Toward Better Wireload Models in the Presence of Obstacles*

Toward Better Wireload Models in the Presence of Obstacles* Chung-Kuan Cheng, Andrew B. Kahng, Bao Liu and Dirk Stroobandt† UC San Diego CSE Dept. †Ghent University ELIS Dept. e-mail: {kuan,abk,bliu}@cs.ucsd.edu, dstr@elis.rug.ac.be *This work was supported in part by the MARCO Gigascale Silicon Research Center and a grant from Cadence Design Systems, Inc.

Presentation Outline • Motivation and Background • Wirelengths and Obstacles • Two-terminal Nets with a Single Obstacle • Two-terminal Nets with Multiple Obstacles • Model Applications • Conclusion

Motivation and Background • Impasse of interconnect delay and placement • To break impasse: use wireload models • Wireload models benefit from wirelength estimation techniques • IP blocks in SOC design form routing obstacles • Wirelength estimation cannot be blind to routing obstacles

A Priori or Online Wirelength Estimation Synthesis A Priori WLE Placement Online WLE Global Routing Detailed Routing OK? done

Presentation Outline • Motivation and Background • Wire Lengths and Obstacles • Two-terminal Nets with a Single Obstacle • Two-terminal Nets with Multiple Obstacles • Model Applications • Conclusion

Problem Formulation • Given obstacles and n terminals uniformly distributed in a rectangular routing region that lie outside the obstacles • Find the expected rectilinear Steiner minimal length of the n-terminal net M N

Effects of Routing Obstacles on Expected Wirelength • Detours that have to be made around the obstacles • Changes due to redistribution of interconnect terminals M N

Definitions of Wirelength Components • Intrinsic wirelength Li is average expected wirelength without any obstacle • Point redistribution wirelength Lp is average expected wirelength with transparent obstacles • Resultant wirelength Lr is average expected wirelength with opaque obstacles

W P1 P1’ P2 H P2’ Intrinsic Li Resultant Lr Point Redistribution Lp Wirelength Components M N

Summary of Wirelength Components • Redistribution effect equals Lp-Li (in the presence of transparent obstacles) • Blockage effect equals Lr-Lp (in the presence of opaque obstacles) Resultant Lr Blockage effect Redistribution Lp Redistribution effect Intrinsic Li

M N Intrinsic Wirelength of Two-terminal Nets • Average expected wirelength between two terminals is one third of the half perimeter of the layout region without obstacles

Point Redistribution WL of Two-terminal Nets W where a=f(M,N,a,b) • Observation 1: The redistribution effect Lp-Li (the difference of average expected wirelength with and without transparent obstacles) mainly increases with the obstacle area (a,b) H M N

yP2 p2 p2 p2 Detour WL Detour Wirelengthof Two-terminal Nets • Detour WL dependence on position of, e.g., P2 • Linear for P2 with y coordinate b-H/2 < yP2 < yP1 and 2b-yP1 < yP2 < b+H/2 • Constant for all P2 with y coordinate yP1 < yP2 < 2b-yP1 W (a,b) H M p1 N

Resultant Wirelength of Two-terminal Nets where a=f(M,N,W,H,a) and b=g(M,N,W,H,b) • Observation 2: The blockage effect Lr-Lp (the difference of average expected wirelength with transparent and opaque obstacles) mainly increases with the largest obstacle dimension

Experimental Setup • Random point generator • Visibility graph • each terminal and obstacle corner as a vertex • each “visible” pair of vertices is connected by an edge • Graph Steiner minimal tree heuristic* p1 1 p2 p3 1 * L.Kou, G.Markowsky and L.Berman,“A Fast Algorithm for Steiner Trees”, Acta Informatica, 15(2), 1981, pp.141-145

Redistribution Effect vs. Obstacle Dimension • Observation 1: The redistribution effect Lp-Li mainly increases with the obstacle area

Blockage Effect vs. Obstacle Dimension • Observation 2: The blockage effect Lr-Lp mainly increases with the largest obstacle dimension

Redistribution Effect of Ten-terminal Nets • Observation 3: For multi-terminal nets the redistribution effect increases with the number of terminals and with the obstacle area

Blockage Effect of Ten-terminal Nets • Observation 3: For multi-terminal nets the blockage effect increases with the number of terminals and with the difference between obstacle dimensions

0.2 0.2 0.2 0.2 0.2 0.5 0.5 0.5 0.5 0.5 Experiment Setting for Obstacle Displacement 1 1

Redistribution Wirelength vs. Obstacle Displacement • Observation 4: The closer the obstacle is to the routing region boundary the smaller is the redistribution effect

Blockage Effect vs. Obstacle Displacement • The closer the obstacle is to the routing region boundary the smaller is the blockage effect • Observation 5: Displacement along the longest obstacle side has little effect on blockage effect

Effect of Layout Region Aspect Ratio • Observation 6: The redistribution effect does not depend on the aspect ratios of the region and the obstacle: it dominates when the aspect ratios are similar • The blockage effect is very dependent on the aspect ratios: it dominates when the aspect ratios are different

Experimental Setting forL-shaped Routing Region P1 1 W H P2 1 • Observation 7: L-shaped region has negative redistribution effect (Lp < 0.67) and no blockage effect (Lr = Lp)

Effect of L-shaped Routing Region • Observation 8: The more the L-shaped region deviates from a rectangle the less its total wirelength

Experimental Setting forC-shaped Routing Region W H 1 1

Blockage Effect in a C-shaped Region Comparing with: • The blockage effect doubles when an obstacle with a high aspect ratio touches the routing region boundary (compared with before it touches the boundary)

Blockage Effect in a C-shaped Region • In a C-shaped region the blockage effect mainly increases with the obstacle dimension that is not on the routing region boundary

Redistribution Effect in a C-shaped Region • The redistribution effect does not generally increase with the obstacle area when the obstacle is on the routing region boundary

Additive Property for Multiple Obstacles • Redistribution effect can be obtained by “polynomial expansion” = - = x x x - - + = x x x x + ) + - 2 ( x x x x + + 2 x x x

Additive Property for Multiple Obstacles • Blockage effect for m WixHi obstacles with non-overlapping x- and y-spans: x + x

Experiment Setting for Additive Property Region 2' Region 1 1 Region 2 1

Additive Property in Region 1 • Observation 9: The redistribution effect is additive for obstacles with small areas • Observation 10: The blockage effect is additive if there is no x- or y-span overlap between any obstacle pair

Non-Additive Property in Region 2 • Observation 9: The redistribution effect is additive for obstacles with small areas • Observation 10: The blockage effect is not additive for obstacles with overlapping x- or y-spans

Effect of Obstacle Number • Randomly generating a given number m of obstacles with a prescribed total obstacle area A • Observation 11: The total wirelength increases as the number of obstacles increases while the total obstacle area remains the same

Analyze Individual Wires • Redistribution effect is an average effect over all wires • Blockage effect is different for each wire with different length • Which wire of what length suffers blockage effect the most?

Blockage Distribution • Blockage effect makes a lot of differences for medium-sized wires (~30% wires make detour, up to a 60% increase in wirelength) • Can be combined with different wirelength distribution models

Conclusion • The first work to consider routing obstacle effect in wirelength estimation • Distinguish two routing obstacle effects • Theoretical expressions for 2-terminal nets and a single obstacle • Lookup table for multi-terminal nets and additive property for multiple obstacles • Help to guide SOC design and improve wireload models

Future Directions • Continuous study on multi-obstacle cases for finding equivalent obstacle relationships • Combination with different wirelength distributions which count placement optimization effect • Effects of channel capacity and routing sequence • Wirelength estimation for skew-balanced clock trees

Discrete Analysis Approach • Site density function f(l) is the number of wires with length l • generating polynomial V(x)=S f(l)xl • Complete expression for intrinsic, redistribution and resultant wirelenghts

Multiple Obstacle Analysis • Two obstacles with disjoint spans • Two obstacles with identical x- or y-spans • Two obstacles with covering x- or y-spans

Two obstacles with covering x- or y-spans • Number of medium-length wires decreases as any of the obstacle widths increases.

Toward Better Wireload Models in the Presence of Obstacles*