Placement Problem

Let Gi (V i,, Wi) and G2 (V 2 - 2 1 2) be two complete edge-weighted graphs with weight functions Wi and W2. A bij ective function ip V2- Viis called a placement of G2 on Gi. The cost of placement is defined as 

The optimal placement problem is to find a placement of G2 on Gi of minimal cost. 

For optimal mask design problem the Nx N vertices of the grid graph Gi correspond to a NxiV array. The distance between grid cells and(z2, 2)inthe 

Manhattan metric is defined as i2 - ii + j2 ji I The vertices of the grid are neighbors if the distance between them is 1. The weight Wi (x,y) is defined as 1 if x and y are neighbors and 0 otherwise. The vertices of the probe graph G2 correspond to NxN probes to be synthesized on a chip. The weight function W2(p q) is de- 

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Recirculation (Recycle)—Hot exhaust air forced downward and back into the cooling tower raises the wet-bulb temperature of the entering air above dry-bulb temperature, impairing tower performance. It is usually caused by design, wind or placement problems. 

The primary interest in the pole placement literature recently has been in finding an analytical solution for the feedback matrix so that the closed loop system has a set of prescribed eigenvalues. In this context pole placement is often regarded as a simpler alternative than optimal control or frequency response methods. For a single control (r=l), the pole placement problem yields an analytical solution for full state feedback (e.g., (38), (39)). The more difficult case of output feedback pole placement for MIMO systems has not yet been fully solved(40). 

The objective of the 2.5-D placement problem is to map a cell netlist (pure standard cell or mixed macro/standard cell) to unique positions in a layered space as illustrated in Fig. 6.1. The inter-chip contacts are assumed to be placed on top of the chip with no need to consume substrate area. We need to differentiate two scenarios hierarchical and flattened design styles. In a hierarchical design set up, after the floorplanning step, cells in a block need to be placed. As mentioned in the last chapter, a random-logic based block could be split into two chips. The 2.5-D placement problem is to assign the cells within such a block to unique positions on two chips. On the other hand, in a flattened design style, the 2.5-D placement problem is to place both standard cell macros onto stacked chips. 

In the following sections of this chapter, we studied the 2.5-D placement problem under the above mentioned three formulations pure standard cell designs with inter-chip contacts consuming substrate area, pure standard cell designs with inter-chip contacts on top of die surface, and mixed standard cell and macro designs corresponding to a flattened design style. 

D stack. We define such a space as a super-block. For the 2.5-D placement problem, a recursive partitioning procedure is carried out on the super-blocks. The process can be explained using the cube model illustrated in Fig. 6.2. For a... 

Compared with pure standard cell placement problem, the difficulty of handling mixed macro/standard cell is how to efficiently remove overlap among cells and macros. Macros are large physical objects with arbitrary shape, and one macro usually interconnects with many small cells. As a result, moving one macro may change the solution structure of many small cells. On the other hand, small standard cells are more flexible and can accommodate the arbitrarily shaped space unoccupied by macros. Consequently, macros and standard cells should be placed concurrently in an interleaved manner. 

We show that a linear-wire-delay model is sufficient to model the impact of buffering for the latch placement problem. 

Table 3.6 RUMBLE deployed in a physical design flow on circuits that have pipeline latch placement problems, cktl has 2.92M objects and 629k latches and ckt2 has 4.74M objects and 247k latches, old reports values before RUMBLE new reports results after and diff reports their difference. FOM is reported in nanoseconds...

From these eausal paths, the components involved in the residuals r and ra are obtained as Ki = [Qp, Ti, Vt, -Pi, P2] and K2 = [Vb, P2, Vo, Pi, P2], respectively. The fault signatures obtained from causal path analysis are identical to the ones obtained before (Table 7.3). Thus, sensor placement problem is reduced to a... 

Gluck et al. (1996) adapted optimal control theory (OCT) to the damper placement problem. OCT is used to minimise the performance objective by optimising the location of linear passive devices. Since passive dampers cannot provide feedback in terms of optimal control gains, three approaches (response spectrum approach, single mode approach, and truncation approach) are proposed to remove the off-diagonal state interactions within the gain matrix and allow approximation of floor damping coefficients. Combination of these methods with OCT and passive devices achieves an equivalent effect compared to active control. 

The solution of the simultaneous decoupling and pole placement problem using global optimization... 

The simultaneous decoupling and pole placement problem is one application where the use of global numerical optimization is needed. The problem as such requires the determination of all solutions of a system of nonlinear equations and may not always have a solution for a decouplable and controllable system without cancelUng invariant zeros. It is of particular interest to develop an approach for the problem without the cancellation of unstable invariant zeros. 

Oscar H. Sendin, Carmen G. Moles, Antonio A. Alonso, Julio R. Banga D5 The solution of the simultaneous decoupling and pole placement problem using 582... 

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