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Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C009 Finals Page 182 24-9-2008 #23 182 Handbook of Algorithms for Physical Design Automation w d h (x,y,z) (c) 1 2 3 4 5 X XY Z241 53 (a) (b) (d) XXY Z 2 2 4 4 1 1 5 5 3 36789 Parent links Indices FIGURE 9.15 (a) 3D slicing floorplan with five modules. (b) Corresponding slicing tree. (c) Dimensions of a 3D block. (d) Static array. where n is the number of modules. Each of the first n −1 elements in the array represents an internal node, and the first element always represents the root of the tree. The last n elements represent the leaves. Each element of the array is associated with a ten-tuple (t, p, l, r, x, y, z, w, d, h). The t is the tag information and its value is X, Y,orZ for each internal node and a module n ame for a leaf. p, I,andr denote the element indices in the array for the parent, the left child, and the right child of a node, respectively. The x, y, z, w, d, h are the dimensional information of a module or a subfloorplan, and (x, y, z) is called the base point (see Figure 9.15c). Figure 9.15d is the static array for the slicing tree shown in Figure 9.15b, where only the parent link of each element is drawn for simplicity. Given the static array of a slicing tree, the position of each module and the dimensions of the corresponding floorplan can be calculated by a recursive p rocedure starting from the root. Two kinds of moves, exchangeand rotation, are used during the annealing process for generating neighboringsolutions. Anexchangemoverandomlychoosestwo subtreesand swapsthem;asaresult, the two corresponding elements in the static array will be updated accordingly. On the other hand, a rotation move randomly selects a subtree and rotates the corresponding subfloorplan along x-, y-, or z-axis; as a result, the elements of the static array corresponding to the internal nodes containedin the subtree will be updated accordingly. It can be proved that the two neighborhood moves are complete in the sense that each slicing tree can be reached from another one via at most 10n −6moves. This 3D floorplanner can be specialized to solve the 2D problem as well, and according to Ref. [28], it is able to produce 2D slicing floorplans with the smallest areas for the two largest MCNC benchmarks, ami33 and ami49, among all 2D slicing and nonslicing floorplanning algorithms ever reported in the literature. Besides, this 3D floorplanner can be extended to handle various types of placement constraints and thermal distribution. 9.8 CONCLUSION In this chapter, wehaveintroduced twoslicing floorplan representations,that is, slicing treeandPolish expression, on which many existing slicing floorplan design/optimization algorithms are based. We have presented efficient/effective area and power optimization algor ithms for slicing floorplans. These optimization algorithms are typically embedded into a slicing floorplanner.We have discussed the problem of slicing floorplan design with or without placement constraints, and highlighted existing solutions. Finally we have described some more recentresults inslicingfloorplansfor FPGAs Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C009 Finals Page 183 24-9-2008 #24 Slicing Floorplans 183 and 3D ICs, in addition to the mathematical analysis on area upper bound and the completeness of slicing tree representation. Before concluding this chapter, we would like to point out that since 1970s, slicing floorplans have been an active research topic, and therefore it is very difficult to discuss all existing solutions of this area in a single chapter. Instead, we have chosen to present the results, including the most recent ones in the past ten years, which we think will interest readers the most. REFERENCES 1. R. H. J. M. Otten. Automatic floorplan design. Proceedings of Design Automation Confer ence, pp. 261–267, 1982. 2. D. F. Wong and C. L . Liu. A new a lgorithm for floorplan design. Proceedings of Design Automation Conference, Las Vegas, N evada, pp. 101–107, 1986. 3. L. J. Stockmey er. Optimal orientation of cells in slicing floorplan designs. Information and Control, 57(2):91–101, 1983. 4. W. Shi. An Optimal algorithm for area minimization of slicing floorplans. Proceedings of International Conference on Computer-Aided Design, San Jose, California, pp. 480–484, 1995. 5. R. H. J. M. Otten. Efficient floorplan optimization. Proceedings of International Conference on Computer Design, Las Vegas, Nevada, pp. 499–502, 1983. 6. G. Zimmermann. A new area and shape function estimation technique for VLSI layouts. Proceedings of Design Automation C onference, Atlantic City, New Jersey, pp. 60–65, 1988. 7. K Y. Chao and D. F Wong. Floorplanning for low power designs. Proceedings of International Symposium on Circuits and Systems, Seattle, Washington, pp. 45–48, 1995. 8. R. E. Ta rjan. Data Structures and Network Algorithms. SIAM Press, P hiladelphia, Pennsylvania, 1983. 9. T. Corman, C. E . Leiserson, and R. Rivest. Introduction to Algorithms. MIT Press, Cambridge, Massachu- setts, 1990. 10. U. Lauther. A min-cut placement algorithm for general cell assemblies based on a graph representation. Pr oceedings of Design Automation C onference, San Diego, California, pp. 1–10, 1979. 11. D. P. La Potin and S. W. Director. Mason: A global floorplanning approach for VLSI design. IEEE Transactions of Computer-Aided Design of Integrated Circuits and Syst ems, CAD-5(4):477–489, 1986. 12. G. J. Wipfler, M. Wiesel, and D. A. Mlynski. A combined force and cut algorithm for hierarchical VLSI layout. Proceedings of Design Automation Conference, pp. 671–677, 1982. 13. B. W. Kernighan and S. Lin. An efficient heuristic procedure for partitioning graphs. Bell Syst em Technical Journal, 49(2):291–307, 1970. 14. D. G. Schweikertand B. W. Kernighan.A propermodel forthe partitioning of electrical circuits. Pr oceedings of Design Aut omation Workshop, pp. 56–62, 1972. 15. C. M. Fiduccia and R. M. Mattheyses. A linear-timeheuristic for improving network partitions. Proceedings of Design Aut omation Conference , pp. 175–181, 1982. 16. B. Quinn. A force directed component placement procedure for printed circuit boards. IEEE Transactions on Circuits and Systems, CAS-26(6):377–388, 1979. 17. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi. Optimization by simulated annealing. Science, 220, 671–680, 1983. 18. F. Y. Young and D. F. Wong. Slicing floorplans with boundary constraints. Proceedings of Asia and South Pacific Design A utomation Conference, Hong Kong, pp. 17–20, 1999. 19. E C. Liu, T H. Lin, and T C. Wang. On accelerating slicing floorplan design with boundary constraints. Pr oceedings of International Symposium on Circuits and Systems, Geneva, Switzerland, pp. III-339–III-402, 2000. 20. E C. Liu, M S. Lin, J. Lai, and T C. Wang. Slicing floorplan design with boundary-constrained modules. Pr oceedings of International Symposium on Physical Design, Sonoma County, California, pp. 124–129, 2001. 21. F. Y. Young and D. F. Wong. Slicing floorplans with range constraints. Proceedings of International Symposium on Physical Design, Monterey, California, pp. 97–102, 1999. 22. F. Y. Young, H. H. Yang, and D. F. Wong. On extending slicing foorplans to handle L/T-shaped modules and abutment constraints. IEEE Tr ansactions of Computer-Aided Design of Integrated Circuits and Systems, 20(6):800–807, 2001. Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C009 Finals Page 184 24-9-2008 #25 184 Handbook of Algorithms for Physical Design Automation 23. W. S. Yuen and F. Y. Young. Slicing floorplan with clustering constraints. Proceedings of Asia and South Pacific Design Automation Conference, Yokohama, Japan, pp. 503–508, 2001. 24. F. Y. Young and D. F. Wong. Slicing floorplans with pre-placed modules. P roceedings of International Conference on Computer-Aided Design, San Jose, California, pp. 252–258, 1998. 25. F. Y. Young and D. F. Wong. How good are slicing floorplans? Proceedings of International Symposiu m on Physical Design, Napa Valley, California, pp. 144–149, 1997. 26. M. Lai and D. F. Wong. Slicing tree is a complete floorplan representation. Proceedings of Design, Automation, and Test in Europe, Munich, G ermany, pp. 228–232, 2001. 27. L. Cheng and D. F. Wong. Floorplan design for multi-million gate FPGAs. Proceedings of International Conference on Computer-Aided Design, San Jose, California, pp. 292–299, 2004. 28. L. Cheng, L. Deng, and D. F. Wong. Floorplanning for 3-D VLSI design. Proceedings of Asia and South Pacific Design Automation Conference, Shanghai, China, pp. 405–411, 2005. 29. P. Guo, C K. Cheng, and T. Yoshimura. An O-Tree representation of non-slicing floorplan and its applications. Proceedings of Design Automation Confer ence, New Orleans, L ouisiana, pp. 268–273, 1999. 30. J. M. Emmert and D. Bhatia. A methodology for fast FPGA floorplanning. Proceedings of International Symposium on Field Programmable Gate Arrays, Monterey, California, pp. 47–56, 1999. 31. Xilinx Inc. http://www.xilinx.com. 32. K. Banerjee, S. J. Souri, P. Kapur, and K. C. Saraswat. 3-D ICs: A novel chip design for improving deep submicrometer interconnect performance and systems-on-chip integration. Proceedings of t he IEEE, 89(5):602–633, 2001. Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C010 Finals Page 185 24-9-2008 #2 10 Floorplan Representations Evangeline F.Y. Young CONTENTS 10.1 Introduction 185 10.2 Corner Block List 187 10.2.1 Corner Block 187 10.2.2 Definition of Corner Block List 188 10.2.3 Transformation between Corner Block List andFloorplan 189 10.2.4 FloorplanningAlgorithm 189 10.2.5 ExtendedCorner Block List Structure 189 10.3 Q-Sequence 191 10.3.1 ExtendedQ-Sequence 192 10.3.1.1 New Move Based on Parenthesis Constraint Tree 192 10.4 Twin BinaryTrees 194 10.5 Twin BinarySequences 196 10.6 Placement Constraints in Floorplan Design 199 10.7 Concluding Remarks 201 References 201 10.1 INTRODUCTION A floorplan representation is a data structure that captures the relative positions of the rooms in a dissection of a rectangular region. It differs from a packing representation, which captures the relative positions of the blocks to be packed into a rectangular region. In this chapter, we will study different rectangulardissectionrepresentations(with theexceptiono f theslicingrepresentation which was discussed in the last chapter). There are several graph-based representations for rectangular dissections in the early floorplanning literature, e.g., polar graph, neighborhood graphs, etc. In this chapter, we will only focus on recent representations for floorplans, which are mainly string-based. Unlike the slicing representation, these repr esentations can characterize any dissection o f a rectangle into rooms. We begin by formally defining a floorplan (which is also often referred to as a mosaic floorplan in the literature). A rectangular dissection is a floorplan if and only if it observes the following three properties: 1. Each room is assigned exactly one block. 2. The internal line segments (segs) of the dissection are only permitted to form T-junctions (Figure10.1). “+”-shaped junctions, where two distinct T-junctions meet at the same point, are considered to be degenerate. The representational power of floorplans is not impacted by this because the two T-junctions can be separated by sliding the noncrossing segment of one of the two T-junctions by a very small distance (Figure10.2). 185 Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C010 Finals Page 186 24-9-2008 #3 186 Handbook of Algorithms for Physical Design Automation B A C E F A C EF G D Mosaic floorplan with nonslicing structure Mosaic floorplan with slicing structure D B (a) (b) FIGURE 10.1 Examples of floorplans, showing (a) a mosaic floorplan with slicing structure and (b) a mosaic floorplan with nonslicing structure. 3. The topology is defined in terms of room–seg adjacency relationships rather than room– room adjacency relationships. The distinction between the two is described below. A floorplan can be defined in terms of relationships between adjacent rooms (i.e., rooms whose boundaries share a line segment) or in terms of adjacency relationships between a room and a seg (one of the boundaries of a room is the seg). In either case, it is necessary to specify the nature of the adjacency relationship ( e.g., room A is to the left of room B or room B is above segment s in Figure10.3). Two floorplans are equivalent with respect to the room–seg relation if and only if it is possible to label the rooms and the segments in such a way that the two sets of room–seg relations are identical. Similarly, two floorplans are equivalent with respect to the room–room relationship if and only if there is a labeling of the rooms such that the two sets of room–room relations are identical. Figure10.3 shows two floorplans that are identical with respect to the room–seg relationship, but Sliding slightly one T-junction horizontally Sliding slightly one T-junction vertically Two T-junctions meet at the same point Or FIGURE 10.2 Degenerated case modeled b y slightly moved T-junctions. A B C E D v w s t D B E C A v s w t FIGURE 10.3 Room–seg relation and room–room relation. Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C010 Finals Page 187 24-9-2008 #4 Floorplan Representations 187 not with respect to the room–room relationship. In this chapter, floorplans are defined with respect to the room–seg relation. 10.2 CORNER BLOCK LIST The corner block list (CBL) [1] was one of the first representations proposed for the class of mosaic floorplans. CBL is a topological representation and the topological relationship between two rooms as described by a CBL is independent of the modules contained in those rooms. We will see later in this section that the time complexity to transform a CBL to a placement is O(n) where n is the number of rooms. It just takes n(3 +lg n) bits to describe a packing in CBL and the size of the solution space is O(n!2 3n−3 ). 10.2.1 CORNER BLOCK In a mosaic floorplan F, the room * in the top-right cor ner is called the corner block. The orientation for a corner block B is defined according to the T-junction at the bottom-left corner of the room containing B. There are only two kinds of T-junctions, a T rotated by 90 ◦ anticlockwise or a T rotated by 180 ◦ . In the first case, B is said to be vertically oriented and is d enoted by a “0” bit. For the other case, B is said to be horizontally oriented and is denoted by a “1” bit. The two possible orientations of a corner block are illustrated in Figure 10.4. To obtain the CBL of a floorplan, or to construct the floorplan from a CBL, the concepts of deleting and inserting a corner block are needed. If the corner block B of a givenmosaic floorplan F is vertically oriented, B is deleted by sliding the bottom segment of B up along its left segment until it reaches the upper boundary of F. Similarly, if B is horizontally oriented, it isdeleted by sliding its leftsegmentalong its bottomsegmentuntil reaching the right boundary of F. An example to illustrate this deletion process is shown in Figure 10.5. It is CC Vertical Horizontal FIGURE 10.4 Different orientations of a corner block. C F G H D H FE C G FE C G Delete a vertically oriented corner block Delete a horizontally oriented corner block E ABAB AB FIGURE 10.5 Deletion of the corner block in a mosaic floorplan. * The term “block” is a phys ical entity with a width and a hei ght, whereas a “room” is a topological entity without specified dimensions. We use the t wo terms interchangeably in this section to be c onsistent with the original CBL paper. Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C010 Finals Page 188 24-9-2008 #5 188 Handbook of Algorithms for Physical Design Automation D B CA E C B A D A B C E D F Insert a horizontally oriented corner block (covering one T-junction) Insert a vertically oriented corner block (covering two T-junctions) FIGURE 10.6 Insertion of a corner block t o a mosaic f loorplan. not difficult to see that a mosaic floorplan will remain mosaic after deleting the corner block. The corner block insertion process is the reverse of the deletion process. To insert a new corner block B  to a mosaic floorplan F horizontally, a vertical segment (covering a certain number of 270 ◦ rotated T-junctions) at the right boundary of F is pushed to the left to create a room at the top-right corner of F for B  . Similarly, to insert a new corner block B  to a mosaic floorplan F vertically, a horizontal segment (covering a certain number of 0 ◦ rotated T-junctions) at the upper boundary of F is pushed downward to create a room at the top-right corner of F for B  . An example to illustrate this insertion process is shown in Figure 10.6. 10.2.2 DEFINITION OF CORNER BLOCK LIST The CBL of a mosaic floorplan F containing n blocks is a three-tuple (S, L, T ) where S = s 1 ···s n is a sequence of the block names, L = l 1 ···l n−1 is a bit string denoting the corner block orientations, and T = t 1 t 2 ···t n−1 is a bit string denoting some T-junction information. The CBL of F is constructed by recursively deleting corner blocks from F until only one block is left. We define a sequence of mosaic floorplans F i , i = 0, , n − 1, where F 0 = F and F i+1 is obtained from F i by deleting the corner block in F i . Then, s i is the c orner block of F n−i ; l i is a bit denoting the orientation of s i+1 (“0” [“1”] for a vertically [horizon tally] oriented block); and t i is a sequence o f k i “1”s followed by a “0,” where k i is the number of T-junctions covered by the bottom (left) segment of the vertically (horizontally) oriented corner block of s i+1 . Notice that F n−1 has only one block and the orientation of or the number of T-junctions covered by it is undefined, so the indices i of l i and t i only run from 1ton −1. An example of the CBL of a mosaic floorplan is shown in Figure10.7. It takes no more than n(3 +lg n) bits to represent a floorplan by a CBL where n ×lg n bits are used to record the block names in the list S, n − 1 bits are used to recor d the orientations in the list L and no more than 2n − 1 bits are used to record the T-junction information in the list T. D C B A S = (FEACDB) L = (10110) T = (01001010) F E FIGURE 10.7 Corner block list of a mosaic floorplan. Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C010 Finals Page 189 24-9-2008 #6 Floorplan Representations 189 Transformation from floorplan to CBL 1. While there is a corner block B, repeat 2. Delete B. 3. If B is not the last block, record (B, orientation, T-subsequence). 4. Add the last block to the block name list and concatenate all records in a reversed order to obtain the lists (S, L, T ). FIGURE 10.8 Transformation from floorplan to CBL. Transformation from CBL to floorplan 1. Initialize the floorplan with block S[1]. 2. For i = 2ton do 3. Add block S[i] to the floorplan with orientation L[i −1], covering a number of T-junctions according to list T. 4. If the number of T-junctions required to be covered is more than the number of T-junctions available, report error and exit. FIGURE 10.9 Transformation from CBL to floorplan. 10.2.3 TRANSFORMATION BETWEEN CORNER BLOCK LIST AND FLOORPLAN The linear-time transformations between CBL and floorplan are described in Figures 10.8 and 10.9. A drawback of CBL is that an arbitrary three-tuple (S, L, T ),whereS is a permutation of n block names, L is an n −1-bit string, and T is a bit string starting and ending with “0”s and having n −1 “0”s in total, may not correspond to a floorplan, because of the constraints on the composition of the list T. 10.2.4 FLOORPLANNING ALGORITHM This CBL representation can be used in search-based optimization technique like simulated annealing. Neighboring solutions can be generated by the following moves: 1. Randomly exchange two blocks in S 2. Randomly toggle a bit in L 3. Randomly toggle a bit in T 4. Randomly pick a block and rotate it by 90 ◦ , 180 ◦ , or 270 ◦ 5. Randomly pick a soft block and change its shape Notice that the second and the third move may result in an infeasible CBL, i.e., one that does not correspond to any floorplan. Therefore, checking and appropriate correction steps are n eeded. 10.2.5 EXTENDED CORNER BLOCK LIST STRUCTURE The extended CBL, ECBL λ , was proposed by Zhou et al [2] to represent general floorplans that may include empty rooms. Like the CBL, ECBL λ represents a rectangular dissection and assigns blocks Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C010 Finals Page 190 24-9-2008 #7 190 Handbook of Algorithms for Physical Design Automation Transformation from floorplan to ECBL 1. Insert a false block to each empty room. 2. While there is a corner block B, repeat 3. Delete B. 4. If B is not a false block, record (B, orientation, T-subsequence), else record (false block, orientation, T-subsequence). 5. Add the last block to the block name list and concatenate all records in a reversed order to obtain the lists (S, L, T). FIGURE 10.10 Transformation from floorplan to ECBL. Transformation from ECBL to floorplan 1. Initialize the floorplan with block S[1]. 2. For i = 2toλn do 3. If all real blocks are added already, output the floorplan and exit, 4. else 5. Add block S[i] to the floorplan with orientation L[i − 1] covering a number of T-junctions according to list T. 6. If the number of T-junctions required to be covered is more than the number of T-junctions available, report error and exit. FIGURE 10.11 Transformation from ECBL to floorplan. to rooms, but it contains more rooms than there are blocks, leaving some of the rooms empty. In block assignment, a false block of zero width and height is assigned to an empty room. An extended CBL, ECBL λ , is defined as follows: Definition 1 Givennblocks,anextended corner block list with an extending factor λ, denoted by ECBL λ , is a corner block list (S, L, T) of a floorplan with λn rooms, of which λn −n rooms are empty and occupied by false blocks and the remaining n rooms hold the n given blocks. The transformation algorithms between floorplan and ECBL are updated to account for the intro- duction of false blocks (Figures 10.10 and 10.11). Similar to the analysis of CBL, the complexities of these algorithms are both O(λn), and the number of combinations of ECBL λ is O(C n λn n!2 3λn−4 ). There is an additional factor of C n λn in the total number of combinations because there are C n λn ways to select n rooms from λn rooms to accommodate the n real blocks. The solution space of ECBL λ is guaranteed to contain the optimal solution when λ = n. It can be shown that the bounded sliceline grid (a packing representation discussed in the next chapter) BSG n×n can be represented by an ECBL n , i.e., λ is set to n. From the optimum solution theorem of bounded sliceline grids, there exists a BSG n×n -based packing corresponding to the optimal solution, so the solution space of ECBL n must also contain the optimal solution. However, setting λ to n will significantly increase the size of the solution space and the complexities of the transformation algorithms from linear to quadratic. Fortunately, it has been shown experimentally that fairly good results can be obtained b y setting λ to a real number constant in the range [1.5,3], which agrees with a fact proven later that (n − 2 √ n) empty rooms are enough to generate any packing. Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C010 Finals Page 191 24-9-2008 #8 Floorplan Representations 191 10.3 Q-SEQUENCE A new data structure called the Quarter-state sequence (abbreviated as Q-sequence) was proposed by Sakanushi et al. [3,4] to represent a floorplan. A Q-sequence is a string of room labels and two positional symbols with a total length of 3n where n is the number of rooms. Both encoding and decoding of the Q-sequence representation can be done in linear time. To construct the Q-sequence of a floorplan, the following terms are defined. A room is called the tail room if it lies at the bottom-right corner of the floorplan. A room r that is not a tail room has a bottom-right corner, which is either a 180 ◦ or a 270 ◦ rotated T-junction. I n either case, the noncrossing segment of the T-junction is called the prime seg of r. If the prime seg l of a room r is vertical (horizontal), the rooms that touch l from the right (below) are called the associated rooms of r, and the topmost (leftmost) associated room is called the next room of r. The Q-state o f room r is a string starting with the room label r followed by n r “R”s (“B”s) if the prime seg of r is vertical (horizontal), where n r is the number of associated rooms o f r. The subQ-sequence Y is constructed by concatenating the Q-states of all the rooms in the order o f r 1 , r 2 , , r n ,wherer 1 is the room at the top-left corner of the packing, and r i+1 is the next room of r i ,fori = 1, , n − 1. We define string X as consisting of p “R”s with q “B”s, where p and q are the numbers of rooms touching the left-wall and the top-wall of the whole floorplan. The final Q-sequence is obtained by concatenating X with the subQ-sequence Y. An example is shown in Figure10.12. Henceforth, we assume that the rooms are always labeled from 1 to n in the Q-sequence (i.e., r i = i). Given a Q-sequence Q, we can obtain an RQ-sequence by deleting all the “B”s and replacing every “R” by an open p arenthesis and every room label by a close parenthesis. We can construct a BQ-sequence similarly by interchanging the roles of “B” and “R”. There are two necessary and sufficient properties: 1. Single: The subsequence of Q between any two rooms contains a string with at least one “R” or at least one “B.” 2. Parenthesis: The RQ-sequence and BQ-sequence of Q are well formed. It can be shown that the number F(n) of distinct Q-sequences for n blocks is upper bounded by 2 (3n−1) ×n!. Given a Q-sequence, vertical and horizontal constraint graphs can be constructed directly from the sequence and floorplan realization can be done in linear time. A decoding algorithm will be given in the next section. Besides, boundary constraint can also be handled efficiently by using the Q-sequence representation. 6 2 1 3 54 RRBB1RR2BB3BB4R5R6 FIGURE 10.12 Example of the Q-sequence representation. . assigns blocks Alpert /Handbook of Algorithms for Physical Design Automation AU7242_C010 Finals Page 190 24-9-2008 #7 190 Handbook of Algorithms for Physical Design Automation Transformation from floorplan. (Figure10.2). 185 Alpert /Handbook of Algorithms for Physical Design Automation AU7242_C010 Finals Page 186 24-9-2008 #3 186 Handbook of Algorithms for Physical Design Automation B A C E F A C EF G D Mosaic. original CBL paper. Alpert /Handbook of Algorithms for Physical Design Automation AU7242_C010 Finals Page 188 24-9-2008 #5 188 Handbook of Algorithms for Physical Design Automation D B CA E C B A D A B C E D F Insert

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