NuCAD ACG - Adjacent Constraint Graph

for General Floorplans

Hai Zhou and Jia Wang

ICCD 2004, San JoseOctober 11-13, 2004

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Outline

• Floorplan and Constraint Graph

• Adjacent Constraint Graph

• Data Structure and Operations

• Experimental Result

• Summary

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Floorplan

• A floorplan contains non-overlapping modules.

a

bd

c

e

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Constraint Graph

• Horizontal graph represents “left to” relation.

a

b

d

ce

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Constraint Graph

• Vertical graph represents “below” relation.

a

b d

ce

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Constraint Graph

• There is at least one relation between any pair of modules.

a

b

d

ce

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a

b

d

ce

Redundancy in constraint graphs.

• Transitive edges.

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a

b

d

ce

Redundancy in constraint graphs.

• Over-specification: more than one relation between two modules.

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Reduce the redundancy.

• Less edges means less running time and less storage requirement.

• Allow exactly one relation between any pair of modules.

• Remove all the transitive edges.

• Combine horizontal graph and vertical graph together. Two types of edges: H and V.

The remaining edges form a total order!

b a d c e

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Cross

• There are totally O(n2) edges in the two constraint graphs where n is the number of modules.

• Can we get rid of that quadratic boundary by the previous steps?

V H H

H V V

• No. In the graph to the right, we have more than k2 edges where there are only 2k modules.

1

2 k+2

k+1

... ...

... k+2 ...

k k+2 2k• The basic structure could be identified as crosses in the graph.

a b c d

a b c d

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ACG-Adjacent Constraint Graph

• Definitions of ACG Constraint graph.

Exactly one relation between every pair of modules.

No transitive edges.

No cross.

b a d c e

a b c d a c b d

• Observation: in VHH cross, either ‘a’ is below ‘d’ or ‘c’ is below ‘b’. Has similar result for HVV cross.

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Properties• Symmetry

Still an ACG when edge directions reversed. Still an ACG when edge types exchanged (H vs V).

• Constraint graph Packing is as simple as longest path computations.

• Number of edges Conjecture: O(n). Experimental result: < 10n up to n = 10000.

• Close to adjacency graph Preserve geometrical adjacency information.

a

bd

c

e

a

b

d

ce

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Data structure

• Vertices representing modules are linked as the total order.

• Every vertex maintains 4 lists of edges. Correspond to 4 geometrical relationships (left/right, below/above).

• Every edge keeps its two end vertices.

• Every edge belongs to 2 lists (incoming/outgoing). Edges from one list have a common end vertex. An edge with a closer end vertex to the common vertex comes first.

b a d c e

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Operations

• Guideline Maintain a valid ACG through operations to control the n

umber of edges.

Operation cost (time/space) should be linear to the edges added/deleted.

• Append We could construct an ACG by appending vertices.

• Swap Will introduce crosses into the graph.

Swapping is complete when we allow crosses in ACG.

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new a b c

Append

• Step 1: add edges with the same type. Always follow the first edge of the other type to determine the

next new edge.

new a b c d

• Step 2: add the first edge of the other type. Still follow the first edge of the other type.

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new a b c d e f

Append

• Step 3: add edges whose types alternating. Follow the edge after the edge connecting the other end points

of the two previous added edges.

• Stop when we cannot proceed with step 3.

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Swap

• Step 1: exchange the positions of the two vertices. This won’t affect the edge lists. Suppose we are going to swap b and c.

a b c d e

• Step 2: change the edge type. Need to change the edge direction too, i.e., from b->c to c->b.

a bc d e

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Swap

• Step 3: remove transitive edges. Check c’s outgoing V neighbors and b’s incoming V neighbors.

a bc d e

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Swap

• Step 3: remove transitive edges. Check c’s outgoing V neighbors and b’s incoming V neighbors.

a bc d e

• Step 4: add edges to keep original relations. Connect c to b’s incoming H neighbors. Connect b to c’s outgoing H neighbors.

• Refer to Lemma 4 for exceptions.

a bc d e

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Experiment Setup

• MCNC benchmarks: apte, hp, xerox, ami33, ami49.

• GSRC benchmarks: n100, n200, n300.

• Create ACG with appending operations.

• Employ the simulated annealing algorithm. Adopt an annealing scheme similar to TCG-S.

Perturbations• Change the orientation of a module.

• Exchange two modules.

• Swapping two adjacent vertices.

Allow crosses in ACG during perturbations.

• Compare running time as well as solution quality to FAST-SP, TCG, TCG-S, and Parquet-2 on area optimization.

• Running on a Sun Ultra 10, running time is in seconds.

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Experimental Result• Compare to FAST-SP, TCG, and TCG-S on all the benchmarks.

• Compare to Parquet-2 on the GSRC benchmarks.

• We cut the number of perturbations by half in our SA.

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Experimental Result (Cont.)

• Number of edges. Min/max edge give the

minimal/maximal number of edges in the graph during SA.

Even we allow crosses, the number keeps small during SA.

• Floorplan for xerox with ACG edges. Most edges are

between adjacent modules.

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Summary

• ACG is a constraint graph.

• ACG is like an adjacency graph due to the fact that the redundancies in the traditional H/V constraint graphs are reduced.

• We have a neat data structure plus efficient operations.

• Future directions. Enforce the no-cross rule in operations. Interconnect estimation and planning (ASP-DAC 2005). Is the number of edges really O(n)?

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Thanks!