Optimization of Linear Placements for Wirelength Minimization with Free Sites

1
Optimization of Linear Placements for Wirelength
Minimization with Free Sites

• A. B. Kahng, P. Tucker, A. Zelikovsky
• (UCLA UCSD)
• Supported by grants from Cadence Design
  Systems, Inc.
• http//vlsicad.cs.ucla.edu

2
Outline

• Single-Row Problem
• Cell Cost Function
• Exact Algorithms for Single-Row Problem
• Dynamic Programming Algorithm
• Prefix Algorithm
• Clumping Algorithm
• Swapping Heuristic for Cell Ordering
• Experimental Results
• Conclusions and Future Directions

3
Single-Row Problem
fixed cells
movable cells
C1
C2
C3
C4
C5
C6
C7
fixed cells
4
Single-Row Problem

• Given
• single cell row with n movable cells Ci with
  fixed left-to-right order (but variable
  positions) and integer lattice of k sites (k gt n)
  
• m signal nets N j containing fixed cells from
  other rows
• Find
• non-overlapping placement of n movable cells at k
  sites minimizing the total bounding-box
  half-perimeter of all m nets.

5
Net with Movable and Fixed Cells
fixed cells
net N
fl(N)
fr(N)
single row with movable cells
ml(N)
mr(N)
span (N)
fixed_span (N)
minimize
6
Cell Cost Function

• Cell cost function of Ci sum over all nets N
  of contributions of Ci to span(N) -
  fixed_span(N)
• Given position x of cell Ci, cell cost function
  
• costi(x) ? maxmr(N) -
  fr(N),0
• Ci rightmost movable
  on net N
• ? maxfl(N) -
  ml(N),0
• Ci leftmost movable on
  net N
• Total linear pieces ? 2 ? pins 2 ? nets
  2m

7
Properties of Cell Cost Function

• Cost function of multi-pin cell is
  piecewise-linear and convex

• If each cell is placed in its minimum segment,
• total bounding box half-perimeter is
  minimized

8
Exact Algorithms for Single-Row Problem

• Dynamic Programming Algorithm
• based on pre-computed cell cost functions
• Prefix Algorithm
• based on piecewise-linearity of cell cost
  function
• Clumping Algorithm
• based on convexity of cell cost function

9
Dynamic Programming Algorithm

• Optimum constrained prefix placement Pi,j of
  C1, ..., Ci subject to Ci being left of
  site sj
• Pi,j is selected from Pi,j-1 and
• Pi-1,j-wi-1 extended by Ci at sj
• wi-1 width of Ci-1
• Cost of prefix placement increased by
  costi(sj)
• Runtime (i-range) ?(j-range)
• n ? (k - ? wi)
• ? O(n2)

10
Dynamic Programming Algorithm
Pi,j has either Ci exactly at sj
(extend Pi-1,j-wi-1)
Ci-1
Ci
sj
sj-wi-1
or Ci to left of sj (use already-computed
Pi,j-1)
Ci
sj-1
11
Prefix Algorithm

• Prefix cost function pcosti(x) optimal
  placement cost of first i cells subject to Ci
  being left of x
• pcosti(x) is piecewise-linear decreasing
• Each linear segment is tuple a,b, min,max
• Computing pcosti from pcosti-1 and costi
• ? merging sorted tuple sequences of sizes
• ?jlti pinj and pini (pini pins
  on Ci)
• Runtime O(m2)
• Note error in proceedings (missing costi
  term)

12
Prefix Algorithm
cost
pcosti-1
costi
pcosti
x
13
Clumping Algorithm

• For each cell Ci, find
• list of coordinates where costi changes slope
• Cis minimum segment
• To each cell in order, apply PLACE(Ci)
• Output positions of cells
• Procedure PLACE(Ci)
• if Ci-1 and Ci cannot be both in their
  minimum segments
• then COLLAPSE(Ci-1,Ci) and PLACE(Ci-1)
• else place Ci at leftmost optimal available
  position

14
Clumping Algorithm

• Procedure COLLAPSE(Ci-1,Ci)
• shift positions from the list of Ci by
  width(Ci-1)
• merge the list for Ci with the list for Ci-1
• find minimum segment for merged list
• width(Ci-1) width(Ci-1) width(Ci)
• delete cell Ci
• Using red-black trees for representation of cell
  lists, achieve runtime O(m log m), m nets

15
Clumping Algorithm
directions to minimum segments of individual cells
clumped cell
clumped cell
optimal positions for cells
16
Swapping Heuristic for Cell Ordering

• Cell-Ordering Problem the Single-Row Problem
  where the left-to-right order of cells is not
  fixed
• Swapping Heuristic
• Repeatedly iterate down the row until no pairs
  swap
• for every adjacent pair of cells that overlap or
  change order when placed at respective min
  points, swap their order if placement cost
  improves

17
Experimental Results
18
Conclusions

• First optimal algorithms for single-row cell
  placement with free sites, fixed order of cells,
  and fixed positions of cells in all other rows
• New iterative algorithm to improve the cell
  ordering within a given row
• Iterative row-based placement algorithm that
  applies single-row cell placement to each row in
  turn, with optional cell ordering improvement in
  the given row
• Average of 6.5 improvement in total wirelength

19
Extensions

• Incorporate cell flipping into DP solution
• Linear programming formulation for Cell Ordering
  Problem
• Extend exact DP solution to k rows simultaneously
• Incorporate routability into objective function