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 fromCadence 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 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.

5. 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

6. 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

7. 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

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 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)

10. 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]

11. 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)

12. Prefix Algorithm cost pcost[i-1] cost[i] pcost[i] x

13. 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

14. 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

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