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VLSI Placement

VLSI Placement. Prof. Shiyan Hu shiyan@mtu.edu Office: EERC 731. Problem formulation. Input: Blocks (standard cells and macros) B 1 , ... , B n Shapes and Pin Positions for each block B i Nets N 1 , ... , N m Output: Coordinates (x i , y i ) for block B i .

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VLSI Placement

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  1. VLSI Placement Prof. Shiyan Hu shiyan@mtu.edu Office: EERC 731

  2. Problem formulation • Input: • Blocks (standard cells and macros) B1, ... , Bn • Shapes and Pin Positions for each block Bi • Nets N1, ... , Nm • Output: • Coordinates (xi , yi ) for block Bi. • The total wire length is minimized. • Subject to area constraint or the area of the resulting block is minimized

  3. Placement can Make A Difference Random Initial Placement Final Placement

  4. Objective: Partitioning: Given a set of interconnected blocks, produce two sets that are of equal size, and such that the number of nets connecting the two sets is minimized.

  5. FM Partitioning: Initial Random Placement list_of_sets = entire_chip; while(any_set_has_2_or_more_objects(list_of_sets)) { for_each_set_in(list_of_sets) { partition_it(); } /* each time through this loop the number of */ /* sets in the list doubles. */ } After Cut 1 After Cut 2

  6. Moves are made based on object gain. Object Gain: The amount of change in cut crossings that will occur if an object is moved from its current partition into the other partition FM Partitioning: -1 2 0 - each object is assigned a gain - objects are put into a sorted gain list - the object with the highest gain is selected and moved. - the moved object is "locked" - gains of "touched" objects are recomputed - gain lists are resorted 0 -1 0 -2 0 0 -2 -1 1 -1 1

  7. FM Partitioning: -1 2 0 0 -1 0 -2 0 0 -2 -1 1 -1 1

  8. -1 -2 -2 0 -1 -2 -2 0 0 -2 -1 1 -1 1

  9. -1 -2 -2 0 -1 -2 -2 0 0 -2 -1 1 1 -1

  10. -1 -2 -2 0 -1 -2 -2 0 0 -2 -1 1 1 -1

  11. -1 -2 -2 0 -1 -2 -2 0 -2 -2 1 -1 -1 -1

  12. -1 -2 -2 -1 -2 0 -2 0 -2 -2 1 -1 -1 -1

  13. -1 -2 -2 -1 -2 -2 0 0 -2 -2 1 -1 -1 -1

  14. -1 -2 -2 1 -2 -2 0 -2 -2 -2 1 -1 -1 -1

  15. -1 -2 -2 1 -2 -2 0 -2 -2 -2 1 -1 -1 -1

  16. -1 -2 -2 1 -2 -2 0 -2 -2 1 -2 -1 -1 -1

  17. -1 -2 -2 1 -2 -2 0 -1 -2 -2 -2 -3 -1 -1

  18. -1 -2 -2 1 -2 -2 0 -1 -2 -2 -2 -3 -1 -1

  19. -1 -2 -2 1 -2 -2 0 -1 -2 -2 -2 -3 -1 -1

  20. -1 -2 -2 -1 -2 -2 -2 -1 -2 -2 -2 -3 -1 -1

  21. Analytical Placement • Write down the placement problem as an analytical mathematical problem • Quadratic placement: • Sum of squared wire length is quadratic in the cell coordinates. • So the wirelength minimization problem can be formulated as a quadratic program. • It can be proved that the quadratic program is convex, hence polynomial time solvable

  22. Example: x=100 x=200 x1 x2

  23. Example: x=100 x=200 x1 x2 Interpretation of matrices A and B: The diagonal values A[i,i] correspond to the number of connections to xi The off diagonal values A[i,j] are -1 if object i is connected to object j, 0 otherwise The values B[i] correspond to the sum of the locations of fixed objects connected to object i

  24. Quadratic Placement • Global optimization: solves a sequence of quadratic programming problems • Partitioning: enforces the non-overlap constraints

  25. Solution of the Original QP

  26. Partitioning • Use FM to cut.

  27. Applying the Idea Recursively • Perform the Global Optimization again with additional constraints that the center of gravities should be in the center of regions. Center of Gravities

  28. Process of Gordian (a) Global placement with 1 region (b) Global placement with 4 region (c) Final placements

  29. Pros: - mathematically well behaved - efficient solution techniques Cons: - solution of Ax + B = 0 is not a legal placement, so generally some additional partitioning techniques are required. - solution of Ax + B = 0 is minimizes wirelength squared, not linear wire length. Quadratic Techniques:

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