How Accurately Can We Model Timing In A Placement Engine?

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1 How Accurately Can We Model Tmng In A Placement Engne? Amt Chowdhary, Karth Raagopal, Satsh Venatesan, Tung Cao, Vladmr Tourn, Yegna Parasuram, Bll Halpn Intel Corporaton Serra Desgn Automaton Synplcty, Inc. Santa Clara, CA Santa Clara, CA Sunnyvale, CA {amt.chowdhary, arth.raagopal, satsh.venatesan, tung.d.cao, vladmr.tourn}@ntel.com, yegna@serra-da.com, bhalpn@synplcty.com 48. ABSTRACT Ths paper presents a novel placement algorthm for tmng optmzaton based on a new and powerful concept, whch we term dfferental tmng analyss. Recognzng that accurate optmzaton requres tmng nformaton from a sgnoff statc tmng analyzer, we propose an ncremental placement algorthm that uses tmng nformaton from a sgnoff statc tmng engne. We propose a set of dfferental tmng analyss equatons that accurately capture the effect of placement perturbatons on changes n tmng from the sgnoff tmer. We have formulated an ncremental placement optmzaton problem based on dfferental tmng analyss as a sngle lnear programmng (LP problem whch s solved to generate the new tmng-optmzed placement. Our experments show that the worst negatve slac (WNS mproves by an average of 30% and the total negatve slac (TNS mproves by 33% on average for a set of crcuts from a 3.0 GHz mcroprocessor that were already syntheszed and placed by a leadng ndustral physcal synthess tool. We also show that multple teratons of our engne gve further TNS mprovements an average mprovement of 5%, whch mples that our placer wll sgnfcantly speed up tmng convergence. Categores and Subect Descrptors B.7. [Integrated crcuts]: Desgn ads placement and routng. General Terms Algorthms, Desgn, Performance. Keywords Tmng-drven placement, statc tmng analyss, lnear programmng, dfferental tmng analyss.. INTRODUCTION Placement s an ntegral part of a tmng convergence flow. It determnes the length of nets on tmng-crtcal paths, whch Permsson to mae dgtal or hard copes of all or part of ths wor 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, or republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. DAC 005, June 3 7, 005, Anahem, Calforna, USA. Copyrght 005 ACM /05/0006 $5.00. drectly affects the delay of cells and nets on these crtcal paths. The problem of tmng drven placement s extremely complcated due to the fact that t s dffcult to accurately model tmng as a functon of the placement of cells. We brefly explan how tmng s modeled n exstng placement algorthms. o Exstng global placement engnes convert tmng nformaton nto net weghts or net constrants whch are then used n the global placement formulaton [3][4]. Few methods [6] nterleave tmng analyss and global placement. Kahng et al. [6] use a mn-max tmng optmzaton approach that moves cells n order to mnmze the maxmum of all weghted edge delays, where an edge s the combnaton of a net and ts drver cell; edge delay s modeled by Elmore model. The mn-max tmng optmzaton step s nterleaved wth global placement based on recursve bsecton. The tmng model n mn-max optmzaton calculates weghts on edges usng tmng nformaton of current placement, but does not model the change n arrval tmes and hence slacs, whch are needed to accurately model tmng smlar to a statc tmng analyss engne. o Exstng ncremental placement engnes [5][7] mprove tmng by movng cells on a few tmng-crtcal paths. Cho and Bazargan [7] teratvely assgn net constrants on nets of top few crtcal paths and move cells to meet these constrants. Aam and Pedram [5] present an teratve technque that models nets wth movable Stener ponts n a statc tmng analyss based ncremental placement framewor. However, ths formulaton s non-convex, whch can only be solved for top few paths, even after approxmaton. Exstng placers cannot accurately model tmng for more than a few paths. Also, ther tmng models do not capture the complextes of statc tmng such as arrval tme propagaton, slope (transton tme effects, transparent latches, etc. As a result, these methods leave much room for further optmzaton. Recent wor has shown that there s sgnfcant scope for mprovement n the state of the art n placement technology [8]. We found that expert desgners can easly mprove tmng of a placement generated by state of the art tmng-drven placement engnes. Desgners do so by movng a few crtcal cells, because they have a good nowledge of the mpact of cell movement on the overall tmng of the desgn. Durng these tmng mprovement steps, desgners do not worry about removal of cell overlaps, because movng a small number of cells results s a small amount of overlaps that could be removed by a legalzaton step at the end. Desgners need to run a tmng analyss engne after movng 80

2 a few cells to fnd the new tmng. The man drawbac of manual ncremental placement by desgners s that they can focus on only a few cells at a tme. Therefore, multple teratons are needed to converge tmng and every teraton requres tmng analyss mang such a manual flow very tme-consumng. It would be very benefcal to accurately automate such a placement step.. MOTIVATION Ths research s motvated by the realzaton that optmzatons need to be based on an accurate sgnoff tmng engne to be successful. We start wth tmng report from a state of the art tmng analyss engne whch models all the complextes of modern desgn. Reference tmng from an accurate tmer s the bass for our placement optmzaton. Rather than fully modelng statc tmng analyss, we use the accurate tmng nformaton and a novel dfferental tmng analyss model to drect placement optmzaton. A tmng model n the placer that calculates changes n tmng wth respect to an accurate reference tmng, whch we call dfferental tmng, wll be much more precse than tmng models used n current placers to estmate absolute tmng numbers. We bound the movement of cells to mprove the accuracy of our dfferental tmng model. Key contrbutons of our approach are: A dfferental tmng analyzer that computes dfferences n arrval and requred tmes at all pns of a crcut, relatve to a reference statc tmng analyss, gven changes n cell placement. Ths dfferental analyzer s almost exact n the neghborhood of the reference statc tmng, ncludng modelng of setup tme and latch transparency. A lnear programmng formulaton of the dfferental tmng model that optmzes tmng of the nput placement. To mantan the valdty of our dfferental tmng model, we lmt the placement changes to a local neghborhood. 3. PROBLEM STATEMENT The problem of ncremental tmng-drven placement can be stated as follows: Gven an ntal placement, ts tmng nformaton from a statc tmng analyss engne and a crtcal subcrcut, fnd a new placement of cells n the subcrcut such that overall tmng s mproved. Tmng of a crcut s measured n terms of two metrcs: worst negatve slac (WNS and total negatve slac (TNS. Slac at any pn of a standard cell or any pad of the crcut s defned as the dfference between the tme sgnal s requred (requred tme and the tme sgnal arrves (arrval tme. A negatve slac mples that sgnal s arrvng later than requred. WNS s defned as the worst slac among all tmng endponts of the crcut, where a tmng endpont s ether the data nput pn of a latch or a flp-flop, or an output pad of the crcut. TNS s the total sum of negatve slacs at the tmng endponts (postve slacs are gnored. We select a crtcal subcrcut of the nput crcut for ncremental placement. Fgure llustrates a small subcrcut that we use as an example of nput to ncremental placement. Cells B, C, D, E, F and G are movable cells. We consder combnatonal as well as sequental cells as movable cells n our ncremental placement approach. Fxed cells (or pads that drve movable cells are called start cells. Cells A and H, and nput pad I are start cells. Fxed cells (or pads drven by movable cells are called end cells. Output pad J and cell K are end cells. 4. Proposed Algorthm We now descrbe dfferental tmng analyss that models changes n tmng as a functon of changes n cell locatons. The ncremental placement problem can be naturally modeled as a lnear programmng (LP problem usng dfferental tmng analyss, as we show next. Current LP solvers can optmally and qucly solve very large LP problems []. We now descrbe the dfferental tmng analyss and the resultng LP problem formulaton of ncremental placement. We frst descrbe the modelng of changes n net length and load capactance, and then use these changes to descrbe our dfferental tmng model. 4. Model for net length and load capactance We defne x and y varables for new x and y locatons of cell for every movable cell. Length of a net s modeled as half-permeter of the boundng box of all cells connected to t. We defne varables leftx, rghtx, lowery and uppery for the four boundares of the boundng box of net. For every cell connected to net, leftx = mn ( x ; rghtx = max ( x lowery = mn ( y ; uppery = max ( y A These mn and max functons are converted to lnear constrants below. leftx x ; rghtx x lowery y ; uppery y Even though these constrants allow leftx to be much less than mn (x, the fnal LP soluton that optmzes TNS wll guarantee that leftx s set to mn (x. We model net by half-permeter of ts boundng box. The change n the length of net s gven below. l = ( rghtx leftx + ( uppery lowery old_l n B H n d C o Fg.. A small subcrcut used to explan our lnear programmng formulaton. Cells B, C, D, E, F, G are movable. I c n4 n5 n3 Here, old_l s the length of net n the ntal placement. The load capactance cload of cell s the sum of the nterconnect capactance and the total pn capactance cpn of all recever pns connected to the net drven by cell. Here, c s the nterconnect capactance per unt length and l s the total length of the net. cload = c l + cpn The change n load capactance s then a lnear functon of change n net length. We currently use a sngle value for c regardless of the metal layer on whch the net s routed. E D n7 n6 F G n9 n8 J K 80

3 cload = c l The maxmum load capactance that can be drven by cell s bounded by Cmax. The Cmax constrant s lnear as gven below. The only varable n ths constrant s l. c old_l + c l + cpn Cmax We lmt cell movement by M to reduce placement perturbaton and to mprove accuracy of dfferental tmng model. Here, old_x and old_y are x and y locatons of cell n the ntal placement. old_x M x old_x + M old_y M y old_y + M 4. Model for dfferental tmng analyss 4.. Delay and slope (transton tme across cells The delay from an nput pn to the output pn of a cell can be modeled as a lnear functon of the load capactance at the output pn and the slope (transton tme at the nput pn, wth a reasonably hgh degree of accuracy. The slope at the output pn of cell can be defned by a lnear functon n a smlar fashon. delay slope,, = A = B A cload + A nslope + B cload + B nslope Here, nslope, s the slope at the nput pn of cell and cload s the load capactance at the output pn of cell. The constants A 0, A, A, B 0, B, B are determned by characterzaton of the standard cell lbrary. We defne delay and output slope constrants for every feasble sgnal transton for the cell. For example, an nverter has only two transtons (nput rse, output fall and (nput fall, output rse, whle a two-nput XOR has all four possble transtons. To smplfy our dscusson n ths paper, we wrte constrants hereafter for only one transton for a cell, but our LP formulaton ncludes all possble transtons for every cell. Change n delay and slope can be modeled by lnear constrants. delay, = A + A nslope, slope = B + B nslope, It s very mportant to note that lnear modelng of delay and slope has a hgher accuracy than lnear modelng of absolute delay and slope. Thus, the use of dfferental tmng analyss wth respect to reference tmng from an accurate statc tmer s more precse than drectly usng statc tmng model. Pror wor has used a smple, but naccurate, modelng of absolute tmng. 4.. Delay and slope across net segments For a net wth m recever pns, we ndvdually consder tmng for m net segments, where a net segment s the connecton from the drver pn to a recever pn of the net. We use Elmore model [] for estmatng delay across a net segment of length l. c l delay = KD r l ( + cpn Here, r s the nterconnect resstance per unt length, K D s a constant wth a value of 0.69, and cpn s the pn capactance of the recever pn of the net segment. For lac of smple modelng, we do not consder the capactance of sde branches when,,, modelng delay from drver pn to a recever pn. We need to enhance our tmng model to nclude capactance of sde recevers, at least recevers close to the recever n queston. Smlarly, slope at the recever pn of a net segment wth drver cell and recever cell s gven below, where K S s a constant wth a value of. for transton from 0% to 90% of V DD. c l nslope, = KS r l ( + cpn + slope We model the change n length of a net segment, smlar to the modelng of change n length of a net. When the length of net segment changes by l, we derve the change n delay and slope as a functon of l as gven below. delay = KD r nslope, = KS r + slope r c ( c old_l + cpn l + ( l r c ( c old_l + cpn l + ( l The above equatons are lnear, except for a quadratc term ( l. Because ( l s a convex functon, we can lnearze t usng a set of lnear constrants as shown n Fg.. We bound the change n wrelength by L n order to mae the lnear approxmaton sq_ l close to the quadratc term ( l. L ± l 3L L ± l sq_ l, ( l L L /4 Even though these constrants allow sq_ l to be larger than the smallest value from these constrants, the optmal LP soluton wll ensure that sq_ l s set to the smallest value from these constrants. Here, we have approxmated ( l by a set of four lnear constrants. We can mprove accuracy of lnear approxmaton by usng a larger set of lnear constrants. In our experments, we have approxmated ths quadratc functon by a set of 0 lnear constrants wthout any sgnfcant mpact on runtme of LP solver Arrval tme propagaton Changes n delay and slope across cells and net segments affects the arrval tme at pns of these cells. We now defne the change n arrval tme at nput and output pns of all cells n the crtcal subcrcut. The arrval tme at an nput pn of a cell s calculated from the arrval tme at the output pn of the drvng -L -L/ L/ L l Fg.. Lnear approxmaton sq_ l of the squared change n wrelength ( l, gven a bound L on change n wrelength. 803

4 cell and the delay across the net segment connectng the two cells. We defne two dfferent arrval tmes at every pn one for rsng and another for fallng transton. However, we state only one arrval tme constrant here for ease of dscusson. arrval, arrval, = arrval = arrval + delay The arrval tme at the output pn of a cell depends on the last arrvng sgnal amongst all nput pns of cell. arrval arrval = max ( arrval = max,, (old_arrval, old_arrval + delay, + arrval, The above max constrant can be lnearzed as follows. arrval old_arrval,, + arrval, old_arrval, + old_delay + old_delay The old_arrval,, old_arrval and old_delay, are respectvely the arrval tme at nput pn, arrval tme at output pn and delay from nput pn to output pn of cell from the reference tmng determned by a state of the art statc tmng analyss engne Sequental cells We allow sequental cells to move durng ncremental placement. Movement of sequental cells can gve large mprovements n tmng, because t allows tradeoff of slac between paths endng and startng at the sequental cells. We defne varables for x and y locatons of sequental cells (latches and flp-flops, smlar to the combnatonal cells. However, we treat a sequental cell as a start cell as well as an end cell. We consder the data nput pn of a flp-flop or a closed latch as a tmng endpont n our formulaton. The setup tme of a gven sequental cell can be modeled as a lnear functon of the nput slope at the data and cloc pns, and the load at the output pn. We assume an deal cloc, whch translates to the slope at the cloc pn to be unchanged,.e. nslope,c. Change n setup tme results n an equal and opposte change n the requred tme at the data nput pn of the sequental cell. setup = S0 + S cload + S nslope, d + S3 nslope, c setup = S + S nslope, d requred = setup We consder the cloc nput pn as the tmng startpont n our formulaton, thus modelng the change n cloc-to-out delay due to the movement of sequental cell. We treat the specal case of transparent latch dfferent from a closed latch. We consder a transparent latch as a combnatonal cell wth a tmng arc gong from data nput pn to output pn. Thus, we model the change n delay from data pn to output pn of transparent latches. We assume that a transparent latch stays transparent durng a sngle teraton of our ncremental placer. The modelng of transparent latches allows our placer to optmze paths that span one or more transparent latches Boundary constrants For a start cell, we set the change n nput slope as well as arrval tme at all nput pns to 0. Even though the start cells are fxed,,, the delay from nput to output pn can change due to the change n ts load capactance. The change n delay for a start cell then changes the arrval tme at the output pn of the start cell. We set the followng boundary constrants for all nput pns of start cells. nslope arrval,, For an end cell, the requred tme at every nput pn s assumed to be unchanged. Thus, the change n slac of an nput pn of an end cell or a pad s smply gven by the negatve of the change n ts arrval tme. slac = requred arrval slac = arrval In case of a sequental end cell, requred tme changes wth the change n setup tme. As a result, change n slac s gven below. slac = requred arrval 4..6 Tmng metrcs We calculate the two tmng metrcs WNS and TNS from the change n slac at the nput pns of end cells or at the output pads. WNS s defned as the worst new slac among all end cells (or pads. Here, old_slac s the slac of pn (or pad from statc tmng analyss based on the ntal placement. WNS = mn( old_slac + slac WNS old_slac + slac, For a pn wth a negatve slac of old_slac, the mpact of ths pn on TNS s bounded by old_slac. Even though the slac at pn could mprove by more than old_slac, the mpact on TNS s stll bounded by old_slac. We call the contrbuton of change n slac of pn on TNS as negslac. negslac = mn( old_slac, slac The mn functon can be modeled n LP problem as follows. negslac negslac old_slac slac The change n TNS can be smply evaluated as the sum total of change n negatve slac for nput pns of end cells or output pads. TNS = negslac 4.3 Lnear programmng formulaton We now state the ncremental placement problem as an LP problem: Maxmze TNS, subect to the lnear constrants stated above. Fgure 3 presents the LP problem for the subcrcut of Fg.. Alternate obectve functons can be used, such as a combnaton of WNS and TNS. We use a fast, commercal LP solver cplex from ILOG to solve the above LP problem []. 4.4 Legalzaton Output of our LP problem s a new placement wth mproved tmng, but can have overlaps. We use a legalzaton engne to resolve cell overlaps n the fnal placement. We use two calls of the legalzaton engne. Frst, we legalze only crtcal cells (cells moved by our placer, whle gnorng remanng cells. Next, we fx crtcal cells that were legalzed n the frst step and then 804

5 legalze remanng cells. The motvaton for two-step legalzaton s to mnmze the change n tmng by lmtng the movement of tmng-crtcal cells at the expense of less crtcal cells. Maxmze TNS subect to Model for net length ( n {n, n9} leftx x; rghtx x cell connected to net lowery y; uppery y l = ( rghtx leftx + ( uppery lowery old_l Tmng for cells ( n {A,,I}, net drven by cell cload delay c old_l, = c l = A + c l + cpn + A Cmax nslope, slope = B + B nslope, Pn d of flpflop C s not used n these constrants pn of cell Tmng for nets (segment from drver to pn of recever delay nslope, = K D = K S r + slope ( r ( c old_l + cpn l + ( c old_l + cpn l r c + 3L L ± l, L ± l 9 Arrval tme propagaton (cell n {A,,K}, s drver of arrval, arrval Sequental cell C = arrval old_arrval, C + arrval nput pn of cell, old_arrval, setupc = S C + S nslopec, d requred = setup + old_delay, nput pn Boundary constrants (start cell n {A,C,H,I},end cell n {C,J,K} nslope arrval slac,, = requred arrval Tmng metrcs (nput pn of end cells C,J,K negslac negslac slac WNS old_slac old_slac + slac TNS = negslac nput pn of end cells C,J,K Fg. 3. LP problem for ncremental placement of subcrcut of Fg.., 5. Accuracy of our dfferental tmng model Dfferental tmng model n our engne s modeled strctly on the concept of statc tmng analyss, whch results n sgnfcant tmng mprovements shown later by our experments. We lst below several lmtatons of our tmng model, and dscuss technques we use to overcome these lmtatons. Fnal legalzaton wll worsen tmng to some extent. However, we bound cell movement and wor on a small subcrcut, resultng n only a few cell overlaps. Also, we frst legalze crtcal cells followed by non-crtcal cells to reduce mpact on tmng. The boundng box model of net length may not correlate well wth fnal routng of the net. We are worng on usng net parameters, such as aspect rato and fanout, to more accurately model net length. The quadratc term n the change n Elmore delay of net can not be precsely modeled n the LP problem. However, we closely approxmate the quadratc term by a large set of lnear constrants. We also bound the change n net length to help n the lnear approxmaton of the quadratc term. In case of long nets, actual load seen by a cell s smaller than the total load capactance due to resstve sheldng. We should use effectve capactance, nstead of total load capactance, whle evaluatng cell delay. We are worng on a heurstc that models change of effectve capactance as a lnear functon of change n net length. 6. Expermental Results We have mplemented the algorthm for formulatng ncremental tmng-drven placement as a lnear programmng problem n C++ on LINUX. We solve the LP problem usng a leadng ndustral LP solver cplex from ILOG []. The soluton of LP problem gves the new mproved placement, whch could have overlaps. We then remove overlaps by a two-step legalzaton usng an nternally-developed legalzer engne. Instead of usng MCNC benchmars, we used crcuts from a recent mcroprocessor, snce the effect of ncremental tmngdrven placement on crcut tmng s more accurately studed by usng data from a recent manufacturng process and standard cell lbrary, and by usng state of the art RC estmaton and tmng analyss engnes. For our experments, we used a set of sx crcuts from a 3.0 GHz mcroprocessor desgned on 0.3 mcron process. Crcuts range from a few thousand cells to 40,000 cells, as lsted n Table. Intal placements of these crcuts were generated usng a leadng ndustral physcal synthess tool. Our results wll show that placements generated from a leadng tmng-drven placement tool leave a lot of room for tmng mprovement, whch could be recovered by an ncremental placer that models tmng more accurately. We used an nternallydeveloped state of the art statc tmng analyss engne to generate tmng report for the ntal placement of crcuts. Tmng report contans slacs and slopes at the pns of all cells n the crcut. Cells are selected as movable based on slac at ther output pns. We defne a slac cutoff, such that all cells wth slac worse than the cutoff are selected. We also select cells n the transtve fanout of crtcal cells (cells wth slac worse than the cutoff, because these cells drectly affect the load on crtcal cells. We also have an upper bound on the number of cells selected, because selectng too many cells mght lead to a lot of overlaps, resultng n a tmng degradaton durng legalzaton (we have mostly used an 805

6 upper bound of 500 cells or 5% of total cells, whchever s smaller. Runtme of our engne s wthn -3 mnutes. Bggest LP problem has 30,000 varables and 60,000 constrants []. Table shows tmng mprovements by placng a small set of cells usng our ncremental placer. The number of cells moved by our placer s wthn 5%, yet we were able to mprove WNS and TNS on average by 30% and 33%, respectvely. (Note that the ntal and fnal tmng numbers were generated by the statc tmng analyss engne. These results show that our ncremental placer can substantally mprove tmng of the ntal placement, ust by movng a small set of cells. The amount of tmng mprovement depends on the tmng qualty of the ntal placement. In case of the largest crcut ct6 wth 40K cells, we found that movng a small set of 50 cells gave huge mprovement n TNS, whch was due to the optmzaton of a few tmng crtcal nets wth hgh fanout that were not correctly optmzed durng global placement. It should be noted that these tmng mprovements are obtaned only by our ncremental placement, wthout dong any buffer nserton or crcut szng. Based on our experence, further mprovement could be obtaned usng placement coupled wth buffer nserton and crcut szng. We found that other placement characterstcs le routablty and wrelength are largely unaffected by ths optmzaton snce these are global parameters of the desgn and don t get perturbed much by movng a few cells. Total wrelength of the fnal placement was wthn +/-% of the wrelength of ntal placement. We ran global routng on these placements and found smlar congeston maps n the ntal and fnal placements. We also found that the changes n slacs and slopes reported by our placement engne correlate well wth these changes reported by statc tmng engne. Crcut Total cells Cells moved WNS (ps Intal TNS (ns WNS (ps Fnal TNS (ns ct, ct, ct3 3, ct4 4, ct5 5, ct6 40, Table : Tmng mprovement from usng our ncremental placer. We also ran multple runs of ncremental placement on the same crcut to llustrate the full extent of optmzaton that can be done usng our ncremental placer. Chart shows the results of multple placer teratons. In every teraton, we select cells n dfferent slac ranges to allow placement optmzaton of dfferent subcrcuts of the same crcut. The number of cells moved per teraton of ncremental placer s bounded by 5% of the total number of cells. Chart shows that TNS mproved by 4-87% for these four crcuts an average mprovement of 5%. Each teraton of ncremental placer was able to further reduce TNS sgnfcantly, because t wored on dfferent subcrcuts. The frst teraton for ct6 mproved TNS a lot, because the startng placement had some very hgh fanout nets whch were not optmzed for tmng, resultng n a huge mprovement n TNS by movng ust 50 cells. We found that results on these sx benchmar crcuts are representatve of results on other crcuts. 7. Conclusons and Future Drectons We have developed a novel dfferental tmng analyss model that uses reference tmng from a state of the art statc tmer and models tmng changes as a result of changes n placement wth a hgh degree of accuracy. We have desgned a powerful algorthm for ncremental placement optmzaton based on our dfferental tmng model. We formulated placement optmzaton problem as an LP problem whch s then solved optmally and qucly by an LP solver. The man strength of our algorthm s that the tmng model s based closely on a sgnoff tmng analyss. We acheved mprovements n WNS and TNS on average of 30% and 33%, respectvely, for a set of sx crcuts by ncrementally placng only 5% of total cells, even when the startng placement was generated by a leadng tool for tmng-drven synthess and placement. We ran several teratons of our ncremental placer on some crcuts and got huge mprovements n TNS by selectng dfferent subcrcuts n each teraton. Our ncremental placer can be even more benefcal when coupled wth szng and buffer nserton TNS (ns ct3 ct4 ct5 ct6 Intal placement st teraton nd teraton 3rd teraton 4th teraton Chart : Iteratve run of our placer: Improvements n TNS. 8. REFERENCES [] W. C. Elmore, The transent response of Damped Lnear networ wth partcular regard to wdeband amplfer, Journal of Appled Physcs, pp.55-63, 948. [] ILOG, ILOG CPLEX 8.0 User s Manual. ILOG, 00. [3] B. Halpn, C. Y. R. Chen, N. Sehgal, Tmng drven placement usng physcal net constrants, Proc. Desgn Automaton Conf., pp , 00. [4] K. Raagopal, T. Shaed, Y. Parasuram, T. Cao, A. Chowdhary, B. Halpn, Force drected tmng drven placement wth physcal net constrants, Proc. Intl Symp. on Physcal Desgn, pp. 47-5, 003. [5] A.H. Aam, M. Pedram, Post-layout tmng drven cell placement usng an accurate net length model, Proc. Desgn Automaton Conf., pp , 00. [6] A. B. Kahng, S. Mant, I. L. Marov, Mn-max placement for large-scale tmng optmzaton'', Proc. Intl. Symp. of Physcal Desgn, pp , 00. [7] W.Cho, K.Bazargan, Incremental Placement for Tmng Optmzaton, Proc. Intl Conf. on CAD, 003. [8] C.-C. Chang, J.Cong, M. Xe, Optmalty and scalablty study of exstng placement algorthms, Proc. of the ASP- DAC, Jan

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