Here is a matlab script to find the eigenvalues and

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Here is a MatLab script to find the eigenvalues and eigenvectors of B :
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FIGURE 16.25 Eigenvalue placement. (a) An example network. (b) The one-dimensional placement.The small black squares represent the centers of the logic cells. (c) The two-dimensional placement. The eigenvalue method takes no account of the logic cell sizes or actual location of logic cell connectors. (d) A complete layout. We snap the logic cells to valid locations, leaving room for the routing in the channel. C=[0 0 0 1; 0 0 1 1; 0 1 0 0; 1 1 0 0] D=[1 0 0 0; 0 2 0 0; 0 0 1 0; 0 0 0 2] B=D-C [X,D] = eig(B) Running this script, we find the eigenvalues of B are 0.5858, 0.0, 2.0, and 3.4142. The corresponding eigenvectors of B are 0.6533 0.5000 0.5000 –0.2706 –0.2706 0.5000 –0.5000 –0.6533 –0.6533 0.5000 0.5000 0.2706 0.2706 0.5000 –0.5000 0.6533
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(16.20) For a one-dimensional placement ( Figure 16.25 b), we use the eigenvector (0.6533, –0.2706, – 0.6533, –0.2706) corresponding to the smallest nonzero eigenvalue (which is 0.5858) to place the logic cells along the x -axis. The two-dimensional placement ( Figure 16.25 c) uses these same values for the x -coordinates and the eigenvector (0.5, –0.5, 0.5, –0.5) that corresponds to the next largest eigenvalue (which is 2.0) for the y -coordinates. Notice that the placement shown in Figure 16.25 (c), which shows logic-cell outlines (the logic-cell abutment boxes), takes no account of the cell sizes, and cells may even overlap at this stage. This is because, in Eq. 16.11 , we discarded all but one of the constraints necessary to ensure valid solutions. Often we use the approximate eigenvalue solution as an initial placement for one of the iterative improvement algorithms that we shall discuss in Section 16.2.6 . 16.2.6 Iterative Placement Improvement An iterative placement improvement algorithm takes an existing placement and tries to improve it by moving the logic cells. There are two parts to the algorithm: The selection criteria that decides which logic cells to try moving. The measurement criteria that decides whether to move the selected cells. There are several interchange or iterative exchange methods that differ in their selection and measurement criteria: pairwise interchange, force-directed interchange, force-directed relaxation, and force-directed pairwise relaxation. All of these methods usually consider only pairs of logic cells to be exchanged. A source logic cell is picked for trial exchange with a destination logic cell. We have already discussed the use of interchange methods applied to the system partitioning step. The most widely used methods use group migration, especially the Kernighan–Lin algorithm. The pairwise-interchange algorithm is similar to the interchange algorithm used for iterative improvement in the system partitioning step: 1. Select the source logic cell at random. 2. Try all the other logic cells in turn as the destination logic cell.
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3. Use any of the measurement methods we have discussed to decide on whether to accept the interchange.
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  • Fall '15
  • prasad
  • Gate, Clock signal, Logic gate, Electronic design automation, Application-specific integrated circuit, logic cells

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