Engineering Multilevel Graph Partitioning Algorithms

Transcription

Engineering Multilevel Graph Partitioning
Algorithms
Peter Sanders, Christian Schulz
Institute for Theoretical Computer Science, Algorithmics II
1
Sept. 6, 2011
Vitaly Osipov, Peter Sanders, Christian Schulz – Institute for Theoretical Computer Science, Algorithmics II
KIT – die Kooperation vonEngineering
Forschungszentrum
Karlsruhe
GmbH
und Universität Karlsruhe (TH)
Multilevel Graph
Partitioning
Algorithms
KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universität Karlsruhe (TH)
Overview
Introduction
Multilevel Algorithms
Flow Based Improvement
More Localized FM Search
Global Search
Experiments
Future Work
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Simulation - FEM
Simulation space is
discretized into a mesh
Solution of partitial differential
equations are approximated
by linear equations
Number of vertices can
become quite large → time
and memory
Parallel processing required
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The Common Parallel Approach
Mesh partitioned via dual graph
1. Each volume (data, calculation) is
represented by a vertex
2. Interdependencies are represented by
edges
All PE’s get same amount of work
Communication is expensive
Graph Partitioning Problem:
Partition a graph into (almost) equally sized blocks, such that the
number of edges connecting vertices from different blocks is
minimal.
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e-Balanced Graph Partitioning
Partition graph G = (V , E , c : V → R>0 , ω : E → R>0 )
into k disjoint blocks s.t.
1+e
total node weight of each block ≤
total node weight
k
total weight of cut edges as small as possible
Applications:
finite element simulations, VLSI design, route planning, . . .
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Multi-Level Graph Partitioning
Successful in existing systems:
Metis, Scotch, Jostle,. . . , KaPPa, KaSPar, KaFFPa
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Why Yet Another MGP Algorithms?
scalable parallel algorithm for the case k =# of processors
large inputs
high quality (better (even parallel) than other seq. algorithms)
understand / engineer multi level method
bridge gaps theory ↔ practice
embedding into global search strategies
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Graph Partitioning
Matching Selection
Goals:
1. large edge weights
sparsify
2. large #edges
few levels
3. uniform node weights
“represent” input
4. small node degrees
“represent” input
unclear objective
gap to approx. weighted matching
which only considers 1.,2.
input
graph
...
match
contract
Our Solution:
Apply approx. weighted matching to general edge rating function
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Graph Partitioning
Edge Ratings
ω ({u , v })
ω ({u , v })
c (u ) + c (v )
ω ({u , v })
expansion∗ ({u , v }) :=
c (u )c (v )
expansion({u , v }) :=
ω ({u , v })2
c (u )c (v )
ω ({u , v })
innerOuter({u , v }) :=
Out(v ) + Out(u ) − 2ω (u , v )
expansion∗2 ({u , v }) :=
where c = node weight, ω =edge weight,
Out(u ) := ∑{u ,v }∈E ω ({u , v })
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Global Paths Algorithm
Approx. Weighted Matching
[Maue Sanders 2007]
Sort edges according to their weight/rating
Grow a set of paths and even-length cycles
Find optimum matching for every path and cycle
Running time O (m + sort(m ))
Approximation 12 OPT
outperforms HEM, SHEM
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Graph Partitioning
Initial Partitioning
KaPPa → Compared PARTY, P M ETIS, and S COTCH
S COTCH yielded best results
input
graph
output
partition
...
local improvement
...
match
contract
initial
uncontract
partitioning
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Sequential Graph Partitioning
Karlsruhe Fast Flow Partitioner
Focus: Local and Global Search
Local Search:
1. Flow Based Improvement Methods
2. More Localized FM Algorithm (inspired by KaSPar)
Global Search:
1. V-cycles and F-cycles similar to methods used in multigrid
input
graph
...
G
s
V1
∂1 B
∂2 B
B
t
V2
local improvement
...
...
...
local improvement
match
match
contract
initial
uncontract
contract
uncontract
partitioning
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Flows as Local Improvement
Two Blocks
G
s
V1
∂1 B
∂2 B
B
t
V2
area B, such that each (s, t )-min cut is e-balanced cut in G
e.g. 2 times BFS (left, right)
stop the BFS, if size would exceed (1 + e) n2 − ω (V2 )
⇒ ω (V2new ) ≤ ω (V2 ) + (1 + e) n2 − ω (V2 )
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Two Blocks
G
s
V1
B
t
V2
obtain optimal cut in B
since each cut in B yields a feasable partition
→ improved two-partition
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Adaptive Search
search in larger areas for feasable cuts
adaptively control the size of corridor B
heuristic for Most Balanced Minimum Cuts [Picard et al. 1980]
We need: SCC’s, DAGs, Closed Vertex Sets, Topological Sorting
s
G
V1
t
B
V2
G
s
V1
B
t
V2
the maximal upper bound factor is called α0
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Local Improvement for k -partitions
Using Flows?
on each pair of blocks
input
graph
output
partition
...
local improvement
...
match
contract
initial
uncontract
partitioning
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Local Improvement for k -partitions
Using Flows?
on each pair of blocks
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More Localized Local Search
Idea: KaPPa, KaSPar ⇒ more local searches are better
Typical: k-way local search initialized with complete boundary
Localization:
1. complete boundary ⇒ maintained todo list T
2. initialize search with single node v ∈rnd T
3. iterate until T = ∅
each node moved at most once
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Localization:
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Localization:
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Localization:
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Localization:
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Localization:
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Localization:
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Localization:
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Localization:
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Localization:
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Localization:
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Localization:
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Global Search
Iterated Multilevel [Walshaw 2004]
input
graph
...
local improvement
...
...
...
local improvement
match
match
contract
initial
uncontract
contract
uncontract
partitioning
don’t contract cut edges
no new initial partitioning needed
random tiebreaking needed
local improvement has to assure ≤ cuts
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Global Search
V-Cycles
Coarsening
Uncoarsening
Graph partitioned
Graph not partitioned
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Global Search
W-Cycles
Coarsening
Uncoarsening
Graph partitioned
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Global Search
F-Cycles
Coarsening
Uncoarsening
Graph partitioned
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Experiments
Testset
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Medium sized instances
graph
n
m
rgg17
217
1 457 506
rgg18
218
3 094 566
Delaunay17
217
786 352
Delaunay18
218
1 572 792
bcsstk29
13 992
605 496
4elt
15 606
91 756
fesphere
16 386
98 304
cti
16 840
96 464
memplus
17 758
108 384
cs4
33 499
87 716
pwt
36 519
289 588
bcsstk32
44 609
1 970 092
body
45 087
327 468
t60k
60 005
178 880
wing
62 032
243 088
brack2
62 631
733 118
finan512
74 752
522 240
bel
463 514
1 183 764
nld
893 041
2 279 080
af shell9
504 855 17 084 020
Experimental Evaluation
Flows
Var.
α0
16
8
4
2
1
Ref.
(+F, -MB, -FM )
∆ cut %
t[s]
−1.88 4.17
−2.30 2.11
−4.86 1.24
−11.86 0.90
−19.58 0.76
(-F, -MB, +FM)
(+F, +MB, -FM)
∆ cut %
t[s]
0.81 3.92
0.41 2.07
−2.20 1.29
−9.16 0.96
−17.09 0.80
2 974 1.13
(+F, -MB, +FM)
∆ cut %
t[s]
6.14 4.30
5.99 2.41
5.27 1.62
3.66 1.31
1.64 1.19
final score of different configurations
α0 flow region upper bound factor
all value are improvements rel. to Ref.
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+/- F
+/- MB
+/- FM
(+F, +MB, +FM)
∆ cut %
t[s]
7.21
5.01
7.06
2.72
6.21
1.76
4.17
1.39
1.74
1.22
↔ +/- Flow
↔ +/- Most Bal. H.
↔ +/- FM Algorithm
Flows - Effectiveness
Effectiveness
α0
16
8
4
2
1
(-F, -MB, +FM)
(+F, +MB, -FM)
∆ cut %
−1.29
−1.12
−3.05
−8.26
−16.41
2 833
(+F,-MB, +FM)
∆ cut %
3.70
4.16
4.04
3.02
1.62
2 831
(+F,+MB,+FM)
∆ cut %
4.28
4.74
4.63
3.36
1.65
2 827
each configuration has the same amount of time
all value are improvements rel. to Ref.
+/- F
+/- MB
+/- FM
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↔ +/- Flow
↔ +/- Most Bal. H.
↔ +/- FM Algorithm
Global Search
Algorithm
2 F-cycle
3 V-cycle
2 W-cycle
1 W-cycle
1 F-cycle
2 V-cycle
1 V-cycle
∆ cut %
2.69
2.69
2.91
1.33
1.09
1.88
2 973
t[s]
2.31
2.49
2.77
1.38
1.18
1.67
0.85
Eff. Avg.
2 806
2 810
2 810
2 815
2 816
2 817
2 834
relative fast basic configuration
only global search is varied
red significantly < blue
red and blue significantly < reference
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Experiments
Remove Components
Remove components step by step
Repetition
KaFFPa Strong
-KWay
-MoreLocalizedSearch
-FCycle
-Flow
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Sept. 6, 2011
Avg.
2 683
2 682
2 729
2 748
2 934
t
8.93
9.23
5.55
3.27
1.66
Eff. Avg.
2 636
2 636
2 668
2 669
2 799
Experimental Results
Comparison with Other Systems
Geometric mean, imbalance e = 0.03:
11 graphs (78K–18M nodes) ×k ∈ {2, 4, 8, 16, 64}
Algorithm
KaFFPa strong
KaSPar strong
KaFFPa eco
Scotch
KaFFa fast
kMetis
large graphs
Best
Avg.
t[s]
12 053 12 182 121.22
12 450
+3%
87.12
12 763
+6%
3.82
14 218
+20%
3.55
15 124
+24%
0.98
15 167
+33%
0.83
Repeating Scotch as long as KaSPar strong run and choosing
the best result
12.1% larger cuts
Walshaw instances, road networks, Florida Sparse Matrix
Collection, random Delaunay triangulations, random geometric
graphs
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Outlook
Walshaw Benchmark
runtime is not an issue
614 instances (e ∈ {1%, 3%, 5%})
focus on partition quality
Algorithm/Entries
KaPPa
KaSPar
KaFFPa
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<
131
155
317
≤
189
238
435
Future Work
1.
2.
3.
4.
30
look at initial partitioning
look at large k
look at e = 0
graph clustering
Sept. 6, 2011
Thank you!
Contact:
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Sept. 6, 2011
christian.schulz@kit.edu
sanders@kit.edu

Engineering Multilevel Graph Partitioning Algorithms

Transcription

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