TAILIEUCHUNG - Handbook of algorithms for physical design automation part 41

Handbook of Algorithms for Physical Design Automation part 41 provides a detailed overview of VLSI physical design automation, emphasizing state-of-the-art techniques, trends and improvements that have emerged during the previous decade. After a brief introduction to the modern physical design problem, basic algorithmic techniques, and partitioning, the book discusses significant advances in floorplanning representations and describes recent formulations of the floorplanning problem. The text also addresses issues of placement, net layout and optimization, routing multiple signal nets, manufacturability, physical synthesis, special nets, and designing for specialized technologies. It includes a personal perspective from Ralph Otten as he looks back on. | 382 Handbook of Algorithms for Physical Design Automation Finally consider minimization of q s in Equation at the coarser level under these assumptions 1 and 2 . min q ec q s Pec 1 5 Pec TQ s Pec - bT 5 Pec It is easily shown that ec minimizes q ec if and only if it satisfies P QPec PTr Galerkin coarsening thus defines the linear coarsening operator PT R R as the transpose of the linear interpolation operator P as simple substitution of Equation into Equation yields Equation on premultiplication by PT. By confining coarse-level iterative improvement to the perturbation e Pec to s relaxation at the coarse-level is in effect decoupled from relaxation at the adjacent finer level. The combined approximation s e is thus not restricted to the range of P and can represent a broader set of solution candidates for fixed P than is possible if relaxation at the coarser level and subsequent interpolation are applied to the entire coarse-level solution sc rather than just its perturbation ec. In both theory and practice coarsening and interpolation are so closely related that defining one essentially characterizes the other even if nonlinear or discrete methods are used. Simple Examples of Interpolation To give the simplified model concreteness for placement consider the sample netlist H illustrated in Figure . The coarsening of H on the right-hand side of the figure shows modules 1 2 and 3 mapped to cluster 0 module 4 mapped to cluster 1 and modules 0 and 5 mapped to cluster 2. For the moment we ignore the algorithm used to define the coarsening and concentrate just on the question of how to place the modules once a placement of their parent clusters has been computed. Because in this example there are six modules at the finer level and three modules at the adjacent coarse level linear interpolation operators for this example are represented as constant 6 x 3 matrices. Let Pconst denote the matrix for piecewise-constant linear .

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