ECE 6133 Project OverviewSpring 2021

Sung Kyu Limlimsk@ece.gatech.eduGeorgia Institute of Technology

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• Circuit partitioning using matrices and eigenvectors

1/4. EIG Partitioning

Laplacian matrix L

2nd smallest eigenvector of Lgives linear placement

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• IBM18 benchmark– 210,613 cells

– Laplacian matrix fill rate: 0.01%!

– 178 GB of memory required to store in a normal matrix

– Sparse matrix can bring that down to 2.5 MB

Laplacian Sparsity

4Cutsize vs. Ratio Cut Landscape

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• Use linear equations to formulate floorplaning

2/4. ILP Floorplanning

…..

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• Off-the-shelf solver gives:

Obtaining ILP Solutions

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• Can size soft modules– Linear approximation used: overlap (or whitespace) may occur

Handling Hard/Soft Modules

floorplan with overlapred: soft, yellow: rotated using “super blocks”

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• Steiner points– Reduce the overall wirelength– Finding Steiner points is NP-hard….– So, we rely on heuristics

Recti-linear Routing with Steiner Points

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• Strategy: build separable MST and improve with L-edges

3/4. L-Shaped Steiner Routing

upper upper

lower lower upper

lower lower/upper upper

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• Strategy: build MST and add single Steiner point at a time

3/4. Kahng/Robins Routing

11Sample Routing Result

initial MST(WL = 242)

final RST (WL = 218)22 Steiner points added

Took 3 min

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• Cost minimization problem– How do we ship the units so that the overall cost is minimized?– Assume the capacity of each edge is 5 units

Multi-Commodity Flow (MCF) Problem

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• Set up ILP for MCF routing– Capacity of each edge in G is 2– Each edge in G becomes a pair of bi-directional arcs in F– n1 = {a,l}, n2 = {i,c}, n3 = {d,f}, n4 = {k,d}, n5 = {g,h}, n6 = {b,k}

MCF-based Multi-net Routing

14ILP Formulation

objective function

demand constraints

capacity constraints

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• Using off-the-shelf ILP solver

ILP-based Routing Solution

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• Based on shortest paths and rip-up-and-reroute

Heuristic: MM Algorithm

17Sample Routing Results

350 net benchmarkusage 1:1849 edgesusage 2:2468 edges

200 net benchmarkusage 1:1298 edgesusage 2: 1412 edges

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• Congestion maps– Thickness and color intensity indicate routing congestion

Sample Routing Results (cont)

30 nets 80 nets 200 nets