Optimization of Linear Placements for Wirelength ... - CECS

Optimization of Linear Placements for Wirelength Minimization with Free Sites Andrew B. Kahng, Paul Tucker † and Alex ZelikovskyUCLA CS Department, Los Angeles, CA 90095-1596† UCSD CSE Department, La Jolla, CA, 92093-0114AbstractWe study a type of linear placement problem arising in detailed placementoptimization of a given cell row in the presence of white-space(extra sites). In this single-row placement problem, the cell order isfixed within the row; all cells in other rows are also fixed. We give thefirst solutions to the single-row problem: (i) a dynamic programmingtechnique with time complexity O(m 2 ) where m is the number of netsincident to cells in the given row, and (ii) an O(mlogm) techniquethat exploits the convexity of the wirelength objective. We also proposean iterative heuristic for improving cell ordering within a row;this can be run optionally before applying either (i) or (ii). Experimentalresults show an average of 6.5% wirelength improvement onindustry test cases when our methods are applied to the final output ofa leading industry placement tool.1 IntroductionThe linear placement problem (LPP) is well-studied in the VLSIphysical design literature, where it has appeared in various guises[15, 10, 11, 6, 2, 9, 13]. Traditionally, linear placement seeks to arrangea number of interconnected circuit modules within a row of locations,to minimize some objective. Applications of LPP are reportedin, e.g., [10, 4, 5]. The LPP problem is NP-hard [8], and so most existingmethods are heuristics (see [12] for a comprehensive review).In this work, we address a variant of the linear placement problemthat arises during detailed placement optimization of standardcelllayouts. Our context is a “successive-approximation” placementmethodology, e.g., (1) mixed analytic (quadratic) and partitioningbasedtop-down global placement (cf. most leading EDA vendortools), (2) row balancing and legalization, (3) detailed optimizationof row assignments (cf., for example, the TimberWolf placement tool[14]), and (4) detailed routability optimization (“white-space” managementand pin alignment) within individual rows. Step (4) in thismethodology arises because of routing hot-spots and limited porosityof the cell library: even with deep-submicron multilayer interconnectprocesses, placers must leave extra space within cell rows so that thelayout is routable.Our problem differs from the traditional LPP formulation in that(i) cells of a given row can be placed with gaps between them, i.e., thenumber of legal locations is more than the number of cells, and thelayout has “white-space”; and (ii) fixed cells from other rows participatein the net wirelength (cost) estimation for a given row.Our contributions include the following. We present the first optimal algorithms for single-row cell placementwith free sites (white-space), where the order of the cellsin the row is fixed, and positions of cells in all other rows arefixed. Our algorithms are based on dynamic programming, andon exploiting convexity of the objective. We present a new iterative algorithm to improve the cell orderingwithin a given row. We present an iterative row-based placement algorithm that appliessingle-row cell placement to each row in turn, with optionalcell ordering improvement in the given row.Research supported by a grant from Cadence Design Systems, Inc. We achieve an average of 6.5% improvement in the wirelengthobjective 1 on five industry test cases, starting from the final output(i.e., after global and detailed placement) of an industryplacement tool.The remainder of this paper is organized as follows. In Section2, we develop notation and terminology. Section 3 analyzes the cellcost function, which captures the dependency of the wirelength objectiveon the position of a movable cell in the given row. Section4 then proposes a dynamic programming approach. We improve thedynamic programming approach in Section 5 using the piecewise linearityof the cost dependence on cell position, and the monotonicityof a so-called prefix cost function derived from the cell cost function.In Section 6, we develop another, more efficient solution that exploitsthe convexity of the wirelength objective. Section 7 describes ourheuristic for improving the order of cells within a row, and Section 8concludes with experimental results and directions for future work.2 Notation and Problem StatementThe Single-Row Problem. We are given a single cell row with nmovable cells C i , i = 1;:::;n whose left-to-right order is fixed, butwhose positions are variable. All cells in all other rows are fixed cells,i.e., they have fixed positions. We seek a non-overlapping placementof movable cells such that the total bounding-box half-perimeter of allm signal nets N j , j = 1;:::;m is minimized. This is a one-dimensionalproblem: we minimize the total x-span of all nets. All cells haveinteger widths, and must be placed on an integer lattice of sites. 2We use the following notation (see Figure 1).single row withmovable cellsf (N)lnet Nfixed cellsm (N)lspan(N)f (N)rFigure 1: A net N with fixed and movable cells.m (N)l The position of a cell C i is the x-coordinate of the left edge ofthe cell. In general, all positions are x-coordinates. The entire row with movable cells consists of a sequence of k legalpositions of cells (“sites”) s 1 ;:::;s k . A given cell C i occupiesw i consecutive sites when placed. The range S i of a cell C i denotes the set of sites at which C i canbe placed. We use k i = jS i j to denote the number of such sites.1 We use the traditional objective: sum of net bounding box half-perimeters. Sincerow assignments are assumed fixed at this stage of the placement process, the objectivebecomes total horizontal span of all nets.2 It is possible to extend our approaches to address flipping (mirroring about the y-axis)of cells, but we will not discuss this here.

The position of the leftmost (respectively, rightmost) fixed pin ofnet N is denoted by f l (N) (respectively, f r (N)). The leftmost (respectively, rightmost) movable cell on N is denotedby L(N) (respectively, R(N)); these are known because theorder of movable cells is fixed. The leftmost (respectively, rightmost) pin of net N on cell C i isdenoted by l i (N) (respectively, r i (N)). The position of the leftmost (respectively, rightmost) movablepin of net N is denoted by m l (N) (respectively, m r (N)). The span of a net N, denoted span(N), isspan(N) =maxf f r (N);m r (N)g,minf f l (N);m l (N)g The fixed span of a net N, denoted fix span(N), isfix span(N) =f r (N) , f l (N) A placement is a subsequence P = fp 1 ;:::;p n g of the sequenceof sites, where p i is the placed position of the cell C i . Nonoverlappingplacement implies p i + w i p i+1 , i = 1;:::;n , 1. A prefix placement, denoted P i , is a placement of the first i cellsC 1 ;:::;C i .3 Cost Contribution of a Movable CellThe cell cost function, denoted cost i (x), of a movable cell C i is thesum over all nets N of the contribution of cell C i to span(N) ,fix span(N). In other words, for a given position x of the cell C i ,cost i (x) = ∑ maxfm r (N) , f r (N);0gC i =R(N)+ ∑C i =L(N)maxf f l (N) , m l (N);0gThe cost of a prefix placement P i is given by cost(P i ) =∑ i j=1 cost j(p j ), i.e., the sum of the cell costs when each cell in theprefix is placed by P i .The cell cost function cost i (x) is seen to be piecewise linear andconvex, as follows. Create a sorted list containing all f r (N) positionsfor nets with C i = R(N) and all f l (N) positions for nets withC i = L(N). Between any two consecutive positions in this sorted list,cost i (x) is linear and decreasing (respectively, constant or increasing)if the number of f r (N) positions that are to the left of their correspondingl i (N)’s is smaller than (respectively, equal to or greater than) thenumber of f l (N) positions that are to the right of their correspondingr i (N)’s (see Figure 2). This means that the piecewise linear functioncost i (x) is convex, with local minimum either a point or a segment. 3The number of linear pieces in the function cost i (x) will be denotedBound i . Note that Bound i is at most twice the number of pins on C isince C i can be both L(N) and R(N) for the same net N. The total numberof pins on the movable cells that can be the leftmost or rightmostfor some net will be denotednB = ∑ Bound i 2m; (1)i=1where m is the number of signal nets (we should take in account onlynets containing movable cells), because each net contains exactly oneleftmost and one rightmost pin.Observe that for any given placement Pnmm∑ cost i (p i )= ∑ span(N j ) , ∑ fix span(N j )i=1j=1j=13 This description remains valid when there are different pin positions on a given cell.f (1) f (2)r lf (3)lf (3) f (4) f (2) f (5) f (6) f (7)r l r r l lFigure 2: The cell cost function for a multi-pin cell.4 Dynamic Programming AlgorithmWe propose the following dynamic programming solution for theSingle-Row Problem. Let C 1 ;:::;C n be the fixed left-to-right orderof the movable cells. For each legal position s j of each movable cellC i , we will compute the minimum-cost constrained prefix placementof the cells C 1 ;:::;C i , subject to the position of C i being at or to theleft of s j (i.e., p i j). We denote this constrained prefix placementby P i (s j ).The optimum constrained prefix placement P i (s j ) can be computedfrom the cost function cost i (), prefix placements P i,1 () for thepreceding cell, and prefix placements P i (s h ) for h < j, as follows.Consider that in the prefix placement P i (s j ), the position of cell C i iseither strictly to the left of s j or exactly at s j . In the latter case, thepreceding cell C i,1 cannot be placed to the right of the site s j,wi,1because it must not overlap with C i . Thus, the minimum-cost placementP i (s j ) is selected from two alternatives: (i) P i (s j,1 ), and (ii)P i,1 (s j,wi,1 ) extended by p i = s j . Notice that extending the prefixplacement by p i = s j simply means adding the cell cost cost i (s j ).The dynamic programming principle is applied by using the precomputedsolutions for P i,1 () to compute P i (). The outer loop of thealgorithm is over cell indices i, and the inner loop is over correspondingranges of positions j. In the inner loop we can walk along thevalues s j 2 S i , i.e., the domain of cell cost function cost i (), sothatevaluating (ii) requires only constant time per each s j . Then, we canchoose the better option in O(1) time. The total runtime will be ofordernn∑ k i n (k , ∑ w i ) (2)i=1i=1Typically, k , ∑ n i=1 w i n, therefore the runtime is O(n 2 ).5 Exploiting Piecewise LinearityThe optimal prefix cost function gives the minimum possible cost of aprefix placement P i = fp 1 ;:::;p i g, subject to cell C i being placed at orto the left of position x. We write this as pcost i (x) =min pi xcost(P i ).Note that the optimal prefix cost function is monotonically nonincreasing4 and piecewise linear. Piecewise linearity follows from (i)and (ii) in the dynamic programming description above. In intervalswhere (i) is preferable to (ii), pcost i (x) will remain constant. In intervalswhere (ii) is preferable to (i), the cell cost function cost i isdecreasing and will be added pointwise to pcost i,1 . More precisely,we have pcost i (x) =pcost i,1 (x , w i,1 )+cost i (x) while this sum decreases.If the slope of pcost i () reaches 0 at some s j 2 S i ,thenpcost i ()remains unchanged thereafter.As stated above, the DP algorithm essentially tests each feasibleposition of each cell. However, we can exploit the piecewise linearityof the optimal prefix cost function in recursively constructing pcost ifrom pcost i,1 and cost i , so that the total number of points explicitlyexamined may be far less than ∑ n i=1 k i. The key insight is that thefunctions pcost i,1 and cost i can be represented, e.g., as sequences of4 This is because increasing the x argument relaxes the constraints on the prefix placement.

maximal line segments (tuples with minimum coordinate, maximumcoordinate, slope and y-intercept). Then, the sites bounding maximalline segments of pcost i will be a subset of the sites bounding maximalline segments in pcost i,1 (+ w i,1 )andcost i . The representation ofpcost i is computed by simultaneously walking these two componentsequences.Recall that the number of linear segments in cost i is at most thenumber of nets for which a pin on C i may be the leftmost or rightmostpin, and that we use Bound i to denote this number. Giventhat pcost i,1 and cost i are represented as sequences of linear segmentsthat are represented by tuples ha;b;x min ;x max i (indicating thatpcost i,1 (s j )=ax + b for x min x < x max being the x coordinateof s j ), the next function pcost i can be constructed as follows, inO(Bound i,1 + Bound i ) time.Prefix Algorithm1. Create a sequence of tuples that is a shifted, extendedcopy of pcost i,11.1. for x 2 [min(S i );max(S i,1 )+w i,1 ],pcost i (x) =pcost i,1 (x , w i,1 )1.2. for x 2 [max(S i,1 )+w i,1 ;max(S i ],pcost i (x) =pcost i,1 (max(S i,1 ))2. Iterate over the tuples in cost i ,inorder,and sum them into the function representation of pcost i .Note that the slope of pcost i never rises above 0, so the tail ofpcost i may consist of a single long segment where the tails of pcost i,1and cost i may contain multiple segments. Consequently, the time tocompute pcost i for all of the cells in the row (by the algorithm above)is upper bounded by ∑ i j=1 Bound j. In practice, we expect a time complexitycloser to O(n) because Bound i is likely to be upper boundedby a small constant.Basically, computing pcost i+1 from pcost i and cost i+1 is proportionalto the time to merge sorted sequences of linear segments oflength at most ∑ i j=1 Bound j and Bound i . We can use merge sort,which is proportional to ∑ i+1j=1 Bound j. We conclude that the runningtime of the Prefix Algorithm is O(B 2 )=O(m 2 ).6 Clumping AlgorithmThe nature of the dynamic programming solutions in Sections 4 and 5does not exploit the convexity of the function cost i . We now describean approach, called the Clumping Algorithm, based on the convexityof cost i . We then show how to speed up this algorithm using advanceddata structures such as red-black search trees.Below we assume that any cell C i has pins fpin i 1 ;:::;pini r ig sortedfrom left to right. Let us shift C i from the site s j to the right by somesmall ε > 0. We set f j (pin i l )=,1ifpini l is the rightmost pin on somenet N and set f j (pin i l )=1ifpini l is the leftmost pin for some net N.In all other cases we set f j (pin i l )=0.Lemma 1 The right slope of the function cost i at the site s j equalsslope i (s j )= f j (pin i l )l=1Proof. Sliding C i to the right by ε increases cost i by (L , R) ε,whereL is the number of nets with the rightmost pin on C i and R is thenumber of nets with the leftmost pin on C i .The minimum interval [l(C i );r(C i )] of a cell C i is the intervalwhere the slope slope i equals 0.The main step of the Clumping Algorithm, which is an applicationof the convexity of cost i , involves the collapsing of two neighboringcells. By collapsing of two neighboring cells C i and C i+1 we meanreplacing them with a new cell Ci 0. The new cell C0 i has the set ofpins fpin i 1;:::;pin i r i; pin i+11;:::;pin i+1 r l+1g (sorted from left to right) andwidth w(C i )+w(C i+1 ).We now present our Clumping Algorithm based on collapsing ofcells. The set of cells is represented as a double list denoted CELL.r i∑The function cost i (x) is represented as a sorted list of positions wherethe function cost i (x) changes slope, i.e., where a net has its fixed leftmost/rightmostpin just over the corresponding pin of C i . Each suchposition is marked with ,1 if it is the rightmost pin of a net and +1ifit is the leftmost pin of a net. Then slope i (x) equals the sum of ,1’sto the left of x and +1’s to the right of x.Clumping Algorithm1. For i = 1;::;n find the list cost i and two endpoints [l i ;r i ] ofthe minimum interval, i.e., where cost i is minimum.2. x(C 0 ) 0, w(C 0 ) 0, C 0 C 0// first cell C 0 with 0-position and 0-width3. Going along the list CELL apply PLACE(C i )// i.e., for i = 1::n4. Output positions of original cells using positions of collapsedcells and widths of the original cellsProcedure PLACE(C i )if p(C i,1 )+w(C i,1 ) > r(C i )// if C i,1 , C i cannot both be placed in their// minimum intervals, p(C i )= position of C ithen COLLAPSE(C i,1 ;C i ) and PLACE(C i,1 )else place C i at the position p(C i ) which is either theleftmost minimum available position p(C i,1 )+w(C i,1 )or the leftmost minimum position l(C i )// choose the rightmost of themProcedure COLLAPSE(C i,1 ;C i )Shift positions from the list of C i by w(C i,1 )Merge the list for C i with the list for C i,1Find the minimum interval for the merged listsSet w(C i,1 )=w(C i,1 )+w(C i )delete cell C iTheorem 1 The Clumping Algorithm finds the optimal placement.Proof. By induction on n, the number of cells. If n = 1, the cell C 1is placed in the minimum interval and the theorem is obviously true.Now assume that the theorem is true for n,1 cells. We will show thatthe theorem is true for n cells.If in the operator if in the procedure PLACE(C i ) we have the caseelse, then the theorem is true since we optimally place C 1 ;:::;C n,1(by induction) and then optimally place C n . Otherwise, if there is aconflict between cells C n,1 and C n , then we collapse C n with C n,1 .In this case, there is no gap between cells C n,1 and C n in the optimalplacement and we may find the optimal placement for the unionof C n,1 [C n . Then, by induction, the collapsed cell C n,1 is placedoptimally.Now we will show that the runtime of the Clumping Algorithmis O(B 2 )=O(m 2 ). Indeed, collapsing of two cells C i and C i+1 takestime of order Bound i + Bound i+1 . In the worst case, we need to collapseeach next cell with the previous cell representing all previousoriginal cells. The runtime isn∑ Bound i = O(B 2 )=O(m 2 )i=1Unlike the Prefix Algorithm, the Clumping Algorithm can be implementedto run faster using advanced data structures. We show howto speed up the Clumping Algorithm to O(mlogm) using red-blacksearch trees (RBST). 5 We use RBSTs to represent lists correspondingto each function cost i (x) for a cell C i , as follows. For each cellC i ,letSet i be a set of coordinates at which the function cost i changesslope. We supply each element s of Set i with the net weight nw(s) =1if s = f r (N) and nw(s) =,1 ifs = f l (N). 6 Each set Set i is suppliedwith pointers to the left and right ends of the minimum interval[l i = l(C i );r i = r(C i )] (slope i (l i )=0).5 The red-black search tree is a data structure which allows insertion, deletion andfinding of predecessor and successor (i.e. previous or next in the sorted list) in a set of nnumbers in time O(logn) (see, e.g., Chapter 14 in [7]).6 Note that if we have some weights on the nets (e.g. 1/#(cells)) we can also incorporatethem in the definition of the net weight.

Now we analyze how fast we can run procedures PLACE(C i )and COLLAPSE(C i,1 ;C i ). The procedure PLACE(C i ) needs constanttime because we can use pointers to l i = l(C i ) and r i = r(C i ). Theimplementation of COLLAPSE(C i,1 ;C i ) is more tricky. We first findwhich cell has the longer list (C i,1 or C i ) and then the positions fromthe shorter list are shifted left or right by the width of the “longer” celldepending on the relative position of the “longer” cell (left or right).Next, we insert shifted positions of the shorter list into the longer list.Each time a new position is inserted we update the minimum intervalaccording to the net weight of that position. Note that the RBSTimplementation of all lists is very fast. We conclude that the runtimeof the procedure COLLAPSE(C i,1 ;C i ) implemented with RBSTs isO(minfM i ;M i,1 glog(maxfM i ;M i,1 g)).The number of entries in all Set i ’s is 2m. Thus the runtime of CAis O(mlogm) sincenn∑(min logmax) log(2m) ∑ min 2mlog(2m)i=1i=17 Heuristics for Improving the Cell OrderingIn this section we deal with the problem of finding an improved(sub)optimal ordering of cells in a single row when all cells in all otherrows are fixed. As noted above, this is an instance of a well-knownNP-hard problem [8].Cell Ordering Problem = the Single-Row Problem where theleft-to-right order of cells is not fixed.We have implemented several heuristics for the Cell OrderingProblem, e.g., based on the order of middle points of the minimumintervals of cells. In our experience, initial cell orderings found instandard-cell placements by industry tools are quite good, and difficultto improve. The Swapping Heuristic below is the most successfulapproach we have found so far. 7Swapping HeuristicFor each cell, compute its cost curve and cost curve min point.(treat all cells in other rows as fixed)Repeatedly iterate down the row doing:For every adjacent pair of cells that would overlap or crosswhen placed at their min points doIf a superior relative placement is possible when swappedthen swap the order of cells in the pairelse don’t swap, and remember not to try this pair againRepeat previous step until no pairs would swap8 Experimental Results and Future DirectionsWe have implemented both the Prefix and Clumping algorithms forthe Single-Row Problem, as well as the swapping heuristic for improvingthe row ordering. Our testbed consists of industry standard-celldesigns in Cadence LEF/DEF format, which we place with a leadingindustry placement tool.Given the final output of the industry placer, we apply either thePrefix or the Clumping algorithm to optimize wirelength incident toeach single row of the design. An iteration of our approach processesall rows of the design, one row at a time; 8 we continue iterations untilimprovement from an iteration is less than 0.001%. We have also studiedour exact single-row algorithms in combination with the swappingheuristic (i.e., the swapping heuristic is applied to reorder cells in agiven row, just before the wirelength minimization is applied). Table1 shows the percentage wirelength improvement and CPU times on a140MHz Sun Ultra-1 workstation for five industry test cases. We cansee that surprisingly significant wirelength reductions (an average of7 We are currently investigating methods inspired by LP-based heuristics for globalplacement ([3]). E.g., one approach is to apply LP for the single-row placement instead ofthe entire layout, and derive orderings from the result.8 Actually, we process each sub-row of the design in turn; a sub-row is typically delimitedby vertical M2 power stripes. Cells must remain within their original subrows.6.58%) are obtained in very little time. 9 The swapping heuristic addsslight further reduction in wirelength.Our results indicate that large wirelength reductions are possible,even beyond the results of industry placement tools, when optimalsingle-row placement with extra sites (white-space) is appliediteratively to the rows of a standard-cell layout. Our current workseeks to verify the routability of the resulting improved placements,and to define more detailed (e.g., routability-driven) single-row placementobjectives that are also amenable to efficient optimal solutions.More generally, we are investigating the utility of optimal solution approachesat a number of phases within a “successive approximation”methodology for standard-cell placement.Benchmarks Algorithm Prefix Clumping# cells #rows # sites Heuristic None Swap None Swap4155 46 41610 improve % 9.39 13.38 9.49 13.46runtime (sec) 56.8 94.1 31.8 69.411471 42 78036 improve % 6.00 6.18 6.00 6.22runtime (sec) 182.9 289 68.9 19812260 610 128893 improve % 1.13 1.38 1.15 1.40runtime (sec) 13.9 35.1 14.8 40.27309 56 47152 improve % 7.43 8.13 7.45 8.08runtime (sec) 46.5 125.7 35.1 95.98829 60 98760 improve % 3.70 3.54 3.76 3.77runtime (sec) 34.0 43.9 20.7 91.8Table 1: Experimental results for five industry test cases.Acknowledgments. We thank Igor L. 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Yamada, H. Okude, and T. Kasai, “A hierarchical algorithm for one-dimensionalgate assignment based on contraction of nets,” IEEE Trans. Computer-Aided Design,vol. 8, pp. 622-629, 1989.9 Notice that the Prefix and Clumping results are slightly different, even though bothare optimal algorithms for the same problem. This is because there are often multipleoptimal placement solutions for a given cell ordering in a row; our two implementationsare not guaranteed to return the same optimal solution, and thus their solution paths candiverge. Also, while the Clumping algorithm is faster, its runtimes can occasionally begreater because it makes more iterations.