Closing the Smoothness and Uniformity Gap in Area Fill Synthesis Supported by Cadence Design Systems, Inc., NSF, the Packard Foundation, and State of Georgia’s Yamacraw Initiative Closing the Smoothness and Uniformity Gap in Area Fill Synthesis Y. Chen, A. B. Kahng, G. Robins, A. Zelikovsky (UCLA, UCSD, UVA and GSU) http://vlsicad.ucsd.edu Thank you .… Today, I will present our work on “Closing the Smoothness and Uniformity Gap in Area Fill Synthesis". (20 sec)

fixed dissection window The Smoothness Gap Large gap between fixed-dissection and floating window density analysis Gap! fixed dissection window with maximum density floating window with maximum density The motivation of our work is a “smoothness gap” in the fill literature. The very first paper on fill has pointed out that there is potentially large difference between fixed-dissection and floating window density analysis. For example, this is the fixed-dissection window with the maximum density on the given layout. However, by shifting the window up-right, we can find a floating window with a larger density. Clearly, these two features contribute to the smoothness gap. The consequence of the gap is that the fill result will not satisfy the given bounds. However, despite this gap observation in 1998, all existing filling methods fail to consider the potential smoothness gap (50 sec) Fill result will not satisfy the given bounds Despite this gap observation in1998, all existing filling methods fail to consider the potential smoothness gap

Accurate Layout Density Analysis Optimal extremal-density analysis with complexity intractable Multi-level density analysis algorithm shrunk on-grid windows are contained by arbitrary windows any arbitrary window is contained by a bloated on-grid window gap between bloated window and on-grid window accuracy fixed dissection window Prof. Kahng has proposed optimal extremal-density analysis algorithms. However, the algorithm with complexity K square becomes intractable for layout with a large number of fill features. Another method overcoming the intractability is the multi-level density analysis algorithm. It is based on the following facts: Shrunk on-grid windows are contained by arbitrary windows And any arbitrary window is contained by bloated on-grid windows We use the gap between the bloated window and on-grid window as the accuracy of the algorithm (45 sec) arbitrary window W shrunk fixed dissection window bloated fixed dissection window tile

Local Density Variation Type I: max density variation of every r neighboring windows in each row of the fixed-dissection Type II: max density variation of every cluster of windows which cover one tile Type III: max density variation of every cluster of windows which cover tiles We also suggest new methods to measure local uniformity of the layout based on the Lipschitz conditions. We proposed three kind of Local density variations. (click) Type One: the maximum density variation of every r neighboring windows in each row of the fixed-dissection (click) Type Two: the maximum density variation of every cluster of windows which cover one tile (click) Type Three: the maximum density variation of every cluster of windows which cover r/2 x r/2 tiles. (45 sec)

Computational Experience Testcase  LP Greedy MC IGreedy IMC Testcase OrgDen FD Multi-Level T/W/r MaxD MinD DenV Denv L1/16/4 .2572 .0516 .0639 .2653 .0855 .0621 .2706 .0783 .2679 .0756 .084 .0727 L1/16/16 .2643 .0417 .0896 .0915 .0705 .2696 .0773 .2676 .0758 .0755 .0753 The window density variation and violation of the maximum window density in fixed-dissection filling are underestimated case Min-Var LP LipI LP LipII LP Comb LP T/W/r Den V Lip1 Lip2 Lip3 L1/16/4 .0855 .0832 .0837 .0713 .1725 .0553 .167 .1268 .1265 .0649 .0663 .0434 .1143 .0574 .0619 .0409 L1/16/8 .0814 .0734 .0777 .067 .1972 .0938 .1932 .1428 .1702 .1016 .1027 .0756 .1707 .0937 .1005 .0766 L2/28/4 .1012 .0414 .0989 .0841 .0724 .0251 .072 .0693 .0888 .0467 .0871 .0836 .0825 .0242 .0809 .0758 L2/28/8 .0666 .034 .0658 .0654 .0264 .0744 .07 .0331 .0697 .0661 .0747 .0255 .0708 .0656 For the smoothness gap, our experiments show that the window density variation and violation of the maximum window density in fixed-dissection filling are underestimated For the local density variations, our experiments show that the solution with the Min-Var objective value do not always have the best Value in terms of the local smoothness, and the LP with combined objective achieves the best comprehensive solutions. (60 sec) The solutions with the best Min-Var objective value do not always have the best value in terms of local smoothness LP with combined objective achieves the best comprehensive solutions

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