1. Using Multilevel Force-Directed Algorithm to Draw Large Clustered Graph
研究生: 何明彥
指導老師:顏嗣鈞 教授
2018/12/4
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2. OUTLINE Introduction Multilevel Paradigm Experiment result Conclusion
Coarsening Phase
Uncoarsening Phase
Experiment result
Conclusion
Reference
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3. Introduction(1/7) (a) network topology (b) social network
(c) E-R diagram
(d) data flow chart
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4. Introduction(2/7)
Graph is widely-used in computer science recently.
entity Vertex
relationship between entities Edge
ex: 3 4 2 1
0-1,2,3,4
1-0,2,3,4
2-0,1,3,4
3-0,1,2,4
4-0,1,2,3
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5. Introduction(3/7) Good? Aesthetic? crossing symmetric
angular resolution
…….
Graph Drawing
Algorithm
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6. Introduction(4/7) Graph Drawing Algorithm Circular Drawing
Orthogonal Drawing
Straight-Line Drawing
(a) Circular Drawing
(b) Orthogonal Drawing
(c) Straight-Line Drawing
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7. Introduction(5/7) Straight-Line Drawing—Force-Directed Method
Vertex Charged ring
by Newton’s Law,
repulsive force between vertices
Edge Spring
by Hook’s Law,
attractive Force to make vertices closer
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8. Introduction(6/7)
Clustering
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9. Introduction(7/7)
Usually, Circular drawing can express the aesthetic goal of embedded cluster.
Embedded cluster is important.
ex: hosts in which LAN for network topology;
people in which group for social network
try to using Straight-Line Drawing to achieve it.
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10. Multilevel Paradigm(1/4)
Two parts:
Graph Coarsening:
Partition vertices to form cluster and define a new graph
Graph Uncoarsening:
To redraw original graph
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11. Multilevel Paradigm(2/4) Graph Coarsening
given graph GL
Level-L
graph partitioning
method
Coarser graph GL+1
Level-L+1
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12. Multilevel Paradigm(3/4) Graph Uncoarsening
Level-L+1
given graph GL+1
uncoarsen GL+1; use spring algorithm to draw GL
Level-L
drawing graph GL
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13. Multilevel Paradigm(4/4)
(a) original graph
(b) group vertices
(c) coarser graph
(d) uncoarsening stage
(e) final result
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14. Coarsening Phase(1/12)
Partition vertices to form cluster and define a new graph
K-way partitioning:
The number of edges of E whose incident vertices belong to different subsets is minimized.
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15. Coarsening Phase(2/12) Graph Partitioning
The number of edges of E whose incident vertices belong to different subsets is so-called
edge-cut and should be minimized. b f a g c d h e a a f b d b g c f h c g e e d h
Edge-cut=5
Edge-cut=1
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16. Coarsening Phase(3/12) Two stage Matching stage Contraction stage
Using random matching
Contraction stage
A coarser graph is created by contracting former matching vertex
Using Kernighan-Lin algorithm
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17. Algorithm: Random matching (Go) Input:G0
Output: Match [u] and Map [u];
Match [u] store the vertex which is matched by u
Map [u] store the label of v in the coarser graph
begin
visit vertex in random order
If vertex u has not been matched yet then
random select one of its unmatched adjacent vertices v
If v exist then
put v in match [u]
map [u]=map [v]
else match [u]=u
end
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18. Coarsening Phase(5/12)
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19. Properties of K-L Algo.
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20. A B a x b a b A B b x a B A
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21. if Gj>0 then Algorithm: Kernighan-Lin Algorithm (G) Input: G=(V,E)
Output: partition A and B with small edge-cut
begin
partition G into A and B
repeat
compute Dv,
for i=1 t0 n/2 do
find an unmarked pair (ai, bi), that maximizing gab
mark ai, bi
update Dv for all unmarked
find j, such that Gj= is maximized
if Gj>0 then
move a1,…, aj from A to B
move b1,…,bj from B to A
until
end
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22. a d b c e f a b c d e f 1 4 3 2 Initial edge cut=22 Random matching
For “a”: a d b c e f 1 2 4 3
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23. a d b c e f f b a c d e f b a c d e f b e a d
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24. f b Iteration 1 f b a c d e Iteration 2
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25. 2018/12/4
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26. Uncoarsening Phase(1/4)
Two stage
Mapping stage
Put nodes in a region
Refinement stage
Using FR force-directed method
Using Virtual spring for drawing cluster
embedded clustered graph
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27. Algorithm: Uncoarening Input: coarser graph Gi
Output: original graph G0
Begin
using Force-directed method to draw Gi
For each vertex u in Gi
if (vertex v in Gi-1 is coarsened into u)
place v in a region of u
set u as the virtual node
again using spring method to draw original graph Go
end
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28. Virtual-Spring to make nodes in cluster close together
internal-spring:
a spring force between a pair
node which belong to a cluster a b
external-spring:
a spring force between a pair
node which not belong to a cluster c d
virtual node:
each cluster has a virtual node
Create virtual spring e g f
virtual spring:
a spring force between a pair
node along the virtual edge h
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29. Experiment Result(1/6)
|V|=25; |E|=60
Original:
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30. Experiment Result(2/6)
Spring Algorithm
result:
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31. Experiment Result(3/6)
|Cluster|=5
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32. Experiment Result(4/6)
|V|=42; |E|=75
Original:
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33. Experiment Result(5/6)
Spring Algorithm
result:
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34. Experiment Result(6/6)
|Cluster|=6
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35. Conclusion It is suitable for drawing large graph
Multilevel algorithm is fast and no local minimum.
K-L algorithm works well for graph partitioning.
Embedded cluster graph is available.
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36. Reference
Chris Walshaw, ”A multilevel algorithm for Force-Directed Graph Drawing”. Journal of Graph Algorithm and Applications, vol7,no.3,pp (2003)
J. Chuang, C.Lin, and H. Yen, "Drawing Graphs with Nonuniform Nodes Using Potential Fields," in the Proceedings of the 11th International Symposium on Graph Drawing 2003,(LNCS 2912) pp , Perugia, Italy, Sept , 2003
Peter Eades, Mao Lin Huang, “Navigating Cluster Graphs using Force-Directed Methods “, Journal of Graph Algorithm and Applications(2000)
George Karypis and Vipin Kumar ,”A Fast and High Quality Multilevel Scheme for Partitioning Irregular” .SIAM of Journal on Scientific Computing
J.Fruchterman, M.Reingold, “Graph Drawing by Force-Directed Placement”, 1991
B.W.Kernighan, S.Lin, “An efficient heuristic procedure for partitioning graphs.” The Bell System Technical Journal
S.Hachul and M.Junger, ”Drawing Large graphs with a potential-field-based multilevel algorithm” GD2004
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