Handbook of algorithms for physical design automation part 44 pdf

Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C020 Finals Page 412 23-9-2008 #15 412 Handbook of Algorithms for Physical Design Automation 20.3.2 TETRIS-BASED LEGALIZATION Rather than distributing logic elements with flow-like methods, an alternate approach based on packing is possible. The Tetris legalization method by Hill [12] is remarkably simple, and trivial to implement. We first discuss the approach in the context of standard cell design, and then show how it extends to handle a mix of standard cells and macroblocks. For each cell c i ,wehaveadesired position (x i , y i ); the cells must be legalized into standard cell rows R ={r 1 , r 2 , , r k }, and assume that the left-most open position in each row is known. The legalization method first sorts the cells C by their x position, and then inserts them into the left side of a row in a g reedy manner, such that the displacement of each cell is minimized. Algorithm 4 The Tetris legalizer by Hill { C }=All cells to be legalized; Sort the cells in C by their X -coordiates to get L s ; l j = left-most position of each row r j ; for i = 1 to the number of cells do best = lim sup; for j = 1 to the number of rows do cost = displacement of moving cell i in L s to l j ; if cost ≤ best then best = cost ; best _ row = j ; end if end for Move cell i in L s to the row best _ row ; l best _ row = l best _ row + width i ; end for Pseudocode for the approach is shown in Algorithm 4. If one were to rotate the placement region counterclockwise, the legalization process might look very much like a game of Tetris, for which the algorithm is named. Our example code can be made to run more quickly by only considering rows close to the desired position of a cell; in p ractice, runtimes are linear with the number of cells to be legalized. Our example also packs cells to the left, but this is not in general necessary; if space has been reserved for routability, it may be perferable to legalize into a position that is not to the left. Other obvious variants are to legalize from both the left and right sides, and to sort the cells by their leftmost boundary, rather than their center. The method can also be adapted to designs that contain macroblocks; [13] simply added a check for each row spanned by a macroblock, to find the leftmost position that did not result in an overlap. Although the Tetris method works well in practice for global placements that have the logic elements distributed evenly,evensmall areas with overlapcancause significant trouble.When a block must be displaced during legalization, it may cause a ripple effect, resulting in the displacements of many other blocks. In some cases, wirelengths can jump by 30 percent or more; the method is fast and effective for easy legalization problems, but performs poorly on more difficult cases. 20.3.3 SINGLE-ROW DYNAMIC PROGRAMMING-BASED LEGALIZATION Contrast to the simple Tetris legalization method is one which operates on a row-by-row basis [15], single-row dynamic programming-based legalization was developed for standard cell placements, and selects a subset of cells to place in a row using dynamic programming. Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C020 Finals Page 413 23-9-2008 #16 Legalization and Detailed Placement 413 The outline of the technique is given in Algorithm 5. Legalization is done from the bottom row to the top row (although the reverse is also possible). Cells are first sorted by their initial Y-coordinates in nondecreasing order. Let L s denote this sorted list of cells. A set of candidate cells C i cand are then selected for assignment to a particular row, say R i . The total width of cells in C i cand is greater than the capacity of R i and satisfies the following relation: k 1 ∗ width(R i )≤width(c i cand ) = m  j=1 width(c j ) ≤k 2 ∗ width(R i ) where k 1 , k 2 ∈and are constants such that 1 < k 1 ≤ k 2 width(R i ) is the width of row R i width(C i cand ) is the total width of cells in C i cand width(c j ) is the width of cell c j ∈ C i cand m =|C i cand | Note that cells in C i cand are always the lowermost m cells in L s that satisfy the above relation. The Tetris algorithm packs a single cell at a time onto one side of the placement region, whereas this approach finds a set of cells to fill an entire standard cell row. The m e thod used to find this set of cells is similar to the dynamic programming technique of solving the classic 0–1 knapsack problem [16]. In the knapsack problem, there is a limit on the weight of the knapsack; the analogue of this constraint is the limit on total cell width in the row. Similarly, the value of an item in the knapsack problem is modeled by the physical displacement of each cell. After identifying the set of candidate cells C i cand , they are sorted by their X-coordinates in nondecreasing order. These cells are considered for assignment to R i in this sorted o rder. The knapsack-like problem is solved using dynamic programming, with the selected cells being packed in, and the remaining cells being considered for legalization in the next row. This process is repeated until all cells have been legalized. Algorithm 5 illustrates the approach. Algorithm 5 Row-by-row dynamic programming based legalization. (The overall approach is a variation of the classic method for the 0–1 knapsack problem) { C }=All cells to be legalized; Sort the cells in C by their Y -coordinates to get L s ; for i = 1to N r-1 do Select a set of candidate cells ( C i cand ) from L s ; Assign a subset of cells C i assign from C i cand to row R i ; L s = L s \ C i assign ; end for Assign the remaining cells in L s to row R N r ; Every cell c j has two factors associated with it: (1) the cost of assigning c j to row R i , denoted by w i j and (2) the penalty of not assigning c j to R i , denoted by p i j . Obviously, if c j is assigned to R i then, p i j = 0 and if it is not assigned to R i ,thenw i j = 0. If cell c j is to be assigned to row R i , then the cost of assignmentis given by the following equation: w i j = (x i j − x j ) 2 + (y i j − y j ) 2 where (x j i , y j i ) is the location in R i where c j is going to be placed (x j , y j ) is c  j s initial location Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C020 Finals Page 414 23-9-2008 #17 414 Handbook of Algorithms for Physical Design Automation On the other hand, the penalty of not assigning c j to R i is given by p i j = (y i+1 j − y j ) 2 where y i+1 j is the y-coordinate of the center of row R i+1 y j is c  j s initial y-coordinate 20.4 LOCAL IMPROVEMENTS Following legalization, a large number of well-studied techniques can be used to optimize a circuit placement. The computational complexity of many useful algorithms is exponential; O(2 n ) or O(n!) are common, where n is the number of logic elements being optimized. Even on the fastest available computer, the practical values of n are small—far smaller than the number of elements in a placement problem. For this reason, only a small number of elements are processed in any given step: those under the sliding window. If the size of the sliding window can accomodate six cells, one might optimize the first six cells in a single row, and then move on to the third through ninth cell. With only six elements, a O(n!) algorithm is practical. Overlaps between optimization windows help minimize the impact of the local nature of this step. Sliding window optimization is shown in Figure 20.9. 20.4.1 CELL MIRRORING AND PIN ASSIGNMENT The first class of local improvements we consider is cell mirroring; this technique is easy to under- stand. Each cell in a design contains a set of input and output pins, and these are not usually symmetrically placed. By mirroring a cell around its X axis, interconnect lengths can be improved in many cases. An illustration of this is shown in Figure 20.10. An early study of the problem by Cheng [17] showed that finding optimal orientations is in fact NP-complete;assuch, wecannothopeto findanoptimalsolutiontothe problem.Inpractice,however, relatively simple heuristic methods can be quite effective. One approach is to simply evaluate each orientation on a cell-by-cell basis, selecting the better choice in a greedy manner. Within a few passes, solution quality converges. Detailed placement techniques normally operate on a small window into the circuit, making local improvements. Most techniques assume macroblocks are fixed, and optimize around them. Sliding window optimization is common; windows typically overlap slightly. Optimization windows may be single- or multirow. Multiple passes of optimization are also commonplace. Fixed macroblock Fixed macroblock FIGURE 20.9 Detailed placement techniques are normally applied on a subset of the design, moving from one area to another with a sliding window approach. Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C020 Finals Page 415 23-9-2008 #18 Legalization and Detailed Placement 415 A A B B C2 C1 Out Out Out Out Default orientation for C1 Mirrored orientation A A B B OutAB OutAB FIGURE 20.10 Standard cells can normally be mirrored along the Y axis. This can reduce the length of s ome interconnect segments, and may be used to slightly shift routing demand from one area to another. Macroblocks frequently allow mirroring along both axis, as well as r otation. For typical standard cell design, mirroring is only possible along the X axis; power and ground lines are integrated into the cell design, and mirroring along the Y axis might not be possible. When implementing a detailed placement approach, one should consider the nature of the standard cells; similarly, the detailed placement tool should be considered in cell design. For macroblocks, mirroring around both the X and Y axes may be possible; it may also be possible to rotate a block. Related to cell mirroring is the technique of pin assignment. For many logic elements, there may be functionally equivalent pins, for example, the inputs of a NAND gate are equivalent. A circuit designer m ay specify that signal net n i is connected to input pin a, while signal net n j is connected to input pin b; some detailed p lacement tools support optimization of the assignment. Individual pins may have differing delay characteristics (due to the internal layout of transistors), so optimization of a pin assignment may benefit wirelength, delay, or both. Finally, cell mirroring and pin assignment may be beneficial in improving the routability of a design. In channel routing tools, an internal data structure known as a vertical constraint graph is frequently used; the data structure reveals an ordering of interconnect segments that will allow easy routing. Depending on precise locations of individual pins, there may be cycles in the constraint graph—significantly complicating routing. Slight changes to pin locations, which can be achieved with mirroring or pin assignment, may eliminate cycles and simplify routing greatly. Early works by Her [18] and Swartz [19] illustrate a tight coupling between detailed placement and routing. Slight adjustments at a local level can improve routing, and these adjustments are difficult to make without actually performing the detail routing step. 20.4.2 REORDERING OF CELLS A secondcommondetailedplacementmethod,sometimesintegratedwithcell mirroring, isimproving a design throughthe reorderingofsmall groupsofcellsusingaslidingwindowapproach. Figure20.11 shows two cases; in the first, wirelength can be improved by swapping the positions of a pair of a cells. In the second, the optimization window size required to find a better placement is three cells. For standard cell design, it is common to see optimization window sizes ranging from three to eight. Much larger windows carry a heavy runtime cost, while windows of size two have relatively small wirelength benefit. Brute force enumeration is easy to implement; some speed improvements can be obtained using branch-and-bound techniques. Caldwell [20] presented an extensive study of the topic, considering different ways to to enumerate permutations, and to bound the solution space. Pseudocode for a simple branch-and-boundapproach to cell reordering is shown in Algorithm 6. Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C020 Finals Page 416 23-9-2008 #19 416 Handbook of Algorithms for Physical Design Automation A A A A B B B B C C C C D D D D AB CD AB C D An optimization window of size 2 can improve the initial placement (swapping the cells B and C). A slightly different initial placement cannot be improved with an optimization window of size 2. To obtain a better placement, a larger optimization window is required. FIGURE 20.11 Order of cells in a row (or group of rows) can be refined u sing small optimization windows; the size of the window limits the improvement possible, and also impacts runtime considerably. With designs that are densely packed, evaluating any given permutation is relatively simple; cells of different sizes are simply placed in the order, with the position of cell c i being immediately after cell c i−1 . When there is open space, however, the problem is slightly more difficult. Algorithm 6 A recursive implementation of cell reordering; by computing partial wirelengths, the number of solutions explored can be reduced { C }=cells in the optimization window; { F }=empty set; left = left side of optimization window; reorder( C,F,left ) { if C is empty then Evaluate current configuration; Store configuration if improved; else for each c i in C do place c i at left ; reorder( C − c i , F + c i , left + width ( c i )); end for end if } Consider a simple case with cells c i and c j , and a small open space. There are two different possible permutations of the cells (c i c j and c j c i ), but a potentially infinite number of ways to divide the available space between the cells. The optimal amount of space to be inserted before the first cell, between the first and second cell, and after the second cell can be determined (see Section 20.4.4), but this complicates the reordering process. Reordering of cells across multiple rows is also fairly common. When cells are of different sizes, not all permutations may be valid. One method of implementing a multiple row reordering fu nction is to first divide the available space on a row-by-row basis. For any given permutation, each cell is inserted into the top-most open row; when a row is filled, processing moves to the next row. If the permutation does not fit, this will be discovered along the way, and the potential solution can be discarded. When implementing a reordering technique, a number of performance trade-offs must be considered. Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C020 Finals Page 417 23-9-2008 #20 Legalization and Detailed Placement 417 • If the size of the optimization window is small, brute force enumeration may be the best approach. Branch-and-bound methods require incremental wirelength computations, and this has a runtime overhead that may outweigh the benefits of solution space pruning. • If the size of the optimization window is large, branch-and-bound method can have signifi- cant benefit. Many permutations can be eliminated easily, with the savings outweighing the extra cost for incremental wirelength computation. • If the design contains space between cells, one must decide how to handle it. Optimal space allocation can be done, but with a runtime cost; for large optimization windows, this is unlikely to be practical. Space can be treated as a single monolithic unit, or divided equally between cells; what is best may depend on the design itself. • For optimization across multiple rows, many permu tations may be eliminated if cells are of different sizes. This can increase the size of a practical optimization window in some cases. When the design contains space, this may make more permutations possible, thereby lowering the practical optimization window size. 20.4.3 OPTIMAL INTERLEAVING Another method of altering the order of a group of cells is optimal interleaving by Hur [21]. The approach utilizes dynamic programming to allow for large window sizes with low runtimes. We suggest the text by Cormen [16] as a good reference for dynamic programming (Figure 20.12). To begin with, let us assume the following: • C is a given set of cells. • A initial is an overlap free initial linear arrangement of cells in C. By linear arrangement, we mean the placement of cells within a single row. Here we use the terms arrangement and placement synonymously. • C 1 = c 11 , c 12 , , c 1n and C 2 = c 21 , c 22 , , c 2m are two proper subsets of C of size |C 1 |=n and |C 2 |=m such that, C 1 ∪ C 2 = C and C 1 ∩ C 2 =∅. • p ij denotes the position of cell c ij . • p ij < p kl means that c ij precedes c kl in some arragement. Note that c kl may not be immediately next to c ij . ABCX Y Z AB X A X B XAB X Y Z ABC Optimal arrangement for (AB)(X) FIGURE 20.12 Interleaving of two groups of standard cells can be done in an efficient manner w ith dynamic programming. In this figure, we have two sets of cells that are to be interleaved; optimal partial solutions are stored in a matrix. Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C020 Finals Page 418 23-9-2008 #21 418 Handbook of Algorithms for Physical Design Automation We are now ready to define interleaving. An interleaving I n,m of C 1 and C 2 is an arrangement of cells in C 1 and C 2 satisfying the following conditions: If p 1i < p 1j in A initial ,thenp 1i < p 1j in I n,m If p 2k < p 2l in A initial ,thenp 2k < p 2l in I n,m where, i = j and 1 ≤ i, j ≤ n k = l and 1 ≤ k, l ≤ m Note that there are  n +m n  different ways of interleaving C 1 and C 2 . An optimal interleaving is the one with minimum cost. The cost of an interleaving I i,j denoted by w(I i,j )(1 ≤ i ≤ n and 1 ≤ j ≤ m) is its total linear wirelength, ignoring the cells in {C \I i,j }. Because placement rows are typically horizontal, by linear wirelength we mean the wirelength in X-dimension. Hur and Lillis [21] proposed a polynomial time algorithm for solving the optimal interleaving problem that utilizes dynamic programming. The recurrence equation is stated as follows: I 0,0 =∅, w(I 0,0 ) = 0 I i,j =  I i−1,j c 1i if w(I i−1,j c 1i )<w(I i,j−1 c 2j ) I i,j−1 c 2j otherwise Note that a dynamic programming formulation is only possible because for any subsequence I i,j , w(I i,j ) is independent of the ordering of cells in {C \I i,j }. The runtime of the algorithm is O(nm + n p (n + m)) where n p is the number o f pins belonging to the nets of cells in C, which is typically proportional to the number of cells. Optimal interleaving can be applied to the entire legalized placement using the sliding window technique described above. In Ref. [21], the subsets C 1 and C 2 are chosen arbitrarily. 20.4.4 LINEAR PLACEMENT WITH FIXED ORDERINGS The placement problem is NP-complete, even for the one-dimensional case [22]. The one- dimensional problem is commonly known as linear placement. In this problem, candidate cells belonging to a single placement row are given and we need to find a placement that m inimizes the wirelength. Note that we are only allowed to change the X-locations of cells; hence the name linear placement. A restricted variant of this problem, in which we cannot alter the cell o rdering, can be solved optimally in polynomial time. One solution method was proposed by Kahng [23], and later on improved on by Brenner [24]. First, to formally state the problem, we assume that we are given the following: • Set of cells {c 1 , c 2 , , c n } with the width of cell c i denoted by width(c i ) • Placement row of width W : W ≥  n i=1 width(c i ) • Legal initial placement P ={p 1 , p 2 , , p n },wherep i + width(c i ) ≤ p i+1 for i = 1, , n −1, 0 ≤ p 1 and p n ≤[W −width(c n )] The optimization objective is to find a legal placement P ={p 1 , p 2 , , p n } with the minimum possible bounding box wirelength such that if p i < p j in P then p i < p j in P. It should be obvious that to impact wirelength, white space must be present. If the cells are densely packed, there is only a single possible placement, and no improvement is possible. Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C020 Finals Page 419 23-9-2008 #22 Legalization and Detailed Placement 419 20.4.4.1 Notations and Assumptions • We are only concerned with nets with at least one movable cell. Let m be the total number of such nets. • Let c N L and c N R be the leftmost and rightmost movable cells of net N with locations denoted by p N L and p N R (recall that we are only dealing with X-location h ere). Note that c N L and c N R are known beforehand and do not change during the course of solving the problem as we are not allowed to change the ordering of cells. • Let f N L and f N R stand for the locations of the leftmost and rightmost fixed pins of net N (A net without a fixed pin can be broken down into two nets; one that connects c N L with a dummy cell at location W andanother one that connectsc N R with a dummy cell at location 0). • For the sake of simplicity, let us assume that all pins are located at the left edge of their cells. Our objective is the following: minimize  N  max  f N R , p N R  − min  f N L , p N L  or equivalently, minimize  N  f N R − f N L  + n  i=1 w i where w i is the contribution of cell c i to the objective function and is given by: w i =  N:c i =c N L max  f N L − p i ,0  +  N:c i =c N R max  p i − f N R ,0  A careful observation of the function w i indicates that cells that are neither c N L nor c N R for any net N can be placed arbitrarily, as they do not contribute anything to our objective function. These cells can be merged with their predecessors. The remainder of our discussion is valid only for cells that do not fall in this category. Because every net under consideration has a unique c N L and c N R and it is possible to have c N L = c N R for a net (a net that has a single movable cell), we have the relation n ≤ 2m. 20.4.4.2 Analysis of the Cost Function One can observe that the function w i is convex piecewise linear. T his can be deduced as follows: take a cell c i and create a sorted list of the f N L and f N R values of all nets for which c i = c N L or c i = c N R , respectively, and mark them on a number line. For any location x,leta (b) denote the number of f N R (f N L ) locations to the right (left) of x. Three possibilities occur • a < b: w i is linear and decreasing in this interval as x increases • a = b: w i is constant and minimum • a > b: w i is linear and increasing as x increases 20.4.4.3 Dynamic Programming Algorithm Solutions to the linear placement with fixed ordering p roblem are based on dynamic programming. We use the following notations for this section: Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C020 Finals Page 420 23-9-2008 #23 420 Handbook of Algorithms for Physical Design Automation • Let S i denote the set of sites where cell c i can be placed without overlapping with any of its neighboring cells. • Let P j i be an optimal prefix placement for cell c i where all cells to the left of c i have been placed optimally and p i ≤ s j (∈ S i ). • Let W j i be the total cost of P j i and w j i denote the cost of placing c i at s j (∈ S i ). Our goal is to find P W−width(c n ) n . The dynamic programming formulation can be stated as follows: P j 0 =∅, W j 0 = 0 P j i =  P j−width(c i−1 ) i−1 ∪{p i = s j } if W j−width(c i−1 ) i−1 + w j i < W j−1 i P j−1 i otherwise 20.5 LIMITS OF LEGALIZATION AND DETAILED PLACEMENT All of the operations performed in legalization and detailed placement are in some sense local, and thus fail to address global optimization objectives. These algorithms are typically applied in an interative manner, in ad hoc mixtures. Many groups have experimented with different legalizers, window sizes for reordering, methods to perform space allocation, and so forth. There is n o simple method to obtain the right strategy. Fortunately, most placement tools from academic groups can be run in a legalization or detailed placement mode, making it easy to test out different techniques. When compared with the runtimes for global placement, detailed placement times are usually low; it is worthwhile to explore different combinations. The first optimization normally applied would be space insertion to reduce routing congestion. As a rule of thumb, one might wish to keep the routing demand in any area less than about 70 percent of the available routing resource. Dense routing frequently results in the failure to complete all nets, or in large net detours. The achievable routing density depends on the specific routing tool used. The wirelength increase d ue to space insertio n may be less than what one might anticipate. If the logic elements are spread apart, with the placement area increasin g by 10 percent, the actual spreading performed is less than 5 percent in horizontal directions. For this case, the worst one should expect would be a 5 percent increase in total wirelength. Going from an abstract global placement to a legal one typically incurs a small wirelength increase (perhaps a few percent); abstract placements with relatively little overlap generally have smaller increases. If there is excess space for routing, it is quite possible that the legalized placement could actually reduce wirelengths. If there is an increase, reordering, cell mirroring, and optimized linear placement can normally recover some of the wirelength loss. Within a few iterations, however, the improvements o f any technique become asymptotically small. The first few percent of wirelength improvements come easily; afterward, the progress is slow. In many respects, it is a decision of the circuit designer as to how long to carry out detailed placement. For some, a minor improvement in wirelength might have a greatdealofvalue, warranting the extra compute time. For others, a quick-and-dirty solution may be sufficient. It should be stressed that one of the most crucial considerations is to minimize the amount of overlap during global placement, and ensure that there are no large sections of the design that have area demands that exceed the space available. Legalization methods handle space distribution imperfectly; if the problems can be solved in global placement, the results will almost certainly be better.Comparing the locations of placements beforeandafter legalization can providea great deal of insight. There are a number of v isualization tools for placements; any area that changes significantly during legalization should be examined carefully. Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C020 Finals Page 421 23-9-2008 #24 Legalization and Detailed Placement 421 By mapping a graphic image to the cells in a design, it is possible to compare two different placements. The original image on the left is mapped to placements from Capo and Feng Shui for the benchmark Peko01 (which has a known optimal solution). Both placements are clearly suboptimal; as the benchmar k lacks pads, the images are rotated. It is frequently useful to compare images before and after legalization, and at various stages of detailed placement. Large distortions can identify problems with legalization, or areas where global placement has performed poorly. Placement Placement Detailed placement FIGURE 20.13 Detailed placement techniques can resolve local problems, but placement suboptimality is both a local and global phenomena. In experiments with synt hetic designs, one can compare an optimal configuration with the output of a placement tool by mapping an image. The distortions of the image reveal ho w the placement deviates from the desired result. Although there h ave been significant advances in placement, the solutions obtained are far from optimal. Recently, Chang presented a set of benchmark circuits with known optimal configura- tions [25]. These placement examples with known op timal (PEKO) circuits attracted a g reat deal of attention, and the results of many placement tools on the benchmarks was surprising. In Figure 20.13, we illustrate the results by mapping an image onto the optimal placement of a synthetic PEKO benchmark, and then rearranging the placements to match results of a number of academic tools. The PEKO benchma rks contain no pads; thus, there are multiple optimal configurations, cor- responding to mirroring or flipping of the design. From the distortions of the images, it should be clear that many placement tools are globally correct ; at a very high level, the placements resemble the optimal result. At a local level, however, there is a great deal of suboptimality; sections of each placement are stretched or warped, resulting in higher interconnect lengths. While there is some question as to how closely the synthetic PEKO benchmarks resemble real circuitry, it is obvious that placement results could be improved, and that this improvement must span both the local and global levels. REFERENCES 1. J. Lou, S. Krishanmoorthy, and H. S. Sheng. Estimating routing congestion using probabilistic analysis. In Pr oceedings of International Symposium on P hysical Design, pp. 112–117, 2001. 2. C. J. Alpert, G. -J. Nam, P. G. Villarrubia, and M. C. Yildiz. Placement stability metrics. In Proceedings of Asia South Pacific Design Automation Conference, pp. 1144–1147, 2005. 3. N. Viswanathan and C. C. -N. Chu. Fastplace: Efficient analytical placement using cell shifting, iterativ e local refinement and a hybrid net m odel. In Proceedings of International Symposium on Physical Design, pp. 26–33, 2004. 4. Z. Xiu and R. A. Rutenbar. Timing-driven placement by grid-warping. In Proceedings of Design Automation Conference, pp. 585–591, 2005. 5. K. Doll, F. M. Johannes, and K. J. Antreich. Iterative placement improvement by network flow methods. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 13(10):1189–1200, 1994. . location Alpert /Handbook of Algorithms for Physical Design Automation AU7242_C020 Finals Page 414 23-9-2008 #17 414 Handbook of Algorithms for Physical Design Automation On the other hand, the penalty of. following notations for this section: Alpert /Handbook of Algorithms for Physical Design Automation AU7242_C020 Finals Page 420 23-9-2008 #23 420 Handbook of Algorithms for Physical Design Automation •. are stored in a matrix. Alpert /Handbook of Algorithms for Physical Design Automation AU7242_C020 Finals Page 418 23-9-2008 #21 418 Handbook of Algorithms for Physical Design Automation We are now ready