Heterogeneous B*-trees for
Analog Placement with Symmetry
and Regularity Considerations
P. Chou, H. Ou and Y. Chang
National Taiwan University
Taipei, Taiwan
ICCAD 2011
Outline
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Introduction
Preliminaries
Problem Formulation
Heterogeneous B*-tree Representation
Simultaneous Symmetry and Regularity Handling
The Placement Method
Experimental Results
Conclusions
Introduction
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Symmetry constraints and regular structures are
two major considerations for expert analog layout
designer.
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Symmetry constraints are specified to place
matched modules symmetrically with respect to
some common axes to reduce unwanted electrical
effects.
Introduction
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Regular structures like row or arrays are also a common
design style for analog designers.
Regular structures can benefit the routability.
The row structures makes the wire crossing from left to right
shorter and with fewer bends, which can improve both
routability and wirelength.
A placement without
a regular structure
A placement with a
regular structure
Introduction
Preliminaries
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Review of the automatically symmetric-feasible
B*-tree (ASF-B*-tree)
ASF-B*-tree
Preliminaries
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Review of the Hierarchical B*-trees
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A hierarchical B*-tree is designed to represent a cluster of
modules such as symmetry islands.
Hierarchical nodes are used in a B*-tree to model the ASF-B*tree for symmetry groups.
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HB*-tree
Problem Formulation
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The Analog Placement Problem:
Inputs:
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A set of modules
A netlist indicating the connection of modules
A set of placement constraints
Outputs:
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A placement of all modules for minimizing the total area
and wirelength such that no modules overlap with each
other and the placement result satisfies all specified
symmetry constraints.
Heterogeneous B*-tree
Representation
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A heterogeneous B*-tree representation with two
kinds of hierarchical nodes: symmetry nodes and
regularity nodes.
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Regularity nodes are formed by a set of nodes
representing modules of the same dimensions.
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Regularity nodes contains all the nodes in the
corresponding regular structure.
The possible tree structures in regularity nodes are
predefined to have a regular packing result such as row or
arrays.
Heterogeneous B*-tree
Representation
B*-tree to store a row
structure
B*-tree to store an
array structure
Simultaneous Symmetry and
Regularity Handling
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Regular Structures with Symmetry and Nonsymmetry Modules
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Symmetry modules can form regular structures on the
boundaries of symmetry islands with non-symmetry
modules if they have similar dimensions.
Simultaneous Symmetry and
Regularity Handling
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A case that a regular structure is on the top
boundary of a symmetry group.
The Placement Method
Module Classification
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Since the perturbing step in the annealing scheme
is usually executed for a large number of times.
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Module dimension classification is needed and is
performed only once in this preprocessing step.
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After classification, modules of the same size are
marked as the same category for selection to form
regular structures during perturbation.
Regular-Structure Probability
Assignments
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The probability distribution for changing the size of a node
during perturbation should not be uniform.
The score sk of a regularity group of size k can be computed
as follows:
m: the number of possible structures for a regularity node of
size k.
Wi and Hi: the width and height of the regular structure i.
Regular-Structure Probability
Assignments
Regular-Structure Probability
Assignments
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The probability for a node n to become a size-t
regularity node is calculated as follows:
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d: the maximum size change defined manually
The denominator is the sum of all scores of the
possible sizes within the range [k-d, k+d].
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New Types of Moves
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Aside from the original three moves of the B*-tree:
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Swap
Rotate
Delete and insert
The new moves are listed below:
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Merge/Split
Reshape
New Types of Moves
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Merge/Split
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Reshape
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The aspect ratio of the regular structure inside the node is
changed.
If a regularity node contains a 1X4 row at first, it can
become a 2X2 array.
Regularity Cost
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The corner node is referred to as a point that is
touched by at least one corner of a module.
A corner node with k coincident corners is a kcorner node.
2-corner node
Regularity Cost
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Corner nodes can be classified into six categories:
Cost:
2/3
2/3
1
1
2/3
1/2
#corner
nodes:
C4
C2
C1-1
C3
C2
C1-2
Regularity Cost
Regularity Cost
The Placement Method
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The total cost function C:
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A: area
L: wirelength
R: regularity cost
Experimental Results
Experimental Results
Experimental Results
Conclusions
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This paper proposed a heterogeneous B*-tree
representation to consider symmetry and regularity
simultaneously.
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Experiments based on both placement and global
routing results have shown that the method is
efficient.