Temporal Floorplanning Using the Ttree Formulation

1
Temporal Floorplanning Using the T-tree
Formulation

• Ping-Hung Yuh1, Chia-Lin Yang1,
• Yao-Wen Chang2

1 Dept. of Computer Science Information
Engineering 2 Dept. of Electrical
Engineering National Taiwan University, Taiwan
2
Outline

• Introduction
• The T-tree Formulation
• Temporal Floorplanning Algorithm
• Fixed-outline Floorplanning
• Reachability of T-tree
• Experimental Results
• Conclusion

3
Outline

• Introduction
• The T-tree Formulation
• Temporal Floorplanning Algorithm
• Fixed-outline Floorplanning
• Reachability of T-tree
• Experimental Results
• Conclusion

4
Reconfigurable System

• Reconfigurable functional unit (RFU) a device
  that can be reconfigured during program
  execution, say an FPGA
• RFUOP (RFU operation) operations executed on RFU

RFUOP 2
RFUOP 2
RFU
5
Reconfiguration Issues

• Given a set of RFUOPs
• When to load?
• How to allocate?
• There exist additional temporal ordering
  requirements among RFUOPs
• Need an efficient and effective topological
  representation for modules
• Need an efficient algorithm to find a solution to
  meet our requirements

6
Previous Work

• Component graph-based approach
• Teich et al. in Proc. PDPTA, 1999
• Use the graph-theoretical approach
• Assume no data dependency
• Fekete et al. in Proc. DATE, 2001
• Consider the RFUOPs ordering
• Absolute-coordinate based approach
• Bazargan et al. in IEEE Design Test of
  Computers, 2000
• 2D bin-packing and 3D placer
• Topological representation based approach
• Yuh et al. in Proc. ASPDAC, 2004
• Use three transitive closure graphs (TCGs) to
  represent a temporal placement

7
Our Contribution

• Introduce the T-tree representation
• A T-tree is a 3-ary tree to represent a temporal
  placement

na
nb
nd
nf
ne
nc
8
Our Contribution (contd)

• First work to use a tree-based representation to
  handle the temporal floorplanning/placement
  problem
• Three advantages compared with the recent
  state-of-the-art representation
• Faster operations (O(n) time per operation, n
  of modules)
• More efficient packing method
• Smaller solution space
• Consider fixed-outline floorplanning

9
Temporal Floorplanning

• Given a set of tasks V, treat a task vi as a 3D
  box with width (x-coordinate), height
  (y-coordinate) and duration (t-coordinate)
• Find a placement ?, an assignment of for each
  vi , , such that no two boxes
  overlap and satisfy all precedence constraints
• Precedence constraint the temporal relations
  among tasks to guarantee the correct execution
• Optimize a predefined cost metric
• Area x Execution time (Volume)
• Wirelength
• Communication overhead
• Reconfiguration overhead

Height
Duration
Width
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Communication Overhead and Reconfiguration
Overhead

• Communication overhead
• The overhead between two data-dependent tasks
• Reconfiguration overhead
• The overhead when a task is loaded into an RFU

11
Outline

• Introduction
• The T-tree Formulation
• Temporal Floorplanning Algorithm
• Fixed-outline Floorplanning
• Reachability of T-tree
• Experimental Results
• Conclusion

12
The T-tree Representation

• A 3-ary tree representation for temporal
  floorplanning/placement problem
• The T-tree keeps the geometric information as
  follows
• Child j is the left child of parent i tj ti
  Ti
• Child k is the middle child of parent i tk
  ti, yk gt yi
• Child l is the right child of parent i tl ti,
  yl yi

13
An Example
na
nb
nd
nf
ne
nc
The corresponding T-tree
A placement
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From a Placement to a T-tree

• A placement is compacted iff no module can be
  moved along its X-, Y- or T- directions to the
  origin while other modules are fixed
• Construct T-tree recursively based on the DFS
  order

na
nb
nd
nf
ne
nc
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From a T-tree to a Placement

• The t-coordinate can be obtained from the
  structure of T-tree
• The original T-tree can be decomposed into
  several binary trees

T-tree H1
Decompose
na
nb
nd
nf
ne
nc
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From a T-tree to a Placement (contd)

• Pack each node in one binary tree in the DFS
  order
• Apply the contour data structure for each binary
  tree to compute the y-coordinate
• Maintain a list L to store all tasks whose t- and
  y-coordinates are already determined
• The x-coordinate of task vi maxxk the
  projections of vi and vk are overlapped on Y-T
  plane the for k L

Binary trees
B1 B2 B3
1
4
5
nf
nd
6
2
ne
3
17
Outline

• Introduction
• The T-tree Formulation
• Temporal Floorplanning Algorithm
• Fixed-outline Floorplanning
• Reachability of T-tree
• Experimental Results
• Conclusion

18
Precedence Constraint

• To maintain the temporal orderings among tasks,
  all the precedence constraints must be satisfied
• If there exists a precedence constraint between
  task vj and task vi, vj is guaranteed to satisfy
  the precedence constraint if node nj is in the
  left sub-tree of node ni

• If nj is in the left sub-tree of ni
• vj starts after vi finishes
• If nj is in the middle or right sub-tree of ni
• vj may not execute after vi

ni
nj
nj
nj
tj tj
tj tj
tj tj Ti
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Set Definition

• Each node nk is associated with two sets
• Ik set of tasks that must be executed before vk
• Ok set of tasks that must be executed after vk
• For example

ve must be executed after vb va must be executed
after vbvd must be executed before vb
Ib vdOb ve, va
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Feasibility Detection

• Use Simulated Annealing
• Perform feasibility detection after each
  perturbation
• If task vk violates the precedence constraint,
  perform tree-reconstruction by moving node nk to
  the left sub-tree of node np, with vp in has the
  latest ending time in Ik
• Guarantee to satisfy the precedence constraint
  after each perturbation
• Problem Tree-reconstruction overhead

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Tree-reconstruction Overhead Reduction

• Idea avoid the operations that will definitely
  violate the precedence constraint
• Derive a heuristic for operations based on two
  observations

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Two Observations

• Observation 1 ne can only choose a node or a
  position in nbs left sub-tree for operations to
  satisfy the precedence constraint
• Observation 2 nb cannot choose a node or a
  position in the sub-tree rooted by ne in order
  not to violate the precedence constraint

ve must be executed after vb
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Node Selection

• Number nodes in the DFS order starting from its
  right child

• Each node nk is associated with two values,
  DFS_down and DFS_up
• DFS_down the DFS number of the nps left child
  where task vp has the latest ending time in Ik
• DFS_up the DFS number of node nq where task vq
  has the least starting time in Ok
• Heuristically choose a node in the range
  DFS_down, DFS_up)

1
4
2
5
3
8
6
7
ve must be executed after vbnbs DFS_down 1
DFS_up 5nes DFS_down 5 DFS_up
9
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Outline

• Introduction
• The T-tree Formulation
• Temporal Floorplanning Algorithm
• Fixed-outline Floorplanning
• Reachability of T-tree
• Experimental Results
• Conclusion

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Fixed-outline Floorplanning

• In practice, RFU is often chosen first, and thus
  the available area is fixed
• Let Wf/Hf and Wp/Hp denote the width and height
  of a reconfigurable device and a placement,
  respectively
• A feasible placement of fixed-outline
  floorplanning must satisfy the outline
  constraint that is, Wp Wf and Hp Hf
• Two ways for handling the fixed-outline
  floorplanning problem
• Guide the perturbation with outline-violation
  probability
• Add extra penalty cost in the cost function

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Two Methods for Fixed-outline Floorplanning

• Guide the tree perturbation with
  outline-violation probability p
• p k/n, where k is the of outline-violated
  placements in last n iterations
• If p is large, increase the probability of
  placing a task along the t-coordinate to fit into
  the desire outline
• If p is small, increase the probability of
  placing a task along the x- or y- coordinate to
  minimize the execution time
• Modify the cost function
• Add the excessive volume into the cost function
  as the penalty cost
• Consider the smaller excessive volume of two
  orthogonal placements

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Outline

• Introduction
• The T-tree Formulation
• Temporal Floorplanning Algorithm
• Fixed-outline Floorplanning
• Reachability of T-tree
• Experimental Results
• Conclusion

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Reachability of T-tree

• Two placements are said to be equivalent if they
  have the same
• Topology of T-tree
• Labeling of each node
• Orientation of each task
• Three steps to transform a T-tree H1 to another
  T-tree H2
• Make H1 the same topology as H2
• Get the correct label for each node
• Get the correct orientation for each task

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Making H1 the Same Topology of H2

• Find a left-skew T-tree H0 to which both H1 and
  H2 can be transformed via a series of operations,
  and the operation is reversible
• Transform H1 to H2 with H0 as an intermediate
  solution

H2
H0
H1
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Tree Transformation Process step 1

• Recursively move the rightmost leaf to the
  leftmost external position and reverse this
  process
• Need at most 2n-2 moves to obtain the correct
  topology
• Perform at most n-1 swaps to obtain the correct
  labeling
• Perform at most n rotations to get the correct
  orientation
• Theorem It needs at most 4n-3 operations to
  transform a T-tree to another T-tree

Move
Move
Move
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Tree Transformation Process step 2

• Recursively move the rightmost leaf to the
  leftmost external position and reverse this
  process
• Need at most 2n-2 moves to obtain the correct
  topology
• Perform at most n-1 swaps to obtain the correct
  labeling
• Perform at most n rotations to get the correct
  orientation
• Theorem It needs at most 4n-3 operations to
  transform a T-tree to another T-tree

Swap
Swap
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Tree Transformation Process step 3

• Recursively move the rightmost leaf to the
  leftmost external position and reverse this
  process
• Need at most 2n-2 moves to obtain the correct
  topology
• Perform at most n-1 swaps to obtain the correct
  labeling
• Perform at most n rotations to get the correct
  orientation
• Theorem It needs at most 4n-3 operations to
  transform a T-tree to another T-tree

At most 3 rotations
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Tree Transformation Process

• Recursively move the rightmost leaf to the
  leftmost external position and reverse this
  process
• Need at most 2n-2 moves to obtain the correct
  topology
• Perform at most n-1 swaps to obtain the correct
  labeling
• Perform at most n rotations to get the correct
  orientation
• Theorem It needs at most 4n-3 operations to
  transform a T-tree to another T-tree

H1
H2
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Outline

• Introduction
• The T-tree Formulation
• Temporal Floorplanning Algorithm
• Fixed-outline Floorplanning
• Reachability of T-tree
• Experimental Results
• Conclusion

35
Experimental Results

• The algorithm was implemented in C and ran on
  SUN Blade 2000
• Conducted three experiments
• Volume, overhead and wirelength optimizations
  with 3D-MCNC
• Volume, overhead and wirelength optimizations
  with 3D-GSRC
• The fixed-outline experiment with 3D-n100
• Compared with 3D-subTCG (ASPDAC-2004)
• The 3D extension of Transitive Closure Graph
  (TCG)
• Use three transitive closure graphs (one for each
  dimension) to represent a 3D placement

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3D-MCNC Benchmark Statistics

• Circuits were the same as those in Yuh et al,
  ASPDAC 2004

37
Results on 3D-MCNC-1 Benchmarks

• Two sets of 3D-MCNC (3D-MCNC-1, 3D-MCNC-2)
  benchmarks with different precedence constraints

38
Results on 3D-MCNC-2 Benchmarks
T-tree can obtain smaller deadspace (15.61 vs.
12.8) in less running time
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3D-GSRC Benchmark Statistics

• 3D extension of 2D GSRC benchmarks

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Results on 3D-GSRC Benchmarks
T-tree performs even better for larger circuits
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Results for Fixed-outline Floorplanning

• Adopted 3D-n100 with various outline widths and
  heights

Our approach for fixed-outline floorplanning is
effective
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Resulting Placement for 3D-ami49

• Resulting placement with precedence constraints
  and volume, wirelength and overhead optimization

43
Outline

• Introduction
• The T-tree Formulation
• Temporal Floorplanning Algorithm
• Fixed-outline Floorplanning
• Solution Space and Reachability
• Experimental Results
• Conclusion

44
Conclusion

• Presented the first tree-based representation for
  temporal floorplanning/placement problem
• Proposed the T-tree and demonstrated its
  efficiency and effectiveness
• Showed superior properties of the T-tree
• Smaller solution space
• The solution space is well-structured
• Future work
• Module sizing
• Task preemption

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Thank Youfor yourattention