaled placement Recognze mportant adjacent pars Tree-based heurstc LP-based refnement End Estmate cell dstrbuton Calculate the weghts of adjacent pars Generate soluton graph Generate maxmum spannng tree Dynamc programmng Fgure 4: The overvew of our heurstc approach. 4.3 Important adjacent par recognton We use a postve nteger to represent how mportant an adjacent par s. We call ths nteger the weght of adjacent par. Hgher weght means more mportant. For example, as shown n Fg. 3(b), apparently, the adjacent par (B, C) should have the hghest weght. We use weght[][j] to denote the weght of adjacent par (t, t j). t ths stage, we do not know what the fnal colorng s. Therefore, we propose a smple method to estmate the new cell dstrbuton. For any adjacent par (t, t j), we calculate the average mnmal dstance d ave,j to avod lthography conflct. Ths value s gven by the followng formula. d ave,j = n n j u=1 v=1 n n j d u,v,j The mnmal total cell dsplacement can be acheved by LP as follows. Mnmze: n r q j =1 j=1 Subject to: x j xj 1 d ave c j 1,c j, 1 n j r x j oj qj, 1 n 1 j r o j xj qj, 1 n 1 j r j c, 1 n 1 j r W j c, 1 n 1 j r Then we defne shftng drecton of standard cell below. Defnton 3. For the j-th standard cell n row r, ts shftng drecton s left f x j < o j, and rght f xj > o j, otherwse no shftng. We use to denote left shftng, for rght shftng, and = for no shftng. lgorthm 1 gves the method to calculate the weghts of adjacent pars. The dea s that for a par of adjacent cells, f ther mnmal dstance to elmnate conflct s ncreased, the weght of ths par would roughly reflect the ncrement of total cell dsplacement. Let us look at an example. placement row contans sx cells and fve adjacent pars. The shftng drectons of these sx cells are,,,,,. The fve adjacent pars weghts ordered from left to rght are respectvely 5, 4, 3, and 1. The weght of the left-most one s 5, because f ts mnmal dstance s ncreased by 1 unt, the total cell dsplacement would be ncreased by 5 unts roughly. 4.4 Tree-based heurstc fter the weghts of all adjacent pars are computed, a soluton graph can be constructed as follows. In the soluton graph, each node represents a standard cell type. The edge between two nodes represents an adjacent par. Let f be the colorng soluton that standard cell type t uses. The cost cost of node t and the cost cost,j of edge connectng t and t j n the soluton graph are defned as follows. cost [f ] = β weght[][] d f,f, cost,j[f, f j] = β [weght[][j] d f,f j,j + α s f +weght[j][] d f j,f j, ] The purpose of our tree-based heurstc s to fnd the colorng soluton for each standard cell type, so that the total cost ncludng cost of nodes and edges n the soluton graph s mnmzed. It s not hard to see that f soluton graph s

5 lgorthm 1 Method to calculate the weghts of adjacent pars 1: Calculate d ave,j for each par of adjacent par (t, t j ); : Solve the LP to get the shftng drecton of each standard cell; 3: for each placement row do 4: [start, end] s the ndex range of cells (n ascendng order of ther x-coordnate) n ths row; 5: for any adjacent par P = (t, t j ) n the row do 6: ll and rr are the ndexes of t and t j n the row; 7: f the left cell s then 8: for k from ll to start do 9: f the cell whose order s k s or = then 10: weght[][j]+ = 1; 11: else 1: break; 13: end f 14: end for 15: end f 16: f the rght cell s then 17: for k from ll + 1 to end do 18: f the cell whose order s k s or = then 19: weght[][j]+ = 1; 0: else 1: break; : end f 3: end for 4: end f 5: end for 6: end for of a tree structure, then dynamc programmng can be appled to get the optmal colorng soluton. Fortunately, t s observed that soluton graphs for ndustral benchmarks are sparse graphs. Next, we propose a method to leverage ths observaton Maxmum spannng tree generaton The basc dea to leverage the observaton s to gnore some relatvely less mportant adjacent pars and turn the soluton graph nto a tree. The cost of each edge connectng t and t j n soluton graph s replaced by cost,j = α [weght[][j] ( d max,j where d max,j and d mn,j d max,j d mn,j d mn,j ) + weght[j][] ( d max are defned as follows. = max 1 u n max 1 v nj d u,v,j = mn 1 u n mn 1 v nj d u,v,j j, d mn j, It s easy to see that for any edge connectng t and t j, f cost,j s small, then no matter what the fnal colorng solutons for t and t j are, the cost of ths edge n the soluton graph s smlar. Therefore, we use maxmum spannng tree to replace the orgnal soluton graph. Note that, cost,j s only used durng generatng maxmum spannng tree rather than the followng dynamc programmng Dynamc programmng soluton fter maxmum spannng tree s generated, dynamc programmng could be appled to fnd an ntal colorng soluton. We use the node whch has maxmal out-degree as the root to generate the tree topology. Then bottom-up method s adopted to construct optmal solutons n the tree. For any node t, we mantan a vector Best[]. The entry Best[][j] stores the best cost over all possble colorng solutons for the sub-tree rooted at node t f t s choosng colorng soluton p j. Suppose t has m chldren (x1, x,, xm), and the ) ], vectors for these m chldren have already been constructed. The vector for t can be constructed by the followng formula. The fnal total cost s the mnmal element of Best[] f t s the root of the tree. Best[][j] = cost [p j ] + 1 p m mn 1 z n xp ( Best[xp][z] + cost,xp [p j, pz x p ] ) 4.5 LP-based refnement The LP-based refnement technque s presented n lgorthm. The dea s that we enumerate all the colorng solutons for one standard cell type whle others are fxated. The node whose assocated edges costs are larger s gven a hgher prorty. In Lne 4 of lgorthm, once the colorng solutons for all the cells are fxed, t s easy to see that mnmal cell dsplacement can be acheved by solvng the followng LP, where d j 1 c,c j s the mnmal dstance to elmnate conflct for adjacent cells c j 1 and c j n the -th row. Mnmze: n r q j =1 j=1 Subject to: x j xj 1 d j 1 c,c j, 1 n j r, 1 n 1 j r x j oj qj o j xj qj, 1 n 1 j r j c, 1 n 1 j r W j c, 1 n 1 j r lgorthm LP-based refnement 1: Calculate the assocated edges costs of each node; : for each node n descendng order of assocated edges costs do 3: for each colorng soluton for ths node do 4: Mnmze the total cell dsplacement by solvng the LP n Secton 4.5; 5: f the value of cost functon s better than the current best then 6: Update the current best; 7: Update the colorng soluton for ths node. 8: end f 9: end for 10: end for 5. EXPERIMENTL RESULTS Our approach s mplemented n C++ on a Lnux server wth Intel Xeon X GHz CPU, 94GB man memory. The benchmarks are derved from [11] s. Gurob [13] s used to solve MILP and LP. Snce the problem s NP-complete and t cannot be expected to get the optmal solutons for some benchmarks wthn lmted CPU runtme. We lmt the MILP solver to run 700s and report the best solutons wthn the tme lmt of MILP solver. The expermental results are shown n Table I. Compare wth MILP solutons, our heurstc approach acheves the same number of sttches. For total cell dsplacement, the heurstc method s only.9% worse than that of MILP solutons on average. However, the heurstc method gets 07 speed up on average. Besdes, our method only ncreases wrelength by less 1% over the ntal detaled placement.

6 Table 1: Experment results: MILP V.S. Heurstc. benchmark MILP Heurstc dsplacement # of conflcts # of sttches runtme(s) dsplacement # of conflcts # of sttches WL ncrease runtme(s) alu-70.88e E % 1 alu E E % 14 alu E E % 15 byp E E % 1 byp E E % 8 byp E E % 31 dv E E % 8 dv E E % 35 dv E E % 3 ecc-70.76e E % 4 ecc E E % 5 ecc E E % 6 efc-70.84e E % 6 efc E E % 8 efc E E % 8 ctl E E % 10 ctl E E % 1 ctl E E % 13 top E E % 36 top E E % 391 top E E % 48 Norm % 1 6. CONCLUSIONS In ths paper, we are focusng on dsplacement-drven TPL optmzaton n detaled placement stage under colorng constrants. We recognze ths problem as NP-complete, then propose two solutons. The frst one s MILP, the other s heurstc approach. We show that the heurstc approach s very effcent compared wth MILP by experment. The proposed heurstc method can produce compettve soluton qualty wthn very lmted CPU runtme. References [1] B. Yu, K. Yuan, B. Zhang, D. Dng, and D. Z. Pan, Layout decomposton for trple patternng lthography, n Proceedngs of the Internatonal Conference on Computer-ded Desgn, pp. 1 8, 011. [] S.-Y. Fang, Y.-W. Chang, and W.-Y. Chen, novel layout decomposton algorthm for trple patternng lthography, n Proceedngs of the 49th nnual Desgn utomaton Conference, pp , 01. [3] J. Kuang and E. F. Y. Young, n effcent layout decomposton approach for trple patternng lthography, n Proceedngs of the 50th nnual Desgn utomaton Conference, pp , 013. [4] Y. Zhang, W.-S. Luk, H. Zhou, C. Yan, and X. Zeng, Layout decomposton wth parwse colorng for multple patternng lthography, n Proceedngs of the Internatonal Conference on Computer-ded Desgn, pp , 013. [5] H. Tan, H. Zhang, Q. Ma, Z. Xao, and M. D. F. Wong, polynomal tme trple patternng algorthm for cell based row-structure layout, n Proceedngs of the Internatonal Conference on Computer-ded Desgn, pp , 01. cell based trple patternng lthography, n Proceedngs of the Internatonal Conference on Computer-ded Desgn, pp , 013. [7] B. Yu, Y.-H. Ln, G. Luk-Pat, D. Dng, K. Lucas, and D. Z. Pan, hgh-performance trple patternng layout decomposer wth balanced densty, n Proceedngs of the Internatonal Conference on Computer-ded Desgn, pp , 013. [8] Z. Chen, H. Yao, and Y. Ca, Suald: Spacng unformty-aware layout decomposton n trple patternng lthography., n ISQED, pp , 013. [9] Q. Ma, H. Zhang, and M. D. F. Wong, Trple patternng aware routng and ts comparson wth double patternng aware routng n 14nm technology, n Proceedngs of the 49th nnual Desgn utomaton Conference, pp , 01. [10] Y.-H. Ln, B. Yu, D. Z. Pan, and Y.-L. L, Trad: trple patternng lthography aware detaled router, n Proceedngs of the Internatonal Conference on Computer-ded Desgn, pp , 01. [11] B. Yu, X. Xu, J.-R. Gao, and D. Z. Pan, Methodology for standard cell complance and detaled placement for trple patternng lthography, n Proceedngs of the Internatonal Conference on Computer-ded Desgn, pp , 013. [1] M. R. Garey and D. S. Johnson, Gude to the Theory of NP-Completeness. Macmllan Hgher Educaton, [13] Gurob. [6] H. Tan, Y. Du, H. Zhang, Z. Xao, and M. D. F. Wong, Constraned pattern assgnment for standard