Dominatorbased partitioning for Delay Optimization

1
Dominator-based partitioning for Delay
Optimization

• David Bañeres Jordi Cortadella
• Univ. Politècnica de Catalunya
• Spain

Mike Kishinevsky SCL, Intel Corporation Hillsboro
, USA
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Outline

• Introduction
• Vertex dominator
• Dominator-based Partition (DBP)
• Comparisons
• DBP vs DEPART
• DBP vs hMetis

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Outline

• Introduction
• Vertex dominator
• Dominator-based Partition (DBP)
• Comparisons
• DBP vs DEPART
• DBP vs hMetis

4
Partitioning in synthesis
Behavioural Synthesis
- RTL Partitioning
Logic Synthesis

• Logic Minimization

- Placement - Routing
Physical Design
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Boolean Network
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Boolean Network
Represents a Boolean function
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Boolean Network
Primary Outputs
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Boolean Network
Primary Outputs
Primary Inputs
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Boolean Network
OUTPUTS
fanouts
fanins
INPUTS
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Boolean Network
OUTPUTS
Inmediate fanins
INPUTS
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Boolean Network
OUTPUTS
Transitive fanins
INPUTS
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Boolean Network
OUTPUTS
INPUTS
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Why on Logic Synthesis?

• Algebraic minimization Cost O(n2)
• Extraction of algebraic divisors
• Double cube divisor
• Factor extraction
• Boolean minimization Cost O(2n)
• Dont care calculation
• Delay minimization

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Why on Logic Synthesis?
Algebraic minimization Boolean minimization
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Why on Logic Synthesis?
Algebraic minimization Boolean minimization
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Why on Logic Synthesis?
Algebraic minimization Boolean minimization
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Why on Logic Synthesis?
Algebraic minimization Boolean minimization
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Sweep Algebraic methods
Light computational tools
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Sweep Algebraic methods
Light computational tools
Partition
DIVIDE CONQUER
Aggressive tools
Merge
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Sweep Algebraic methods
Light computational tools
Partition
Dont cares Delay optimization
Aggressive tools
Merge
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Partitioning in Synthesis

• Placement
• Iterative by Partitioning (Capo)
• FPGA
• MLC Sze04
• Logic Time-driven Optimization
• REDUCE DEPTH Touati91
• CLUSTER Yang97
• DEPART Aggarwal97

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REDUCE DEPTH Touati91
OUTPUTS
MaxSize 5 nodes
INPUTS
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CLUSTER Yang97
Area-driven optimization
Partition

• Optimization
• Delay-driven
• Area-driven

Optimization Method
Merge
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Partitioning in Synthesis

• Placement
• Iterative by Partitioning (Capo)
• FPGA
• MLC Sze04
• Logic Time-driven Optimization
• REDUCE DEPTH Touati91
• CLUSTER Yang97
• DEPART Aggarwal97

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Outline

• Introduction
• Vertex dominator
• Dominator-based Partition (DBP)
• Comparisons
• DBP vs DEPART
• DBP vs hMetis

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Vertex Dominator

• Code optimization in compilers
• Test pattern generation techniques
• Non-disjoint decomposition of Boolean functions
  Dubrova04

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Vertex Dominator

• Single Dominator
• A vertex x dominates another vertex y in a graph
  if every path from y to root contains x
  Lenguauer79.

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Vertex Dominator
OUTPUTS
Single vertex dominator
INPUTS
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Vertex Dominator
OUTPUTS
Single vertex dominator
INPUTS
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Vertex Dominator
OUTPUTS
INPUTS
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Vertex Dominator
OUTPUTS
INPUTS
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Vertex Dominator
OUTPUTS
INPUTS
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Vertex Dominator

• Single Dominator
• A vertex x dominates another vertex y in a graph
  if every path from y to root contains x
  Languager79.
• K-Vertex Dominator
• A set of vertices Xx1, x2,, xk dominates
  another vertex y in a graph if every path from y
  to root contains some vertex xi ? X Dubrova04.

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Vertex Dominator
OUTPUTS
2-vertex dominator
INPUTS
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Vertex Dominator
OUTPUTS
3-vertex dominator
INPUTS
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Why Vertex Dominator?
OUTPUTS
INPUTS
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Why Vertex Dominator?
OUTPUTS
INPUTS
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Why Vertex Dominator?
OUTPUTS
INPUTS
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Why Vertex Dominator?
OUTPUTS

• High interaction rest of the network

INPUTS
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Why Vertex Dominator?
OUTPUTS

• High interaction rest of the network

• Low freedom for restructuring

INPUTS
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Why Vertex Dominator?
OUTPUTS
INPUTS
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Why Vertex Dominator?
OUTPUTS
INPUTS
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Why Vertex Dominator?
OUTPUTS

• Low interaction rest of the network

INPUTS
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Why Vertex Dominator?
OUTPUTS

• Low interaction rest of the network

• High freedom for restructuring

INPUTS
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Outline

• Introduction
• Vertex dominator
• Dominator-based Partition (DBP)
• Comparisons
• DBP vs DEPART
• DBP vs hMetis

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Critical path
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Flowchart DBP
Select critical Region
Select non-critical Region
Dominator partition
Dominator partition
Delay Optimization
Area Optimization
NO
YES
YES
Improvement?
Improvement?
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Notation
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Notation
Primary outputs
Internal nodes
Primary inputs
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Dominator-based partition
MaxSize 8 nodes
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Dominator-based partition

• Single Dominators

MaxSize 8 nodes
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Dominator-based partition
4

• Single Dominators

MaxSize 8 nodes
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Dominator-based partition
4

• Single Dominators

MaxSize 8 nodes
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Dominator-based partition
4

• Single Dominators

MaxSize 8 nodes
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Dominator-based partition
2
4

• Single Dominators
• Double-vertex Dominators

MaxSize 8 nodes
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Dominator-based partition
NODES 14 !!
2
4

• Single Dominators
• Double-vertex Dominators

MaxSize 8 nodes
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Dominator-based partition
2
4

• Single Dominators
• Double-vertex Dominators

• k-vertex Dominators

MaxSize 8 nodes
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Dominator-based partition
2
4

• Single Dominators
• Double-vertex Dominators

• Cluster remaining single nodes

MaxSize 8 nodes
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Dominator-based partition
2

• Single Dominators
• Double-vertex Dominators

• Cluster remaining single nodes

MaxSize 8 nodes
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Dominator-based partition
2
??

• Single Dominators
• Double-vertex Dominators

• Cluster remaining single nodes

MaxSize 8 nodes
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Dominator-based partition
2

• Single Dominators
• Double-vertex Dominators

• Cluster remaining single nodes

• Cluster small windows

MaxSize 8 nodes
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Dominator-based partition
2

• Single Dominators
• Double-vertex Dominators

• Cluster remaining single nodes

• Cluster small windows

MaxSize 8 nodes
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Dominator-based partition
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Consistent delay propagation
Topological Order
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Preserving topological order
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1
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2
1
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Outline

• Introduction
• Vertex dominator
• Dominator-based Partition (DBP)
• Comparisons
• DBP vs DEPART
• DBP vs hMetis

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DEPART

• DElay optimization using PARTitioning
  Aggarwal97.

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Size of clusters in DEPART
MaxSize 8 nodes
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Size of clusters in DEPART
MaxSize 8 nodes
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Size of clusters in DEPART
MaxSize 8 nodes
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Size of clusters in DEPART
MaxSize 8 nodes

• Disadvantage
• Root nodes are excluded from minimization

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Delay Optimization
DBP
DEPART
Select Critical Region
Select Critical Outputs
50 Nodes 200 Nodes
Transitive partition
Dominator partition
Delay Optimization
Delay Optimization
Speed_up
YES
YES
Improvement?
Improvement?
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Delay Optimization
Delay script speed_up Size clusters DBP -gt 50
nodes DEPART -gt 200 nodes
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Delay Optimization
Delay script speed_up Size clusters DBP -gt 50
nodes DEPART -gt 200 nodes
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Delay Optimization
Delay script speed_up Size clusters DBP -gt 50
nodes DEPART -gt 200 nodes
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Delay Optimization
Delay script speed_up Size clusters DBP -gt 50
nodes DEPART -gt 200 nodes
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Delay Optimization
Even better that internal partitioning of
speed_up!!!
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Outline

• Introduction
• Vertex dominator
• Dominator-based Partition (DBP)
• Comparisons
• DBP vs DEPART
• DBP vs hMetis

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hMetis
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hMetis
??
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hMetis with weighted critical paths
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Time propagation on hMetis
INPUTS
OUTPUTS
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Time propagation on hMetis
DBP
INPUTS
OUTPUTS
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Time propagation on hMetis
Cut of 7 edges
DBP
INPUTS
OUTPUTS
hMetis
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Experimental Results
DBP
hMetis
Select Critical Region
Select Critical Region
hMetis
Dominator partition
50 Nodes
Delay Optimization
Delay Optimization
Speed_up
YES
YES
Improvement?
Improvement?
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DBP vs hMetis
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DBP vs hMetis
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DBP vs hMetis
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Size of clusters
DBP
hMetis
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Conclusions

• DBP produces good partitions of Logic
  minimization
• Selection of dominators
• Topological order to propagate delay information
• Size of clusters is crucial to obtain a trade-off
  between runtime and results
• Future work extend to sequential circuits.

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hMetis with weights
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Importance of Dominators
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Importance of Dominators
DBP
hMetis






a
c
c
a
b
b
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Why on Logic Synthesis?
RTL
Partition
Logic Synthesis
Merge
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Why on Boolean Minimization?
RTL
Partition
Logic Synthesis
Merge
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Vertex Dominator

• K-vertex dominator
• Given a network G (V,E), a subset of nodes X
  x1,x2, ,xk ? V is a dominator of a node u ?
  V - X if
• Every path from u to a primary output contains
  some vertex xi ? X, and
• no proper subset of X is a dominator of u.

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Vertex Dominator
2-vertex dominator
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Vertex Dominator
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DEPART

• DElay optimization using PARTitioning
  Aggarwal97.
• Non-overlapping among clusters (bins).
• Select critical primary outputs.
• Select clusters by cones of influence of the root
  node.
• Order of minimization by criticality of the
  clusters

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Time propagation on hMetis
DBP
hMetis
3
1
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45
15
16
3
3
7
3
7
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45
17
17
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Why on Logic Synthesis?
Cost Minimization Tools O(n2)
Espresso Brayton84 Speed_up Singh88
Full_simplify Savoj91 Time_dec Cortadella03
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Dominator-based partition
2
4

• Single Dominators

MaxSize 8 nodes
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Dominator-based partition
2
4

• Single Dominators

MaxSize 8 nodes
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hMetis

• Graph to Hypergraph

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hMetis without weights
??
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hMetis with weights
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Preserving topological order
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