1. A Parallelization of State-of-the-Art Graph Bisection Algorithms
Nan Dun, Kenjiro Taura, Akinori Yonezawa
Graduate School of Information Science and Technology
The University of Tokyo

2. Problem Description Graph Partition Problem Complexity Solutions
Goal: To minimize cut
K-partition
Bisection (Bipartition)
Problem Complexity
To find best partition or To find approximate partitions: NP-Hard1)2)
Solutions
Heuristics
Non-deterministic
On the Grid
グラフ分割問題
L={1,2,3}
R={4,5,6} 1 1 4 5 2 2 3 6
無向グラフ G=(V,E)が与えられたとき、|L|=|R|を満たすVの分割(L,R)で、LとR間の枝の本数を最小にするものを求める問題。
July 31, Kochi
SWoPP 2006

3. Practical Application
In Mathematics
Analysis of sparse system of linear equations
In Computer Science
Modeling data placement on distributed memory, to minimize communication
In other Various Domains
VLSI Design
Transportation Networks
Communication Networks
July 31, Kochi
SWoPP 2006

4. Bisection Initialization
Bisection Flow
Bisection Initialization
Random Initialization
Half-Half Initialization
Region Growing
Bisection Refinement
Kernighan-Lin3)4)
Tabu Search7)
Fixed Tabu Search
Reactive Tabu Search
Bisection Initialization
Initial Bisection
Bisection Refinement
Final Bisection
July 31, Kochi
SWoPP 2006

5. Min-Max Greedy Growing7)
addset A B A
Max: Breaking ties by maximizing internal connections
Min: Search vertices which cause minimal edge-cut C
July 31, Kochi
SWoPP 2006

6. Swapping Pair of Vertices
Kernighan-Lin3)4) A C
Calculate gain of each vertex
Search a serials of pairs which leads to maximal edge-cut reduction if being swapped
Swap pairs of vertices obtained in 2, lock them from further swap in current pass
Iterate step 1, 2, 3 until edge-cut stops to converge B D
Swapping Pair of Vertices A B C D
gain(B) = -1, gain(C) = -2
ΔCut of swapping B, C = gain(B) + gain(C) + 2 = -1
*gain := # of Internal Edges - # of External Edges
July 31, Kochi
SWoPP 2006

7. Tabu Search7) Kernighan-Lin Like Temporarily Forbidden
Swapping pairs of vertices according to their gains
Temporarily Forbidden
Previously swapped vertices are temporarily forbad to move for a period of time (Tabu Length)
Tabu Length: A fraction (Tabu Fraction) of |V|
E.g.: Tabu Fraction = 0.01, |V| = 1000, Tabu Length = 0.01 x |V| = Previously swapped pairs are allowed to move again after 10 other swaps
To exceed “Local-Minimum”
July 31, Kochi
SWoPP 2006

8. Graph Types – Tabu Lengths
|V| = |E| = Deg: Max 43 Min 3 Avg. 19.8
|V| = |E| = Deg: Max 573 Min 1 Avg. 6.1
Edge-Cut
Tabu Fraction
Number of Vertex Degree
Denser random graphs tend to prefer smaller Tabu lengths, while denser geometric graphs tend to prefer larger tabu lengths8)
Distribution of Vertex Degree
Graphs having uniform distribution of vertex degree tend to have unique fitting tabu length
July 31, Kochi
SWoPP 2006

9. RRTS7) Synthesis of Heuristics Reactive Best Quality Long Running Time
Heuristics perform as complementary for each other
Reactive
Try each Tabu-length to see which is better
Adaptive to various graphs
Best Quality
Beyond “Local-minimum”
Long Running Time
Scoring Phase
REACTIVERANDOMIZEDTABUSEARCH
Scoring each Tabu length by small runs of TS
do I times
Initial bisection by Min-Max
do J times TS with high-scored Tabu length
Refine by Kernighan-Lin runs
R. Battiti and A. A. Bertossi. Greedy, Prohibition, and Reactive Heuristics for Graph Partitioning. IEEE Transactions on Computers, Vol. 48, April 1999.
July 31, Kochi
SWoPP 2006

10. Multi-level for Large Graphs
Coarsen Phase
Coarsen large graphs to smaller one by using “Match Scheme”
Multi-level coarsen
Bisection Phase
Bisecting small graphs is usually very fast
Uncoarsen Phase
Mapping back to original graph
Perform refinement in each uncoarsening phase
METIS5)12)
Matching Scheme
July 31, Kochi
SWoPP 2006

11. Comparison of Heuristics
METIS
RRTS100
FTS10000
cut
time G1
130
0.01
168.11
1.22 G2
366
0.07
353
696.49
354
13.85 G3
311
0.10
935.56
306
32.85 G4
6337
0.04
6257
353.45
6316
3.77 G5
950
0.17
Timeout (1 hour)
929
31.55
Graph
|V|
|E|
Degree
Best Tabu Fraction
Avg
Min
Max
G1:fe_4elt
11143
32818
7.93 15
0.02
G2:fe_pwt
36519
144794
5.89 3 12
G3:fe_body
45087
163734
7.26 28
G4:mem
17758
54196
6.10 1
573
0.14
G5:wing
62032
121544
3.92 2 4
0.01
July 31, Kochi
SWoPP 2006

12. Comparison of Heuristics
METIS
Extremely Fast
Using Multi-level Technique
High-Quality Bisections but worse than RRTS
Multi-level lacks “Global-Optimizing” during coarsen phase
RRTS
Very Slow
Scoring Phase is time costing
“Ever-best” Bisections
Adaptive to kinds of graphs
FTS with Known Tabu-Length
Must faster than RRTS
Comparable result to RRTS
July 31, Kochi
SWoPP 2006

13. A Naive Parallelization
Dispatch Graphs
RRTS100
RRTS100
RRTS100
RRTS100
RRTS100
RRTS100
RRTS100
Synthesize Results
Run RRTS independently on each node
Simply equivalent to scale-up iterations
Generate Different seeds for different nodes
Heuristics are initial sensitive
10% ~ 20% enhanced
July 31, Kochi
SWoPP 2006

14. Statistical Properties of Cut-size
Incidence of Bests
Average quality is good
Only 0.25% is the best
General Property
Distribution becomes “Peak” as |V| grows
Distribution tends towards Gaussian8)
Mean and Variance scales linearly with |V|
Count
Edge-Cut
|V| = |E| = Degree: Max 43 Min 3 Avg 19.80
RRTS100 on 400 nodes provided by Grid Challenge Federation
July 31, Kochi
SWoPP 2006

15. Issues of Parallelizing Heuristics
Hard by Message-Passing Model (MPI)
J.R. Gilbert and E. Zmijewski9): A parallel graph partitioning algorithm for a message-passing multiprocessor. International Journal of Parallel Programming
Par-METIS (Parallel METIS)
Par-METIS only parallelized “coarsen-uncoarsen” part
Hard to Be Efficient (statistic property)
If we could parallelize heuristic efficiently
The fraction of reach the best bisections is still small among overall iterations
If we corporately run independent instance on Grid
How many nodes will leads to best partition
When will a good threshold come
July 31, Kochi
SWoPP 2006

16. Contribution of Phases
Initial Phase
Reduce large portion of Edge-cut
Good initial partitions lead to good final partitions
Consistent time for different running, good initial partitions gain time for refinement
TS and KL Phase
Reductions tend be alike
More iterations, better results
ΔEdge-Cut
Best Edge-Cuts
July 31, Kochi
SWoPP 2006

17. Results from Same Initial Bisections
Given Same Initial Partitions
Best initial partitions leads to best final partitions
FTS and KL tend to be deterministic
Fewer swapping are available
Diversity of edge-cut can be cancelled by distributing only one phase
Run FTS and KL on one node is enough
Count
Perform FTS and KL on same initial partitions, 50 nodes
July 31, Kochi
SWoPP 2006

18. Multi-level Scoring Mainly Used to Adapt Large-Scale Graphs
Edge-Cut
Edge-Cut
Level-1 Tabu Fraction
Level-2 Tabu Fraction
Mainly Used to Adapt Large-Scale Graphs
If |V| = 1000, Tabu = 0.01 x 1000 = 10 If |V| = , Tabu = 0.01 x = 1000
Tuning Tabu-Length to fit specific graphs better
Level-1 Scoring distinguish graphs from their types
Level-2 Scoring test better Tabu-length from specific graphs
July 31, Kochi
SWoPP 2006

19. Final Approaches Not to Use Multi-level Partition
To preserve a “best” quality
Not to Parallelize Heuristics Itself
Not a good trade-off
To Parallelize Scoring Phase
One group of nodes score one tabu length
With multi-level scoring technique
To Parallelize Initial Phase Only
Remove diversity of edge-cut ASAP
Take advantage of running distribution to remove diversity of edge-cut
Reduce computing effort AMAP
Further refinement can be done on single node
To Use GXP Cluster Shell
“mw” command: mw M {{ W }}
July 31, Kochi
SWoPP 2006

20. Full Picture Best Initial Partitions FTS and KL Multi-Level Scoring
High-Scored Level-1 Tabu Fraction
S:0.001
S: 0.002
S: 0.003
S: 0.004
S: 0.005
S: 0.006
S: 0.007
High-Scored Level-2 Tabu Fraction
Initial Phase
Init
Init
Init
Init
Init
Init
Best Initial Partitions
Refinement Phase
FTS and KL
July 31, Kochi
SWoPP 2006

21. Conclusions Bisection Quality Bisection Time
“Ever-Best” partitions
Edge-CutOUR ≤ Edge-CutRRTS≤ Edge-CutMETIS
Bisection Time
Comparable and Reasonable
TimeMETIS < TimeOUR << TimeRRTS
Speed Up 10 comparing to RRTS
Adapted to Grid Environment
Scalable Performance
Convenient usage
Good Fault Tolerant
July 31, Kochi
SWoPP 2006

22. 御静聴ありがとうございました！
July 31, Kochi
SWoPP 2006