1. Local Unidirectional Bias for Smooth Cutsize-delay Tradeoff in Performance-driven Partitioning
Andrew B. Kahng and Xu Xu
UCSD CSE and ECE Depts.
Work supported in part by MARCO GSRC

2. Outline  Motivation Performance driven bipartition problem
New bipartitioning algorithm
Experimental results
Conclusion and future work

3. Partitioning and Performance
The goal of traditional hypergraph partitioning is to minimize cutsize.
To meet the performance requirement of current designs, we need a performance-driven partitioner, which considers both cutsize and delay.

4. Previous Work (I) [Cong et al. ISPD-2002]
Global clustering based algorithm with retiming
Min-delay
Clustering
w/ retiming
Min-cutsize
Clustering
De-clustering
and refinement
Reduces delay by 16% while increasing cutsize by 17%
Requires substantial gate replication

5. Previous Work (II) [Ababei et al. ICCAD-2002] Reweighting based method
Path based
Input
Reweighting
Cutsize oriented partitioner, such
as hMetis,MLPart 1 1
Global timing analysis
Find critical paths 1
Net based 1 2 1
14% reduction of delay with 10% increase in cutsize
139% increase in runtime compared with hMetis

6. Motivating Questions  Can we avoid global timing analysis?
Global timing analysis is extremely time-consuming
Can we improve path delay without significant degrading of cutsize?
Need smooth tradeoff between delay and cutsize
Can we reduce implementation overheads?
Previous methods store thousands of critical paths and continuously update them

7. Outline Motivation Performance driven bipartition problem
New bipartitioning algorithm
Experimental results
Conclusion and future work

8. Delay Model hop Part 0 Part 1 cut [Cong et al. ISPD-2002]
Delay = hop_delay + node_delay
hop
Part 0
Part 1
FF nodes
Combinational nodes
cut
[Cong et al. ISPD-2002]
hop_delay=5
node_delay=1
 Delay = 3x5 + 5x1 = 20
[Ababei et al. ICCAD-2002]
hop_delay=Elmore delay
node_delay=constant

9. Performance Driven Bipartition Problem
Given:
Hypergraph H=(V,E)
Area Balance tolerance s (0<s<1), a parameter to control allowable slack in the area constraint
a, a given parameter which captures tradeoff between cutsize and path delay (hopcount)
Find:
A bipartition (V0|V1) which satisfies:
and minimizes a(cutsize)+(1-a)(Max_hopcount)

10. Outline Motivation Performance driven bipartition problem
 New bipartitioning algorithm
Experimental results
Conclusion and future work

11. Unidirectional Partition
Path delay is minimized with hopcount = 1 if the partition is unidirectional (“acyclic”), that is, all cuts are in the same direction
Part 1
Part 0
Part 0
Problem:
High cutsize
No unidirectional solution
Can we achieve “locally unidirectional” partition?
Max hopcount=5
Max hopcount=3
Part 0
Part 1
Part 0
Part 1

12. V-Shaped Nodes vj vt v V-shaped node
If a combinational node v satisfies:
there exist vj, vt in the other part
and a path from vj to vt that includes only v
then v is a V-shaped node vj vt
Part 0
Part 1 v

13. V-Shaped Nodes in Critical Paths
Empirical observations from study of partitioning solutions:
there are V-shaped nodes in the partitioning solutions
every V-shaped node is included in many critical paths
every critical path contains several V-shaped nodes
For testcase 1:
Number of nets : 16377
Number of critical paths :
On average, one critical path contains 27.6 nodes
On average, one critical path contains 3.4 V-nodes
On average, one V-node belongs to critical paths

14. Key Idea: V-Shaped Nodes Elimination
Part 0 f
Part 0 f a c c a
Move b b
Part 1 b d e d e
Part 1
Move V-shaped node “b” to reduce path hopcount
PATH: abc hopcount=2
PATH: dbc hopcount=1
PATH: ebc hopcount=1
PATH: abc hopcount=0
PATH: dbc hopcount=1
PATH: ebc hopcount=1

15. Distance-k V-Shaped Nodes Elimination
Distance-k V-shaped Nodes (Vk Nodes):
Paths of k combinational nodes with neighbors in the other part.
Part 0
Part 0 d a d a b c
Move b,c
Part 1 b c
Part 1
k = 2: Move V2 node “b, c” reduce path hopcount from 2 to 0
Problems with large k:
Cutsize may be greatly increased

16. New Gain Function v v Gain(v)=δ(0)+ δ(1) After Move Before Move
g(v): traditional FM gain
rj(v): reduction of Vj nodes after moving v

17. Distance-k Unidirectional Algorithm
Calculate initial gains for all nodes and store the gains
Select the node v with maximum gain
/* CLIP-like method: move the cluster that v belongs to */
Reset the gains of all nodes to zero
Move v and update the gains of v and its neighbors
While ( one node not moved)
Select one node v with the maximum updated gain
Move v and update the related gains
Find the point in the move sequence at which the sum of gains is maximum; undo all moves after this point

18. Outline Motivation New bipartitioning algorithm Experimental results
Conclusion and future work

19. Experimental Setup Four industry testcases obtained as LEF/DEF
Model of Ababei et al. (ICCAD-2002) used to calculate delay
Partitioning solutions compared to results of MLPart
strongest multilevel netlist partitioning code
website:
All tests on 600MHz Intel Pentium-III Xeon

20. Biasing against V1 Nodes vs. MLPart
δ(0)=1, δ(1)=10
Testcase
MLPart
MLPart+V-shaped nodes Removal
cutsize h
delay
time(s) 1
820.7
5.3
352.8
11.79
856.1
3.3
266.8
12.58 2
169.9
3.5
220.7
13.45
189.8
2.5
211.2
15.32 3
141.3
291.6
16.67
152.3
2.3
283.6
18.27 4
408.7
302.6
12.43
421.2
3.6
252.7
14.03
Reduction of delay: 4.5%-24.4% average:15.1%
Increase of cutsize: 3.0%-10.0% average: 4.9%
Increase of runtime: 6.3%-11.4% average: 9.7%
Using the delay model in Cong et al. ISPD -2002
Reduction of delay: 4.3%-21.2% average:14.7%

21. Biasing against V2 Nodes vs. MLPart
δ(0)=1, δ(1)=30, δ(2)=3
Testcase
MLPart
MLPart+Vk=2 nodes Removal
cutsize h
delay
time(s) 1
820.7
5.3
352.8
11.79
847.5 3
262.1
13.16 2
169.9
3.5
220.7
13.45
183.2
202.5
15.67
141.3
291.6
16.67
149.2
275.6
18.92 4
408.7
302.6
12.43
416.7
3.4
243.5
14.79
Reduction of delay: 8.9%-30.0% average: 18.7%
Increase of cutsize: 3.1%-7.2% average: 3.5%
Increase of runtime: 11.9%-15.9% average: 13.1%
Using the delay model in Cong et al. ISPD -2002
Reduction of delay: 8.3%-28.7% average: 17.3%

22. Outline Motivation Performance driven bipartition problem
New bipartitioning algorithm
Experimental results
 Conclusions and future work

23. Conclusions
Simple yet efficient timing-driven partitioning that does not require global timing analysis
Negligible implementation, runtime overhead
Significantly reduces path delay with cutsize and runtime almost same as leading-edge MLPart
Similar improvements observed with different path delay metrics
Futures
Impact of new partitioner on placement
Efficient methods for biasing δ(k) k>2

24. Thank you!

25. Future Work Impact of new partitioner on placement
Efficient methods for biasing δ(k) k>2

26. Why Performance Driven Partitioning?
Achieving timing closure becomes increasingly difficult in deep-submicron technologies due to non-ideal scaling of interconnect delay
Routing alone can no longer solve timing problem, even with aggressive optimizations (buffer insertion, buffer/wire sizing,…)
Timing needs to be addressed at all design stages
Partitioning is a critical step in defining interconnect timing properties, but is traditionally driven by cutsize objective

27. Previous Work (I) With Logic Replication Without Logic Replication
Retiming
Replication graph
Without Logic Replication
Net based reweighting
Path based reweighting

28. FM Partitioning and Gain Function
Start with random partition v v
Move the node with the max gain and lock it
Part 0
Before Move
After Move
Part 1
Gain(v)=-1
Gain(v) = Reduction of cutsize
after moving v
Keep moving until all nodes are locked
Part 1
Find the best point in the move sequence
Part 0
Part 0
Part 1

29. Procedure to Calculate rj(v)
Delete all FF nodes and their related edges
In the remaining graph, BFS from v
For each level j from 1 to k
If v is a Vj node before moving, rj’=1
If v is a Vj node after moving, rj’’=1
rj=rj’’-rj’

30. CLIP Algorithm v
CLIP
Reminiscent of CLIP (Deng et al. DAC 1996) in how it
induces movement of clusters across the cutline.

31. Distance-k V-Shaped Nodes
Distance-k V-shaped nodes (Vk-node):
If k combinational nodes vi,1 … vi,k satisfy:
vi,1 … vi,k are in the same part
 vj, vt in the other part
 a path from vj to vt and only passes vi,1 … vi,k
then vi,1 … vi,k are distance-k V-shaped nodes vj vt
Part 0
vi,1
Part 1
vi,k

32. Notation H(V,E)= circuit hypergraph
V = set of nodes representing components of the circuit
E = set of signal nets
A bipartition (V0|V1) of H(V,E) divides V into two disjoint subsets s.t. V= V0V1, which are called Part 0 and Part 1
A = the total area of all the nodes in V
A0 = the area of all the nodes in V0