NuCAD
ACG - Adjacent Constraint Graph
for General Floorplans
Hai Zhou and Jia Wang
ICCD 2004, San Jose
October 11-13, 2004
Outline
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•
•
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Floorplan and Constraint Graph
Adjacent Constraint Graph
Data Structure and Operations
Experimental Result
Summary
(2)
Floorplan
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A floorplan contains non-overlapping modules.
a
c
e
b
d
(3)
Constraint Graph
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Horizontal graph represents “left to” relation.
a
c
e
d
b
(4)
Constraint Graph
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Vertical graph represents “below” relation.
a
b
c
e
d
(5)
Constraint Graph
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There is at least one relation between any pair of
modules.
a
c
e
d
b
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Redundancy in constraint graphs.
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Transitive edges.
a
c
e
d
b
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Redundancy in constraint graphs.
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Over-specification: more than one relation
between two modules.
a
c
e
d
b
(8)
Reduce the redundancy.
• Less edges means less running time and less storage
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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?
•
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No. In the graph to the right, we have
more than k2 edges where there are
only 2k modules.
• The basic structure could be identified
as crosses in the graph.
a
a
b
c
HVV
1
k+1
2
k+2
...
...
...
k+2
k
k+2 2k
b
c
...
d
d
VHH
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ACG-Adjacent Constraint Graph
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Observation: in VHH cross, either ‘a’ is below ‘d’ or ‘c’ is below ‘b’.
 Has similar result for HVV cross.
a
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b
c
d
a
c
b
d
Definitions of ACG
 Constraint graph.
 Exactly one relation between every pair of modules.
 No transitive edges.
 No cross.
b
a
d
c
e
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Properties
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Symmetry
 Still an ACG when edge directions reversed.
 Still an ACG when edge types exchanged (H vs V).
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Constraint graph
 Packing is as simple as longest path computations.
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Number of edges
 Conjecture: O(n).
 Experimental result: < 10n up to n = 10000.
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Close to adjacency graph
 Preserve geometrical adjacency information.
a
c
a
e
b
c
e
d
d
b
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Data structure
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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).
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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
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Guideline
 Maintain a valid ACG through operations to control the
number of edges.
 Operation cost (time/space) should be linear to the edges
added/deleted.
• Append
 We could construct an ACG by appending vertices.
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Swap
 Will introduce crosses into the graph.
 Swapping is complete when we allow crosses in ACG.
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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
• Step 2: add the first edge of the other type.
 Still follow the first edge of the other type.
new
a
b
c
d
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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.
new
a
b
c
d
e
f
• Stop when we cannot proceed with step 3.
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Swap
a
b
c
d
e
• 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.
• Step 2: change the edge type.
 Need to change the edge direction too, i.e., from b->c to c->b.
a
c
b
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
c
b
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
c
b
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
c
b
d
e
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Experiment Setup
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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 FASTSP, 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.
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We cut the number of perturbations by half in our SA.
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Experimental Result (Cont.)
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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.
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Floorplan for xerox
with ACG edges.
 Most edges are
between adjacent
modules.
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Summary
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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!
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