Optimal Slack-Driven Block Shaping Algorithm in Fixed-Outline Floorplanning

Transcription

1 Optmal Slack-Drven Block Shapng Algorthm n Fxed-Outlne Floorplannng Jackey Z. Yan Placement Technology Group Cadence Desgn Systems San Jose, CA 9134 USA zyan@cadence.com Chrs Chu Department of ECE Iowa State Unversty Ames, IA 0010 USA cnchu@astate.edu ABSTRACT Ths paper presents an effcent, scalable and optmal slack-drven shapng algorthm for soft blocks n non-slcng floorplan. The proposed algorthm s called SDS. Dfferent from all prevous approaches, SDS s specfcally formulated for fxed-outlne floorplannng. Gven a fxed upper bound on the layout wdth, SDS mnmzes the layout heght by only shapng the soft blocks n the desgn. Iteratvely, SDS shapes some soft blocks to mnmze the layout heght, wth the guarantee that the layout wdth would not exceed the gven upper bound. Rather than usng some smple heurstc as n prevous work, the amount of change on each block s determned by systematcally dstrbutng the global total amount of avalable slack to ndvdual block. Durng the whole shapng process, the layout heght s monotoncally reducng, and eventually converges to an optmal soluton. We also propose two optmalty condtons to check the optmalty of a shapng soluton. To valdate the effcency and effectveness of SDS, comprehensve experments are conducted on MCNC and HB benchmarks. Compared wth prevous work, SDS s able to acheve the best expermental result wth sgnfcantly faster runtme. Categores and Subect Descrptors B.7.2 [Hardware, Integrated Crcuts, Desgn Ads]: Layout General Terms Algorthms, Desgn, Performance Keywords Block Shapng, Fxed-Outlne Floorplan, Physcal Desgn 1. INTRODUCTION Floorplannng s a very crucal step n modern VLSI desgns. A good floorplan soluton has a postve mpact on the placement, routng and even manufacturng. In floorplannng step, a desgn contans two types of blocks, hard and soft. A hard block s a crcut block Ths work was partally supported by IBM Faculty Award and NSF under grant CCF 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. To copy otherwse, to republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. ISPD 12, March 2 28, 2012, Napa, Calforna, USA. Copyrght 2012 ACM /12/03...$ wth both area and aspect rato 1 fxed, whle a soft one has fxed area, yet flexble aspect rato. Shapng such soft blocks plays an mportant role n determnng the top-level spatal structure of a chp, because the shapes of blocks drectly affect the packng qualty and the area of a floorplan. However, due to the ever-ncreasng complexty of ICs, the problem of shapng soft blocks s not trval. 1.1 Prevous Work In slcng floorplan, researchers proposed varous soft-block shapng algorthms. Stockmeyer [1] proposed the shape curve representaton used to capture dfferent shapes of a subfloorplan. Based on the shape curve, t s straghtforward to choose the floorplan soluton wth the mnmum cost, e.g., mnmum floorplan area. In [2], Zmmermann extended the shape curve representaton by consderng both slcng lne drectons when combnng two blocks. Yan et al. [3] generalzed the noton of slcng tree [4] and extended the shape curve operatons. Consequently, one shape curve captures sgnfcantly more shapng and floorplan solutons. Dfferent from slcng floorplan, the problem of shapng soft blocks to optmze the floorplan area n non-slcng floorplan s much more complcated. Both Pan et al. [] and Wang et al. [6] tred to extend the slcng tree and shape curve representatons to handle non-slcng floorplan. But ther extensons are lmted to some specfc nonslcng structures. Instead of usng the shape curve, Kang et al. [7] adopted the bounded slcelne grd structure [8] and proposed a greedy heurstc algorthm to select dfferent shapes for each soft block, so that total floorplan area was mnmzed. Moh et al. [9] formulated the shapng problem as a geometrc programmng and searched for the optmal floorplan area usng standard convex optmzaton. Followng the same framework as n [9], Murata et al. [10] mproved the algorthm effcency va reducng the number of varables and functons. But the algorthm stll took a long tme to fnd a good soluton. In [11], Young et al. showed that the shapng problem for mnmum floorplan area can be solved optmally by Lagrangan relaxaton technque. Ln et al. [12] changed the problem obectve to mnmzng the half permeter of a floorplan, and solved t optmally by the mn-cost flow and trust regon method. All of the above shapng algorthms for non-slcng floorplan were targetng at classcal floorplannng,.e., mnmzng the floorplan area. But, n the nanometer scale era classcal floorplannng cannot satsfy the requrements of herarchcal desgn. In contrast, fxed-outlne floorplannng [13] enablng the herarchcal framework s preferred by modern ASIC desgns. In [14], Adya et al. ntroduced the noton of slack n floorplannng, and proposed a slack-based algorthm to shape the soft blocks. Such shapng algorthm was appled nsde an annealng-based fxed-outlne floorplanner. There are two problems wth ths shapng algorthm: 1) It s a smple greedy heurstc, n whch each tme every soft block s shaped to use up all ts slack 1 The aspect rato s defned as the rato of the block heght to the block wdth.

2 n one drecton. Thus, the resultng soluton has no optmalty guarantee; 2) It s not formulated for fxed-outlne floorplannng. The fxed-outlne constrant s smply consdered as a penalty term n the cost functon of annealng. Therefore, n non-slcng floorplan t s necessary to desgn an effcent and optmal shapng algorthm that s specfcally formulated for fxed-outlne floorplannng. 1.2 Our Contrbutons Ths work presents an effcent, scalable and optmal slack-drven shapng (SDS) algorthm for soft blocks n non-slcng floorplan. SDS s specfcally formulated for fxed-outlne floorplannng. Gven a fxed upper bound on the layout wdth, SDS mnmzes the layout heght by only shapng the soft blocks n the desgn. If such upper bound s set as the wdth of a predefned fxed outlne, SDS s capable of optmzng the area for fxed-outlne floorplannng. As far as we know, none of prevous work n non-slcng floorplan consders the fxed-outlne constrant n the problem formulaton. In SDS, soft blocks are shaped teratvely. At each teraton, we only shape some of the soft blocks to mnmze the layout heght, wth the guarantee that the layout wdth would not exceed the gven upper bound. The amount of change on each block s determned by systematcally dstrbutng the global total amount of avalable slack to ndvdual block. Durng the whole shapng process, the layout heght s monotoncally reducng, and eventually converges to an optmal soluton. Note that n [14] wthout a global slack dstrbuton, all soft blocks are shaped greedly and ndependently by some smple heurstc. In ther work, both the layout heght and wdth are reduced n one shot (.e., not teratvely) and the soluton s stuck at a local mnmum. Essentally, we have three man contrbutons. Basc Slack-Drven Shapng: The basc slack-drven shapng algorthm s a very smple shapng technque. Iteratvely, t dentfes some soft blocks, and shapes them by a slack-based shapng scheme. The algorthm stops when there s no dentfed soft block. The runtme complexty n each teraton s lnear tme. The basc SDS can acheve an optmal layout heght for most cases. Optmalty Condtons: To check the optmalty of the shapng soluton returned by the basc SDS, two optmalty condtons are proposed. We prove that f ether one of the two condtons s satsfed, the soluton returned by the basc SDS s optmal. Slack-Drven Shapng (SDS): Based on the basc SDS and the optmalty condtons, we propose the slack-drven shapng algorthm. In SDS, a geometrc programmng method s appled to mprove the non-optmal soluton produced by the basc SDS. SDS always returns an optmal shapng soluton. To show the effcency of SDS, we compare t wth the two shapng algorthms n [11] and [12] on MCNC benchmarks. Even though both of them clam ther algorthms can acheve the optmal soluton, expermental results show that SDS consstently generates better soluton on each crcut wth sgnfcantly faster runtme. On average SDS s 23 and 33 faster than [11] and [12] respectvely, to produce solutons of smlar qualty. We also run SDS on HB benchmarks. Expermental results show that on average after 6%, 10%, 22% and 47% of the total teratons, the layout heght s wthn 10%, %, 1% and 0.1% dfference from the optmal soluton, respectvely. The rest of ths paper s organzed as follows. Secton 2 descrbes the problem formulaton. Secton 3 ntroduces the basc slack-drven shapng algorthm. Secton 4 dscusses the optmalty of a shapng soluton and presents two optmalty condtons. Secton descrbes the algorthm flow of SDS. Expermental results are presented n Secton 6. Fnally, ths paper ends wth a concluson and the drecton of future work. 2. PROBLEM FORMULATION In the desgn, suppose we are gven n blocks. Each block (1 n) has fxed area A. Let w and h denote the wdth and heght of block respectvely. The range of w and h are gven as W mn w W max and H mn h H max. If block s a hard block, then W mn = W max and H mn = H max. Let x and y denote the x and y coordnates of the bottom-left corner of block respectvely. To model the geometrc relatonshp among the blocks, we use the horzontal and vertcal constrant graphs G h and G v, where the vertces represent the blocks and the edges between two vertces represent the non-overlappng constrants between the two correspondng blocks. In G h, we add two dummy vertces 0 and n +1that represent the left-most and rght-most boundary of the layout respectvely. Smlarly, n G v we add two dummy vertces 0 and n +1that represent the bottom-most and top-most boundary of the layout respectvely. The area of the dummy vertces s 0. Wehave x 0 =0and y 0 =0. Vertces 0 and n +1are defned as the source and the snk n the graphs respectvely. Thus, n both G h and G v, we add one edge from the source to each vertex that does not have any ncomng edge, and add one edge from each vertex that does not have any outgong edge to the snk. In our problem formulaton, we assume the constrant graphs G h and G v are gven. Gven an upper bound on the layout wdth as W, we want to mnmze the layout heght y n+1 by only shapng the soft blocks n the desgn, such that the layout wdth x n+1 W. Such problem can be mathematcally formulated as follows: PROBLEM 1. Heght Mnmzaton wth Fxed Upper-Bound Wdth Mnmze y n+1 subect to x n+1 W x x + w, (, ) G h W mn H mn y y + h, (, ) G v w W max, 1 n h H max, 1 n w h = A, 1 n x 0 =0 y 0 =0 It s clear that f W s set as the wdth of a predefned fxed outlne, Problem 1 can be appled n fxed-outlne floorplannng. 3. BASIC SLACK-DRIVEN SHAPING In ths secton, we present the basc slack-drven shapng algorthm, whch solves Problem 1 optmally for most cases. Frst of all, we ntroduce some notatons used n the dscusson. Gven the constrant graphs and the shape of the blocks, we can pack the blocks to four lnes,.e., the left (LL), rght (RL), bottom (BL) and top (TL) lnes. LL, RL, BL and TL are set as x =0, x = W, y =0 and y = y n+1, respectvely. Let Δ x denote the dfference of x when packng block to RL and LL. Smlarly, Δ y denotes the dfference of y when packng block to TL and BL. For block (1 n), the horzontal slack s h and vertcal slack s v are calculated as follows: s h = max(0, Δ x ), s v = max(0, Δ y ) In G h, gven any path 2 from the source to the snk, f for all blocks on ths path, ther horzontal slacks are equal to zero, then we defne such path as a horzontal crtcal path (HCP). The length of one HCP s the summaton of the wdth of blocks on ths path. Smlarly, we can defne the vertcal crtcal path (VCP) and the length of one VCP s the summaton of the heght of blocks on ths path. Note that, 2 By default, all paths n ths paper are from the source to the snk n the constrant graph.

3 because we set RL as the x = W lne, f x n+1 <W, then there s no HCP n G h. The algorthm flow of the basc SDS s smple and straghtforward. The soft blocks are shaped teratvely. At each teraton, we apply the followng two operatons: 1. Shape the soft blocks on all VCPs by ncreasng the wdth and decreasng the heght. Ths reduces the lengths of the VCPs. 2. Shape the soft blocks on all HCPs by decreasng the wdth and ncreasng the heght. Ths reduces the lengths of the HCPs. The purpose of the frst operaton s to mnmze the layout heght y n+1 by decreasng the lengths of all VCPs. As mentoned prevously, f x n+1 <Wthen there s no HCP. Thus, the second operaton s appled only f x n+1 = W. Ths operaton seems to be unnecessary, yet actually s crtcal for the proof of the optmalty condtons. The purpose of ths operaton wll be explaned n Secton 4. At each teraton, we frst globally dstrbute the total amount of slack reducton to the soft blocks, and then locally shape each ndvdual soft block on the crtcal paths based on the allocated amount of slack reducton. The algorthm stops when we cannot fnd any soft block to shape on the crtcal paths. Durng the whole shapng process, the layout heght y n+1 s monotoncally decreasng and thus the algorthm always converges. In the followng subsectons, we frst dentfy whch soft blocks to be shaped (whch we called target soft blocks) at each teraton. Secondly, we mathematcally derve the shapng scheme on the target soft blocks. Fnally, we present the algorthm flow of the basc SDS. 3.1 Target Soft Blocks For a gven shapng soluton, the set of n blocks can be dvded nto the followng seven dsont subsets (1 n). 8 Subset I = { s hard} Subset II = { s soft} {s h 0,s v 0} >< Subset III = { s soft} {s h =0,s v =0} Subset IV = { s soft} {s h 0,s v =0} {w W max Subset V = { s soft} {s h 0,s v =0} {w = W max Subset VI = { s soft} {s >: h =0,s v 0} {h H max Subset VII = { s soft} {s h =0,s v 0} {h = H max } } } } Based on the defntons of crtcal paths, we have the followng observatons 3. OBSERVATION 1. If block subset II, then s not on any HCP nor VCP. OBSERVATION 2. If block subset III, then s on both HCP and VCP,.e., at the ntersecton of some HCP and some VCP. OBSERVATION 3. If block subset IV or V, then s on some VCP but not on any HCP. OBSERVATION 4. If block subset VI or VII, then s on some HCP but not on any VCP. As mentoned prevously, y n+1 can be mnmzed by reducng the heght of the soft blocks on the vertcal crtcal paths, and such blockheght reducton wll result n a decrease on the horzontal slacks of those soft blocks. From the above observatons, only soft blocks n subsets III, IV and V are on the vertcal crtcal paths. However, for block subset III, s h =0, whch means ts horzontal slack cannot be further reduced. And for block subset V, w = W max, whch means ts heght cannot be further reduced. As a result, to mnmze y n+1 we can only shape blocks n subset IV. Smlarly, we conclude 3 Please refer to Theorem 1 n [14] for the proof of these observatons. that whenever we need to reduce x n+1 we can only shape blocks n subset VI. For the hard blocks n subset I, they cannot be shaped anyway. Therefore, the target soft blocks are the blocks n subsets IV and VI. 3.2 Shapng Scheme Let δ h denote the amount of ncrease on w for block subset IV, and δ v denote the amount of ncrease on h for block subset VI. In the remanng part of ths subsecton, we present the shapng scheme to shape the target soft block subset IV by settng δ h. Smlar shapng scheme s appled to shape the target soft block subset VI by settng δ v. By default, all blocks mentoned n the followng part are referrng to the target soft blocks n subset IV. We use p to denote that block s on a path p n G h. Suppose the maxmum horzontal slack over all blocks on p s s p max. Bascally, s p max gves us a budget on the total amount of ncrease on the block wdth along ths path. If P p δh >s p max, then after shapng, we have x n+1 >W, whch volates the constrant x n+1 W. So we have to set δ h accordngly, such that P p δh s p max for all p n G h. To determne the value of δ h, we frst defne a dstrbuton rato α p (α p 0) for block p. We assgn the value of αp, such that X α p =1 p LEMMA 1. For any path p n G h, we have X α p sh s p max p PROOF. Because s p max = MAX p(s h ), ths lemma can be proved as follows: X X α p sp max = s p max α p = sp max p α p sh X p Based on Lemma 1, for a sngle path p, t s obvous that f δ h α p sh ( p), then we can guarantee P p δh s p max. More generally, f there are multple paths gong through block (1 n), then δ h needs to satsfy the followng nequalty: p δ h α p sh, p P h (1) where P h s the set of paths n G h gong through block. Inequalty 1 s equvalent to the followng nequalty. δ h MIN(α p h )sh (2) Essentally, Inequalty 2 gves an upper bound on the amount of ncrease on w for block subset IV. For block p, the dstrbuton rato s set as follows: α p = ( P 0 s the source or the snk w otherwse W max w k k ) The nsght s that f we allocate more slack reducton to the blocks that have potentally more room to be shaped, the algorthm wll converge faster. And we allocate zero amount of slack reducton to the dummy blocks at the source and the snk n G h. Based on Equaton 3, Inequalty 2 can be rewrtten as follows (1 n): δ h MAX h ( P w )s h (W max k w k )) (3) (4)

4 From the above nequalty, to calculate the upper bound of δ h, we need to obtan the value of three terms, w ), s h and MAX h( P max (Wk w k )). The frst term can be obtaned n constant tme. Usng the longest path algorthm, s h for all can be calculated n lnear tme. A trval approach to calculate the thrd term s va traversng each path n G h. Ths takes exponental tme, whch s not practcal. Therefore, we propose a dynamc programmng (DP) based approach that only takes lnear tme to calculate the thrd term. In G h, suppose vertex (0 n +1) has n-comng edges comng from the vertces n the set V n, and out-gong edges gong to the vertces n the set V out. Let P n denote the set of paths that start at the source and end at vertex n G h, and P out denote the set of paths that start at vertex and end at the snk n G h. For the source of G h,wehavev0 n = φ and P0 n = φ. For the snk of G h,wehavevn+1 out = φ and Pn+1 out = φ. We notce that MAX h( P max (Wk w k )) can be calculated recursvely by the followng equatons. It s clear that by the DP-based approach, the whole process of calculatng the upper bound of δ h for all takes lnear tme. 3.3 Flow of Basc Slack-Drven Shapng The algorthm flow of basc slack-drven shapng s shown n Fgure 1. In ths flow, for each block n the desgn, we set ts ntal wdth w = W mn (1 n). Based on the nput G h, G v and ntal block shape, we can calculate an ntal value of x n+1. If such ntal value s already bgger than W, then Problem 1 s not feasble. At each teraton we set δ v = β MIN v (α p )sv for target soft block subset VI. By default, β = 100%, whch means we set δ v exactly at ts upper bound. One potental problem wth ths strategy s that the layout heght y n+1 may reman the same,.e., never decreasng. Ths s because after one teraton of shapng, the length of some non-crtcal vertcal path ncreases, and consequently ts length may become equvalent to the length of the VCP n the prevous teraton. Accdentally, such scenaro may keep cyclng forever, and Basc Slack-Drven Shapng Input: w = W mn ( 1 n); G h and G v; upper-bound wdth W. Output: optmzed y n+1, w and h. Begn 1. Set LL, BL and RL to x =0, y =0 and x = W. 2. Pack blocks to LL and use longest path algorthm to get x n If x n+1 >W, 4. Return no feasble soluton.. Else, 6. Repeat 7. Pack blocks to BL and use longest path algorthm to get y n Set TLto y = y n Pack blocks to LL, RL and TL, respectvely. 10. Calculate s h and s v. 11. Fnd target soft blocks. 12. If there are target soft blocks, 13. subset IV, ncrease w by δ h = MIN h(α p )sh ; MAX( X 14. subset VI, ncrease h by δ v = β MIN v (α p )sv. 0 n k w k )) = 0 1. Untl there s no target soft block. MAX n+1( X End out k w k )) = 0 Fgure 1: Flow of basc slack-drven shapng. MAX( X n k w k )) = MAX(MAX( X V n n k w k ))) thus y n+1 would never decrease. Ths ssue can be solved, as long as δ + v s set less than ts upper bound. In ths way, after one teraton of w ) () MAX ( X out k w k )) = MAX ( MAX ( X shapng we can guarantee that the length of the VCP wll be shorter than the one n the prevous teraton. Theoretcally, any β<100% V out out k w k )) can break the cyclng scenaro and guarantee the algorthm convergence. But because n SDS any amount of change that s less than + w ) (6) MAX( X h k w k )) = MAX( X would be masked by numercal error, we can actually calculate a lower bound of β, and obtan ts range as follows. n k w k )) + MAX ( X 0.01 out k w k )) MIN v (α p % <β<100% )sv In the mplementaton, whenever we detect that y n+1 does not change w ) (7) for more than two teratons, we wll set β =90%for the next teraton. For δ h, we always set t at ts upper bound. Based on the equatons above, the DP-based approach can be appled step by step as follows (1 n): Because n each teraton the total ncrease on wdth or heght of the target soft blocks would not exceed the budget, we can guarantee 1. We apply topologcal sort algorthm on G h. that the layout would not be outsde of the four lnes after shapng. 2. We scan the sorted vertces from the source to the snk, and calculate MAX n( P As teratvely we set TL to the updated y = y n+1 lne, y n+1 wll max (Wk w k )) by Equaton. be monotoncally decreasng durng the whole shapng process. Dfferent from TL, because we set RL to the fxed x = W lne, 3. We scan the sorted vertces from the snk to the source, and calculate MAX out( P max durng the shapng process x n+1 may be bouncng.e., sometmes (Wk w k )) by Equaton MIN h( P ncreasng and sometmes decreasng, yet always no more than W. max (Wk w k )) s obtaned by Equaton 7. The shapng process stops when there s no target soft block. 4. OPTIMALITY CONDITIONS For most cases, n the basc SDS the layout heght y n+1 wll converge to an optmal soluton of Problem 1. However, sometmes the soluton may be non-optmal as the one shown n Fgure 2-(a). The layout n Fgure 2-(a) contans four soft blocks 1, 2, 3 and 4, where A =4, W mn =1and W max = 4 (1 4). The gven upper-bound wdth W =. In the layout, w 1 = w 3 = 4 and w 2 = w 4 =1. There s no target soft block on any one of the four crtcal paths (.e., two HCPs and two VCPs), so the basc SDS returns y n+1 =. But the optmal layout heght should be 3.2, when w 1 = w 2 = w 3 = w 4 =2.as shown n Fgure 2-(b). In ths secton, we wll look nto ths ssue and present the optmalty condtons for the shapng soluton returned by the basc SDS. Let L represent a shapng soluton generated by the basc SDS n Fgure 1. All proof n ths secton are establshed based on the fact that the only remanng soft blocks that could be shaped to possbly

5 y y x (a) (b) Fgure 2: Example of a non-optmal soluton from the basc SDS. 2 3 x reduced. But ths ncreases the length of the soft HCP, whch volates x n+1 W constrant. So, none of the blocks can be shaped to mprove L. 3. There s one or multple soft HCPs, and there s one soft VCP (e.g., Fgure 3-(c)) In ths case, L has one or multple ntersecton soft blocks. Gven any one of such blocks, say. Smlarly, t can be proved that x n+1 W constrant wll be volated, f h s reduced. So, none of the blocks can be shaped to mprove L. As a result, for all the above cases, we cannot fnd any soft block that could be shaped to possbly mprove L. Ths means our assumpton s not correct. Therefore, L s optmal. (a) (b) (c) Fgure 3: Examples of three optmal cases n L. mprove L are the ones n subset III. Ths s because L s the soluton returned by the basc SDS and n L there s no soft block that belongs to subsets IV nor VI any more. Ths s also why we need apply the second shapng operaton n the basc SDS. Its purpose s not reducng x n+1, but elmnatng the soft blocks n subset VI. From Observaton 2, we know that any block n subset III s always at the ntersecton of some HCP and some VCP. Therefore, to mprove L t s suffcent to ust consder shapng the ntersecton soft blocks between the HCPs and VCPs. Before we present the optmal condtons, we defne two concepts. Hard Crtcal Path: If all ntersecton blocks on one crtcal path are hard blocks, then ths path s a hard crtcal path. Soft Crtcal Path: A crtcal path, whch s not hard, s a soft crtcal path. LEMMA 2. If there exsts one hard VCP n L, then L s optmal. PROOF. Snce all ntersecton blocks on ths VCP are hard blocks, there s no soft block that can be shaped to possbly mprove ths VCP. Therefore, L s optmal. LEMMA 3. If there exsts at most one soft HCP or at most one soft VCP n L, then L s optmal. PROOF. As proved n Lemma 2, f there exsts one hard VCP n L, then L s optmal. So n the followng proof we assume there s no hard VCP n L. For any hard HCP, as all ntersecton blocks on t are hard blocks, we cannot change ts length by shapng those ntersecton blocks anyway. So we can bascally gnore all hard HCPs n ths proof. Suppose L s non-optmal. We should be able to dentfy some soft blocks and shape them to mprove L. As mentoned prevously, t s suffcent to ust consder shapng the ntersecton soft blocks. If there s at most one soft HCP or at most one soft VCP, there are only three possble cases n L. (As we set TL as the y = y n+1 lne, there s always at least one VCP n L.) 1. There s no soft HCP, and there s one or multple soft VCPs (e.g., Fgure 3-(a)) In ths case, L does not contan any ntersecton soft blocks. 2. There s one soft HCP, and there s one or multple soft VCPs (e.g., Fgure 3-(b)) In ths case, L has one or multple ntersecton soft blocks. Gven any one of such blocks, say. To mprove L, h has to be. FLOW OF SLACK-DRIVEN SHAPING Usng the condtons presented n Lemmas 2 and 3, we can determne the optmalty of the output soluton from the basc SDS. Therefore, based on the algorthm flow n Fgure 1, we propose the slack-drven shapng algorthm shown n Fgure 4. SDS always returns an optmal soluton for Problem 1. Slack-Drven Shapng Input: w = W mn ( 1 n); G h and G v; upper-bound wdth W. Output: optmal y n+1, w and h. Begn Lnes 1 14arethesame as the ones n Fgure Else, 16. If Lemma 2 or 3 s satsfed, 17. L s optmal. 18. Else, 19. Improve L by a sngle step of geometrc programmng. 20. If no optmal soluton s obtaned, 21. Go to Lne Else, 23. L s optmal. 24. Untl L s optmal. End Fgure 4: Flow of slack-drven shapng. The dfferences between SDS and the basc verson are startng from lne 1 n Fgure 4. When there s not target soft block, nstead of termnatng the algorthm, SDS wll frst check the optmalty of L, and f t s not optmal, L wll be mproved va geometrc programmng. The algorthm stops when an optmal soluton s obtaned. As mentoned prevously, f the soluton L generated by the basc SDS s not optmal, we only need to shape the ntersecton soft blocks to mprove L. In ths way, the problem now becomes shapng the ntersecton blocks to mnmze the layout heght y n+1 subect to layout wdth constrant x n+1 W. In other words, t s bascally the same as Problem 1, except that we only need to shape a smaller number of soft blocks (.e., the ntersecton soft blocks). Ths problem s a geometrc program. It can be transformed nto a convex problem and solved optmally by any convex optmzaton technque. However, consderng the runtme, we don t need to rely on geometrc programmng to converge to an optmal soluton. We ust run one step of some teratve convex optmzaton technque (e.g., deepest descent) to mprove L. Then we can go back to lne 7, and appled the basc SDS agan. It s clear that SDS always converges to the optmal soluton because as long as the soluton s not optmal, the layout heght wll be mproved. In modern VLSI desgns, the usage of Intellectual Property (IP) and embedded memory blocks becomes more and more popular. As a result, a desgn usually contans tens or even hundreds of bg hard

6 macros,.e., hard blocks. Due to ther bg szes, after applyng the basc SDS most lkely they are at the ntersectons of horzontal and vertcal crtcal paths. Moreover, n our experments we observe that there s always no more than one soft HCP or VCP n the soluton returned by the basc SDS. Consequently, we never need to apply the geometrc programmng method n our experments. Therefore, we beleve that for most desgns the basc slack-drven shapng algorthm s suffcent to acheve an optmal soluton for Problem EXPERIMENTAL RESULTS Ths secton presents the expermental results. All experments are run on a Lnux server wth AMD Opteron 2.9 GHz CPU and 16GB memory. We use two sets of benchmarks, MCNC [11] and HB [1]. For each crcut, the correspondng nput G h and G v are provded by a floorplanner. The range of the aspect rato for any soft block n the crcut s set to [ 1 3, 3]. After the nput data s read, SDS wll set the ntal wdth of each soft block at ts mnmum wdth. In SDS, f the amount of change on the wdth or heght of any soft block s less than , we would not shape such block because any change smaller than that would be masked by numercal error. Such numercal error, whch s unavodable, comes from the truncaton of an nfnte real number so as to make the computaton possble and practcal. 6.1 Experments on MCNC Benchmarks Usng the MCNC benchmarks we compare SDS wth the two shapng algorthms n [11] and [12]. All blocks n these crcuts are soft blocks. The source code of [11] and [12] are obtaned from the authors. In fact, these three shapng algorthms cannot be drectly compared, because ther optmzaton obectves are all dfferent: [11] s mnmzng the layout area x n+1y n+1; [12] s mnmzng the layout half permeter x n+1 + y n+1; SDS s mnmzng the layout heght y n+1, s.t. x n+1 W. Stll, to make some meanngful comparsons as best as we can, we setup the experment n the followng way. We conduct two groups of experments: 1) SDS v.s. [11]; 2) SDS v.s. [12]. As the crcut sze are all very small, to do some meanngful comparson on the runtme, n each group we run both shapng algorthms 1000 tmes wth the same nput data. For group 1, we run [11] frst, and use the returned fnal wdth from [11] as the nput upper-bound wdth W for SDS. For group 2, smlar procedure s appled. For groups 1 and 2, we compare the fnal results based on [11] s and [12] s obectves respectvely. Table 2 shows the results on group 1. The column ws(%) gves the whte space percentage over the total block area n the fnal layout. For all fve crcuts SDS acheves sgnfcantly better results on the floorplan area. On average, SDS acheves 394 smaller whte space and 23 faster runtme than [11]. In the last column, we report the runtme SDS takes to converge to a soluton that s better than [11]. To ust get a slghtly better soluton than [11], on average SDS uses 23 faster runtme. As ponted out by [12], [11] does not transform the problem nto a convex problem before applyng Lagrangan relaxaton. Hence, algorthm [11] may not converge to an optmal soluton. Table 3 shows the results on group 2. The authors clams the shapng algorthm n [12] can fnd the optmal half permeter on the floorplan layout. But, for all fve crcuts SDS gets consstently better half Table 1: Comparson on runtme complexty. Algorthm Runtme Complexty Young et al. [11] O(m 3 + km 2 ) Ln et al. [12] O(kn 2 mlog(nc)) Basc SDS O(km) (k s the total number of teratons, n s the total number of blocks n the desgn, m s the total number of edges n G h and G v, and C s the bggest nput cost.) permeter than [12], wth on average 10 faster runtme. Agan, n the last column, we report the runtme SDS takes to converge to a soluton that s better than [12]. To ust get a slghtly better soluton than [12], on average SDS uses 33 faster runtme. We beleve algorthm [12] stops earler, before t converges to an optmal soluton. From the runtme reported n Tables 2 and 3, t s clear that as the crcut sze ncreases, SDS scales much better than both [11] and [12]. In Table 1, we lst the runtme complextes among the three shapng algorthms. As n our experments, t s never necessary to apply the geometrc programmng method n SDS, we lst the runtme complexty of the basc SDS n Table 1. Obvously, the basc SDS has the best scalablty. 6.2 Experments on HB Benchmarks Ths subsecton presents the expermental results of SDS on HB benchmarks. As both algorthms [11] and [12] crashed on ths set of crcuts, we cannot compare SDS wth them. The HB benchmarks contan both hard and soft blocks rangng from 00 to 2000 (see Table 4 for detals). For each test case, we set the upper-bound wdth W as the square root of 110% of the total block area n the correspondng crcut. Let Y denote the optmal y n+1 SDS converges to. The results are shown n Table 4. The Convergence Tme column lsts the total runtme of the whole convergence process. The Total #.Iteratons column shows the total number of teraton SDS takes to converge to Y. For fxed-outlne floorplannng, SDS can actually stop early as long as the soluton s wthn the fxed outlne. So n the subsequent four columns, we also report the number of teratons when y n+1 Y Y starts to be less than 10%, %, 1% and 0.1%, respectvely. The average total convergence tme s 1.18 second. SDS takes average 1901 teratons to converge to Y. The four percentage numbers n the last row shows that on average after 6%, 10%, 22% and 47% of the total number of teratons, SDS converges to the layout heght that s wthn 10%, %, 1% and 0.1% dfference from Y, respectvely. In order to show the convergence process more ntutvely, we plot out the convergence graphs of y n+1 for four crcuts n Fgures (a)- (d). In the fgures, the four blue arrows pont to the four ponts when y n+1 becomes less than 10%, %, 1% and 0.1% dfference from Y, respectvely. Fnally, we have four remarks on SDS. 1. As SDS sets the ntal wdth of each soft block at ts mnmal wdth, such ntal floorplan s actually consdered as the worse start pont for SDS. Ths means f any better ntal shape s gven, SDS wll converge to Y even faster. 2. In our experments, we never notce that the soluton generated by the basc SDS contans more than one soft HCP or VCP. So f gnorng the numercal error mentoned prevously, SDS obtans the optmal layout heght for all crcuts n the experments smply by the basc SDS. 3. The expermental results show that after around 1 of the total teratons, the dfference between y n+1 and Y s already consdered qute small,.e., less than 1%. So n practce f t s not necessary to obtan an optmal soluton, we can bascally

7 Table 2: Comparson wth [11] on MCNC Benchmarks ( shows the total shapng tme of 1000 runs and does not count I/O tme). #. Young et al. [11] SDS SDS stops when result Crcut Soft ws Fnal Fnal Shapng ws Fnal Fnal Upper-Bound Shapng s better than [11] Blocks (%) Wdth Heght Tme (s) (%) Wdth Heght Wdth W Tme (s) ws (%) Tme (s) apte xerox hp am33a am49a Normalzed Table 3: Comparson wth [12] on MCNC Benchmarks ( shows the total shapng tme of 1000 runs and does not count I/O tme). #. Ln et al. [12] SDS SDS stops when result Crcut Soft Half Fnal Fnal Shapng Half Fnal Fnal Upper-Bound Shapng s better than [12] Blocks Permeter Wdth Heght Tme (s) Permeter Wdth Heght Wdth W Tme (s) Half Per. Tme (s) apte xerox hp am33a am49a Normalzed set a threshold value on the amount of change on y n+1 as the stoppng crteron. For example, f the amount of change on y n+1 s less than 1% durng the last 10 teratons, then SDS wll stop. 4. Lke all other shapng algorthms, SDS s not a floorplannng algorthm. To mplement a fxed-outlne floorplanner based on SDS, for example, we can smply ntegrate SDS nto a smlar annealng-based framework as the one n [14]. In each annealng loop, the nput constrant graphs are sent to SDS, and SDS stops once the soluton s wthn the fxed outlne. The annealng process keeps refnng the constrant graphs so as to optmze the varous floorplannng obectves (e.g., wrelength, routablty [16] [17], tmng, etc.) n the cost functon. 7. CONCLUSION AND FUTURE WORK Ths work proposed an effcent, scalable and optmal slack-drven shapng algorthm for soft blocks n non-slcng floorplan. Unlke prevous work, we formulate the problem n a way, such that t can be appled for fxed-outlne floorplannng. For all cases n our experments, the basc SDS s suffcent to obtan an optmal soluton. Both the effcency and effectveness of SDS have been valdated by comprehensve expermental results and rgorous theoretcal analyss. Due to the page lmt, we have to reserve some problems on SDS as the motvaton of future work, whch ncludes: 1) To use the dualty gap of Problem 1 as a better stoppng crteron, because t ndcates an upper-bound of the gap between the ntermedate and optmal shapng solutons; 2) To propose a more scalable algorthm as a substtuton of the geometrc programng method n Fgure 4; 3) To extend SDS to handle classcal floorplannng. Also, because of the smlarty between the slack n floorplannng and statc tmng analyss (STA), we beleve SDS can be modfed and appled on buffer/wre szng for tmng optmzaton. Acknowledgment The authors would lke to thank Prof. H. Zhou from Northwestern Unversty for provdng us the source code of algorthms [11] and [12]. 8. REFERENCES [1] L. Stockmeyer. Optmal orentatons of cells n slcng floorplan desgns. Informaton and Control, 7:91 101, May/June [2] G. Zmmermann. A new area and shape functon estmaton technque for VLSI layouts. In Proc. DAC, pages 60 6, [3] J. Z. Yan and C. Chu. DeFer: Deferred decson makng enabled fxed-outlne floorplannng algorthm. IEEE Trans. on Computer-Aded Desgn, 43(3): , March [4] R. H. J. M. Otten. Effcent floorplan optmzaton. In Proc. ICCD, pages , [] P. Pan and C. L. Lu. Area mnmzaton for floorplans. IEEE Trans. on Computer-Aded Desgn, 14(1): , January 199. [6] T. C. Wang and D. F. Wong. Optmal floorplan area optmzaton. IEEE Trans. on Computer-Aded Desgn, 11(8): , August [7] M. Kang and W. W. M. Da. General floorplannng wth L-shaped, T-shaped and soft blocks based on bounded slcng grd structure. In Proc. ASP-DAC, pages , [8] S. Nakatake, K. Fuyosh, H. Murata, and Y. Katan. Module placement on BSG-structure and IC layout applcatons. In Proc. ICCAD, pages , [9] T. S. Moh, T. S. Chang, and S. L. Hakm. Globally optmal floorplannng for a layout problem. IEEE Trans. on Crcuts and Systems I, 43: , September [10] H. Murata and E. S. Kuh. Sequence-par based placement method for hard/soft/pre-placed modules. In Proc. ISPD, pages , [11] F. Y. Young, C. C. N. Chu, W. S. Luk, and Y. C. Wong. Handlng soft modules n general non-slcng floorplan usng Lagrangan relaxaton. IEEE Trans. on Computer-Aded Desgn, 20(): , May [12] C. Ln, H. Zhou, and C. Chu. A revst to floorplan optmzaton by Lagrangan relaxaton. In Proc. ICCAD, pages , [13] A. B. Kahng. Classcal floorplannng harmful? In Proc. ISPD, pages , [14] S. N. Adya and I. L. Markov. Fxed-outlne floorplannng: Enablng herarchcal desgn. IEEE Trans. on VLSI Systems, 11(6): , December [1] J. Cong, M. Romess, and J. R. Shnnerl. Fast floorplannng by look-ahead enabled recursve bparttonng. In Proc. ASP-DAC, pages , 200. [16] Y. Zhang and C. Chu. CROP: Fast and effectve congeston refnement of placement. In Proc. ICCAD, pages , [17] Y. Zhang and C. Chu. RegularRoute: An effcent detaled router wth regular routng patterns. In Proc. ISPD, pages 4 2, 2011.

8 Table 4: Expermental Results of SDS on HB Benchmarks. Crcut #.Soft Blocks Upper-Bound Fnal Fnal Convergence Total #.Iteratons when y n+1 Y becomes Y / #.Hard Blocks Wdth W Wdth Heght (Y ) Tme (s) #.Iteratons < 10% < % < 1% < 0.1% bm01 66 / bm / bm / bm / bm0 64/ bm06 71 / bm / bm / bm / bm / bm / bm12 82 / bm13 30 / bm / bm / bm / bm / bm18 68 / Average % 9.6% 22.3% 47.3% #Iter = 2336, Heght = (a) bm01 #Iter = 17, Heght = (c) bm12 #Iter = 48, Heght = (b) bm02 #Iter = 3770, Heght = (d) bm1 Fgure : Layout-heght convergence graphs for crcuts bm01, bm02, bm12 and bm1. (x-axs denotes the teraton number and y-axs denotes the layout heght.)