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Optimization of Linear Placements for Wirelength Minimization with Free SitesPowerPoint Presentation

Optimization of Linear Placements for Wirelength Minimization with Free Sites

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### Optimization of Linear Placements for Wirelength Minimization with Free Sites

A. B. Kahng, P. Tucker, A. Zelikovsky

(UCLA & UCSD)

Supported by grants fromCadence Design Systems, Inc.

http://vlsicad.cs.ucla.edu

Outline Minimization with Free Sites

- 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

Single-Row Problem Minimization with Free Sites

fixed cells

movable cells

C1

C2

C3

C4

C5

C6

C7

fixed cells

Single-Row Problem Minimization with Free Sites

- Given
- single cell row with nmovable cells C[i] with fixed left-to-right order (but variable positions) and integer lattice of k sites (k > 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.

Net with Movable and Fixed Cells Minimization with Free Sites

fixed cells

fl(N)

net N

fr(N)

single row with

movable cells

ml(N)

mr(N)

span (N)

fixed_span (N)

minimize

Cell Cost Function Minimization with Free Sites

- Cell cost function of C[i] = sum over all nets N of contributions of C[i] to span(N) - fixed_span(N)
- Given position x of cell C[i], cell cost function =
cost[i](x) = max{mr(N) - fr(N),0}

C[i] = rightmost movable on net N

+ max{fl(N) - ml(N),0}

C[i] = leftmost movable on net N

- Total # linear pieces 2 #pins = 2 #nets = 2m

fr(1) Minimization with Free Sites

fl(2)

fl(3)

fr(3)

fr(2)

fl(4)

fr(4)

minimum segment (point)

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

Exact Algorithms for Single-Row Problem Minimization with Free Sites

- 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

Dynamic Programming Algorithm Minimization with Free Sites

- Optimum constrained prefix placementP[i,j] of C[1], ..., C[i] subject to C[i] being left of site s[j]
- P[i,j] is selected from P[i,j-1] and
P[i-1,j-w[i-1]]extended by C[i] at s[j]

w[i-1] = width of C[i-1]

- Cost of prefix placement increased by cost[i](s[j])
- Runtime = (i-range) (j-range)
= n (k - w[i])

O(n2)

Dynamic Programming Algorithm Minimization with Free Sites

P[i,j] has either:

C[i] exactly at s[j] (extend P[i-1,j-w[i-1]])

C[i-1]

C[i]

s[j]

s[j-w[i-1]]

orC[i] to left of s[j] (use already-computed P[i,j-1])

C[i]

s[j-1]

Prefix Algorithm Minimization with Free Sites

- Prefix cost functionpcost[i](x) = optimal placement cost of first i cells subject to C[i] being left of x
- pcost[i](x) is piecewise-linear decreasing
- Each linear segment is tuple = [a,b, min,max]
- Computing pcost[i] from pcost[i-1] and cost[i]
merging sorted tuple sequences of sizes

j<ipin[j] and pin[i] (pin[i] = #pins on C[i])

- Runtime = O(m2)
- Note: error in proceedings (missing +cost[i] term)

Clumping Algorithm Minimization with Free Sites

- For each cell C[i], find
- list of coordinates where cost[i] changes slope
- C[i]’s minimum segment

- To each cell in order, apply PLACE(C[i])
- Output positions of cells
- ProcedurePLACE(C[i])
if C[i-1] and C[i] cannot be both in their minimum segments

thenCOLLAPSE(C[i-1],C[i]) and PLACE(C[i-1])

else place C[i] at leftmost optimal available position

Clumping Algorithm Minimization with Free Sites

- Procedure COLLAPSE(C[i-1],C[i])
- shift positions from the list of C[i] by width(C[i-1])
- merge the list for C[i] with the list for C[i-1]
- find minimum segment for merged list
- width(C[i-1]) = width(C[i-1]) + width(C[i])
- delete cell C[i]

- Using red-black trees for representation of cell lists, achieve runtime = O(m log m), m = # nets

Clumping Algorithm Minimization with Free Sites

directions to minimum segments of individual cells

clumped

cell

clumped cell

optimal positions for cells

Swapping Heuristic for Cell Ordering Minimization with Free Sites

- 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

Experimental Results Minimization with Free Sites

Conclusions Minimization with Free Sites

- 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

Extensions Minimization with Free Sites

- 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

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