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.

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 fixed cells movable cells C1C2C3C4C5C6C7

Single-Row Problem Given –single cell row with n movable 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 single row with movable cells fixed cells net N span (N) fl(N) fr(N) ml(N) mr(N) fixed_span (N) minimize

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 Cost function of multi-pin cell is piecewise-linear and convex fr(1)fl(2)fl(3)fr(3)fr(2)fl(4)fr(4) minimum segment (point) If each cell is placed in its minimum segment, total bounding box half-perimeter is minimized

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 Optimum constrained prefix placement P[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(n 2 )

Dynamic Programming Algorithm C[i-1] C[i] s[j] s[j-w[i-1]] C[i] s[j-1] P[i,j] has either: C[i] exactly at s[j] (extend P[i-1,j-w[i-1]]) or C[i] to left of s[j] (use already-computed P[i,j-1])

Prefix Algorithm Prefix cost function pcost[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<i pin[j] and pin[i] (pin[i] = #pins on C[i]) Runtime = O(m 2 ) Note: error in proceedings (missing +cost[i] term)

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

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 Procedure PLACE(C[i]) if C[i-1] and C[i] cannot be both in their minimum segments then COLLAPSE(C[i-1],C[i]) and PLACE(C[i-1]) else place C[i] at leftmost optimal available position

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 Algorithm clumped cell clumped cell optimal positions for cells directions to minimum segments of individual cells

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

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 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