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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 from Cadence Design Systems, Inc. http://vlsicad.cs.ucla.edu. Outline. Single-Row Problem Cell Cost Function

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optimization of linear placements for wirelength minimization with free sites

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
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
single row problem
Single-Row Problem

fixed cells

movable cells

C1

C2

C3

C4

C5

C6

C7

fixed cells

single row problem1
Single-Row Problem
  • 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
Net with Movable and Fixed Cells

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
Cell Cost Function
  • 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
properties of cell cost function

fr(1)

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
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
dynamic programming algorithm
Dynamic Programming Algorithm
  • 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 algorithm1
Dynamic Programming Algorithm

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
Prefix Algorithm
  • 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)
prefix algorithm1
Prefix Algorithm

cost

pcost[i-1]

cost[i]

pcost[i]

x

clumping algorithm
Clumping Algorithm
  • 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 algorithm1
Clumping Algorithm
  • 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 algorithm2
Clumping Algorithm

directions to minimum segments of individual cells

clumped

cell

clumped cell

optimal positions for cells

swapping heuristic for cell ordering
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
conclusions
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
extensions
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