Block Placement with Symmetry Constraints based on the Otree Nonslicing Representation

1
Block Placement with Symmetry Constraints based
on the O-tree Non-slicing Representation

• Yingxin Pang, Florin Balasa?
• Koen Lampaert?, Chung-Kuan Cheng
• University of California, San Diego
• ?University of Illinois at Chicago
• ?Conexant System Inc.

2
Overview

• Introduction and Motivation
• O-tree representation
• Symmetric constraints placement based on the
  O-tree representation
• Experimental results
• Conclusions

3
Why Symmetry Constraints
In analog circuits

• To match interconnection parasitics and device
  parameters
• To balance thermal effects

4
Symmetry Constraints

• Symmetry Pair
• A pair of blocks which have to be placed
  symmetrically with respect to an axis
• Symmetry Group
• A group of symmetry pairs which share a common
  axis
• A symmetry group

h
5
Previous Work

• Exploring absolute placement configurations
• Operates directly on the absolute coordinates of
  the devices
• Implements the symmetry constraints directly in
  the move set
• Allows overlapped placements in intermediate
  solutions and the placer has to drive the overlap
  to zero during annealing

6
Previous Work

• Employing topological representations
• Slicing structure
• Avoids the overlap problem
• Restricts the set of reachable layout topologies
• Sequence pair
• Allows to explore all placement configurations
• Works well in an industrial environment

7
O-tree Representation
An O-tree is an ordered tree, which can be
encoded by

• is a 2n-bit string which identifies the
  structure of the tree
• is a permutation of n nodes

An Example 00110100011011, abcdefg
8
O-tree to Placement
9
Symmetric X-feasible O-tree

• Symmetric x-feasible O-tree
• If an O-tree can lead to a placement satisfying
  both the horizontal positioning constraints and
  the horizontal symmetric constraints
• X-constraint graph Gx
• Has the same nodes as the given O-tree
• Includes the horizontal constraint edges of the
  O-tree
• Introduces two new arcs (a, b) and (b, a) for
  each symmetry pair (a,b)
• w(a,b)0, w(b,a)0

10
X-constraint graph
A symmetric x-feasible O-tree
h
g
f
e
b
Symmetry pairs (b,h), (e,f)
11
X-constraint graph

A non-symmetric x-feasible O-tree
h
f
g
e
b
Symmetry pairs (b,h), (e,f)
12
Symmetric X-feasible O-tree

• Theorem 1
• If the x-constraint graph does not contain any
  positive cycles, then the corresponding O-tree is
  symmetric x-feasible.
• Theorem 2
• If an O-tree is symmetric x-feasible, a minimum
  width placement satisfying both horizontal
  positioning and symmetric constraints can be
  built in O(n2) time by applying the single source
  longest path algorithm

13
Placement of X-feasible O-tree
h
b
14
Symmetric Y-feasible O-tree

• Symmetric y-feasible O-tree
• If an O-tree can lead to a placement satisfying
  both the vertical positioning constraints and the
  vertical symmetric constraints
• Y-constraint graph
• Has the same nodes as the given O-tree
• Includes the vertical constraint edges of the
  O-tree
• Introduces additional edges modeling the vertical
  symmetric constraints as follows

15
Y-constraint Graph

• For any two symmetry pairs ,
• If there is a path from to
• add a directed edge
• unless the edge exists

h
g
f
d
e
c
a
b
16
Y-constraint Graph

• For any two symmetry pairs ,
• If there is a path from to
• Add a directed edge
• unless the edge exists

h
g
f
d
e
c
a
b
17
Y-constraint Graph
A symmetric y-feasible O-tree
Symmetry pairs (b,h), (e,f)
18
Y-constraint Graph
A non-symmetric y-feasible O-tree

f
g
h
d
e
c
a
b
Symmetry pairs (b,h), (e,f)
19
Y-coordinates

• yi the longest path length from root to node Bi
• execute a topological sort of the nodes in Gy
• record each symmetry pairs (ai, bi) such that bi
  is ordered according the topological sort
• for each symmetry pair (ai, bi)
• if d gt0,
• update
• execute the single-source longest path algorithm,
  consider bi as the source

20
Symmetric Y-feasible O-tree

• Theorem 3
• If the y-constraint graph does not contain any
  cycles, then the corresponding O-tree is
  symmetric y-feasible.
• Theorem 4
• If an O-tree is symmetric y-feasible O-tree, a
  minimum height placement satisfying both vertical
  positioning and symmetric constraints can be
  built in O(n2) time

21
Symmetric Feasible O-tree

• Symmetric feasible O-tree
• Symmetric x-feasible
• Symmetric y-feasible
• Theorem 5
• A symmetric feasible O-tree leads to a feasible
  placement and the placement is optimal under the
  O-tree packing rule
• Theorem 6
• A minimum area rectangle placement with symmetric
  constraints can be represented by a symmetric
  feasible O-tree

22
Symmetric Feasible O-tree to Placement
h
g
h
f
e
b
23
Experimental Results
24
Layout Result
lpf2_b25b
25
Conclusions

• It handles the symmetry constraint efficiently by
  adding the symmetry constraints directly in the
  O-tree constraint graphs
• It is better than other methods in terms of
• Time complexity
• Lower bound of configurations