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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


Experimental results

Experimental Results


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


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