Rectilinear Motion Revisited Objective: We will look at rectilinear motion using Integration.
The Rectilinear Symmetric Crossing Minimization Problem Seokhee Hong The University of Sydney.
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Transcript of The Rectilinear Symmetric Crossing Minimization Problem Seokhee Hong The University of Sydney.
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The Rectilinear Symmetric Crossing Minimization Problem
Seokhee Hong
The University of Sydney
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1. Introduction
Graph drawing algorithm aims to construct geometric representations of graphs in 2D and 3D.
A - B, C, DB - A, C, DC - A, B, D, ED - A, B, C, EE - C, D
The input is a graph with no geometry
A B
D
C
E
The output is a drawing of the graph; the drawing should be easy to understand, easy to remember, beautiful.
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Aesthetics in Graph Drawing
NP-hardness
• minimize edge crossings
• minimize area
• maximize symmetry
• minimize total edge length
• minimize number of bends
• angular resolution
• aspect ratio
• ……
Conflict
Minimize edge crossings
Maximize symmetry
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• To construct symmetric drawings of graphs,
we need to solve two problems.
problem 1. find symmetry in graphs.
• symmetry finding algorithm
problem 2. display the symmetry in a drawing.
• symmetric drawing algorithm
• This talk: we want to construct a straight-line drawing which displays given symmetries with minimum number of edge crossings.
Symmetric Graph Drawing
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Types of Symmetric Drawing in 2D
Types of symmetry in 2D
1. Rotation
2. Reflection
Types of symmetric drawing in 2D
1. Cyclic group of size k: k rotations
2. Dihedral group of size 2k: k rotations and k reflections
3. One reflection
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Example: The Peterson Graph
dihedral group(size 10)• 5 rotations• 5 reflections
5 edge crossings
dihedral group(size 6)• 3 rotations• 3 reflections
3 edge crossings
1 reflection2 crossings
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The Crossing Minimization Problem
A great deal of literature• Mathematicians: Crossing number [Pach]• Graph Drawers : Two layer crossing minimization [the book]
• [Buchheim & Hong 02]: crossing minimization for given symmetries with curve
The Symmetric Crossing Minimization Problem
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8
• An automorphism of a graph G is a permutation of the vertex set which preserves adjacency.– Automorphism problem: isomorphism-complete
2. Background
• [Eades & Lin 00]
geometric automorphism is an automorphism which can be displayed as a symmetry of a drawing D of G.– Geometric automorphism problem: NP-complete
1
5
43
2 1 2 3
4 5
(12)(5)(34) (123)(45)
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9
Geometric Automorphism Group
[Eades & Lin 00]
a group of geometric automorphisms which can be displayed as symmetries of a single drawing D of G.
1
2 3
41 2
3 4
(1234)(123)(4)
Symmetry finding step: find geometric automorphism group of G with maximum size
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Symmetry Finding Algorithm
General graph• NP-hard
• Heuristics
• Exact algorithm: ILP• Exact algorithm: MAGMA
[Manning 90]
[Lipton,North,Sandberg85]
[de Fraysseix 99]
[Buchheim, Junger 01]
[Abelson,Hong,Taylor02]
[Manning, Atallah 88]
[Manning, Atallah 92]
[Manning 90]
[Hong, Eades, Lee 98]
[Hong, Eades 01]
Planar graph• Tree• Outerplanar graph• Plane graph• Series parallel digraph• Planar graph• triconnected planar graph
2D 3D
[Hong, Eades 00]
[Hong, Eades 00]
[Hong01]
[Hong01]
[Abelson, Hong,
Taylor 02]
N/A
N/A
[Hong, McKay,Eades02]
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• Step1: symmetry finding algorithm– orbit: partition of vertex set V under the geometric
automorphism group– For each vertex v, the orbit of v under the group H is
the set { u V: u = p(v) for some p H}.• Step 2: drawing algorithm: [Eades & Lin 00]
– draw each orbit in a concentric circle• different drawings depending on the order of orbits
Output of [Abelson, Hong, Taylor 02]
8
1
2
3
4
5
6
7 9O1: {1}O2: {2,3,4,5}O3: {6,7,8,9}
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Dodecahedron: dihedral 2D
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4-cube: cyclic 2D
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Fourcell: dihedral 2D|V| = 120, |E| = 1440
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PSP44: cyclic 2D|V|=85, |E|=1700
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[Buchheim and Hong02]
Symmetric Crossing Minimization Problem (SCM)
Input: A graph G, and a geometric automorphism p.
Output: A drawing of G which displays p with the minimum number of edge crossings.
[Theorem] The problem SCM is NP-hard.
Symmetric Crossing Minimization Problem
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3. Rectilinear Symmetric Crossing Minimization
Rectilinear Symmetric Crossing Minimization Problem: (REC-SCM)
Input: A graph G, and a geometric automorphism group H.
Output: A straight-line drawing of G which displays H with the minimum number of edge crossings.
[Theorem] The problem REC-SCM is NP-hard.
<proof> We divide into three cases.
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Cyclic Rectilinear Symmetric Crossing Minimization Problem: REC-SCM-Cyclic
Input: A graph G and a cyclic geometric automorphism group C.
Output: A straight-line drawing of G which displays C with the minimum number of edge crossings.
Dihedral Rectilinear Symmetric Crossing Minimization Problem: REC-SCM-Dihedral
Input: A graph G and a dihedral geometric automorphism group D.
Output: A straight-line drawing of G which displays D with the minimum number of edge crossings.
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Axial Rectilinear Symmetric Crossing Minimization Problem: REC-SCM-Axial
Input: A graph G and an axial geometric automorphism group A.
Output: A straight-line drawing of G which displays A with the minimum number of edge crossings.
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[Theorem] The problems REC-SCM-Cyclic, REC-SCM-Dihedral and REC-SCM-Axial are NP-hard.
<proof>• reduce the rectilinear crossing number problem, which
is NP-hard[Bienstock93] to REC-SCM problem.• use the similar argument from [Buchheim & Hong02].• main idea: define a function using disconnected
isomorphic graphs.
G
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SCM : REC-SCM
• symmetric crossing number SCR(G,H): the smallest number of the edge crossings of a drawing of the graph G which displays the geometric automorphism group H.
• rectilinear symmetric crossing number REC-SCR(G,H): the smallest number of the edge crossings of a straight line drawing of the graph G which displays the geometric automorphism group H.
• [Lemma] SCR(G,H) REC-SCR(G,H)
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Example: SCR(K6,H) REC-SCR(K6,H)
Two symmetric drawings of K6 displaying a dihedral group H of size 12.
SCR(K6, H) = 9 REC-SCR(K6,H) = 15
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The Orbit Graphs• intra-orbit edge: if e connects two vertices in the same orbit. • inter-orbit edge: if e connects two vertices in two different
orbits.• The orbit graph G_H of G under H: a graph whose vertices
represent each orbit and edges represent the set of inter-orbit edges.
1
2
3
4
5
6
7 9
8
6,7,8,9
2,3,4,5
1
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Choosing a Representation
• consider a graph with a single orbit.• different drawings displaying a given geometric
automorphism group H, depending on the choice of the representation.
• example: H = <p> = <(12345)>
1
2
3
45
1
2 3
4 5
1
234
5
1
2
3 4
5
p p2 p3 p4
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More than Two Orbits• different drawings displaying a given geometric
automorphism group H, depending on the ordering of orbits.• example: two drawings displaying dihedral group of size 10
5 crossings 10 crossings
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Heuristics for REC-SCM
• the main task: to decide the ordering of orbits to minimize edge crossings.
• The REC-SCM problem is geometric, rather than combinatorial, in contrast to the SCM problem.
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10 crossings 15 crossings
Example: the radius makes a difference for REC-SCM.
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4. Work in Progress
• SCM problem: evaluation of the general framework for SCM of [Buchheim and Hong 02]
• REC-SCM problem: – design and implement good heuristics (approximation
algorithms) to compute the ordering of the orbit graph. – variations of heuristics for the minimum linear
arrangement problem– variations of two layer crossing minimization methods– soft computing method: simulated annealing
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Preliminary results
• Preliminary results showed that for the REC-SCM problem, simple heuristics perform fairly well for reasonably-sized graphs. – the size of the orbit graph is relatively small
compare to the size of the graph G.
• Observations:– there is a trade off between the quality and the run
time in general. – For dense graphs, the resulting drawings do not
appear different to human eyes. – For sparse graphs, it is easier to compare the
result instantly.
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Future Work
• So far, we mainly focused on displaying the cyclic group.
• Further work includes the axial case and the dihedral
case.
• However, this looks challenging, because of the existence of edges that are fixed by the axial symmetry.
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5. Open Problems• Find a tight lower bound of the SCR and REC-SCR:
– Intuitively, a drawing that displays a larger geometric automorphism group needs more crossings.
• Investigate the relationship between the crossing numbers such as CR, SCR, REC-CR, REC-SCR:
– CR(G) REC-CR(G)
– SCR(G,H) REC-SCR(G,H)
– CR(G) SCR(G,H) : CR(G) = SCR(G,H), where H is a trivial automorphism group
– REC-CR(G) REC-SCR(G,H)
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5. Open Problems
• Compute the REC-SCR for Kn displaying k symmetries:
– If H is a dihedral group of size 2n (i.e. maximum size geometric automorphism group), then there are n(n-1)(n-2)(n-3)/24 edge crossings.
– What if we display fewer symmetries?