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a b c d e f g h i (a) (b) a b c d e f g h i (c) a b c d e f g h i CCCG 2014August 11, a b c d e f g h i A segment is maximal path P such that the vertices on P are collinear in the drawing.
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CCCG 2014 August 11, 2014
Drawing Plane Triangulations with Few Segments
Department of Computer ScienceUniversity of Manitoba
Stephane Durocher Debajyoti Mondal
2
k-Segment Drawings
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(a) (b)
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(c)
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CCCG 2014
August 11, 2014
3
k-Segment Drawings
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(a) (b)
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(c)
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CCCG 2014
August 11, 2014
a b
cdef
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iA segment is maximal path P such that the vertices on P
are collinear in the drawing.
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k-Segment Drawings
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def
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(a) (b)
a b
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(c)
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CCCG 2014
August 11, 2014
A 8-segment drawingA 10-segment drawing
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k-Segment Drawings
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def
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(a) (b)
a b
cdef
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(c)
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ef
gh
i
CCCG 2014
August 11, 2014
A 8-segment drawingA 10-segment drawing
Minimization is NP-complete [Durocher, Mondal, Nishat,
and Whitesides, CCCG 2011]
6
Previous Results
CCCG 2014
August 11, 2014
Graph Class Lower Bounds Upper Bounds References
Trees | {v: deg(v) is odd} | / 2 | {v: deg(v) is odd} | / 2Dujmović, Eppstein, Suderman and Wood,
CGTA 2007Maximal Outerplanar n n
Plane 2-Trees and 3-Trees 2n 2n
3-Connected Cubic Plane n/2+3 n/2+4Mondal, Nishat,
Biswas, and Rahman, JOCO 2010
3-Connected Plane 2n 5n/2 (= 2.50n)Dujmović, Eppstein, Suderman and Wood,
CGTA 2007
Triangulations 2n 7n/3 (= 2.33n) This Presentation
4-Conneted Triangulations 2n 9n/4 (= 2.25n) This Presentation
7
Better Upper Bound for Triangulations
CCCG 2014
August 11, 2014
a
b
c
de
f
g
h
i
j
k
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Every rooted tree T has an upward drawing with leaf (T) segments.
Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees
8
Better Upper Bound for Triangulations
CCCG 2014
August 11, 2014
a
b
c
de
f
g
h
i
j
k
l
Every rooted tree T has an upward drawing with leaf (T) segments.
a
b
c
Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees
9
Better Upper Bound for Triangulations
CCCG 2014
August 11, 2014
a
b
c
de
f
g
h
i
j
k
l
Every rooted tree T has an upward drawing with leaf (T) segments.
a
b
cd
Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees
10
Better Upper Bound for Triangulations
CCCG 2014
August 11, 2014
a
b
c
de
f
g
h
i
j
k
l
Every rooted tree T has an upward drawing with leaf (T) segments.
a
b
cd
e
Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees
11
Better Upper Bound for Triangulations
CCCG 2014
August 11, 2014
a
b
c
de
f
g
h
i
j
k
l
Every rooted tree T has an upward drawing with leaf (T) segments.
a
b
cd
e h
f
g
Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees
12
Better Upper Bound for Triangulations
CCCG 2014
August 11, 2014
a
b
c
de
f
g
h
i
j
k
l
Every rooted tree T has an upward drawing with leaf (T) segments.
a
b
cd
e h
f
g
ij
Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees
13
Better Upper Bound for Triangulations
CCCG 2014
August 11, 2014
a
b
c
de
f
g
h
i
j
k
l
Every rooted tree T has an upward drawing with leaf (T) segments.
a
b
cd
e h
f
g
ij
k
l
Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees
14
Better Upper Bound for Triangulations
CCCG 2014
August 11, 2014
a
b
c
de
f
g
h
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Whenever we create a new segment, we ensure that this new segment starts at a non-leaf vertex T and ends at a leaf of T The drawing has leaf (T) segments.
a
b
cd
e h
f
g
ij
k
l
Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees
15
Better Upper Bound for Triangulations
CCCG 2014
August 11, 2014
Whenever we create a new segment, we ensure that this new segment starts at a non-leaf vertex T and ends at a leaf of T The drawing has leaf (T) segments.
a
b
cd
e h
f
g
ij
k
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a
b
cd h
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ij
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Divergence: downward extension of the segments does not create edge
crossings.
Idea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees
Better Upper Bound for TriangulationsIdea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees
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v6 v7
v1 v2
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v6 v7
v8
v1 v2
v3
v4v5
v6 v7
v8
G3
G6 G7 G8
G4 G5
A Canonical Ordering of G[De Fraysseix, Pach, and Pollack 1988]
Better Upper Bound for TriangulationsIdea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees
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v1 v2
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v1 v2
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v6 v7
v1 v2
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v4v5
v6 v7
v8
v1 v2
v3
v4v5
v6 v7
v8
G3
G6 G7 G8
G4 G5
A Canonical Ordering of G[De Fraysseix, Pach, and Pollack 1988]
Better Upper Bound for TriangulationsIdea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees
v1 v2
v3
v4v5
v6 v7
v8
v1 v2
v3
v4v5
v6 v7
v8
A Canonical Ordering of G[De Fraysseix et al. 1988]
v1v2
v3
v4v5
v6
v8
v1 v2
v3
v4v5
v6 v7
v8
v1v2
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v4v5
v6 v7
v8
A Schnyder realizer of G [Schnyder 1990]
Better Upper Bound for TriangulationsIdea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees
v1 v2
v3
v4v5
v6 v7
v8
v1 v2
v3
v4v5
v6 v7
v8
A Canonical Ordering of G[De Fraysseix et al. 1988]
v1v2
v3
v4v5
v6
v8
v1 v2
v3
v4v5
v6 v7
v8
v1v2
v3
v4v5
v6 v7
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A Schnyder realizer of G [Schnyder 1990]
leaf (Tl) =3v7
leaf (Tr) =3
Tm
20CCCG 2014
August 11, 2014
Better Upper Bound for TriangulationsIdea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees
Draw G with at most leaf (Tl) + leaf (Tr) + n segments
v1 v2
v3
v4v5
v6 v7
v8
Better Upper Bound for TriangulationsIdea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees
Draw G with at most leaf (Tl) + leaf (Tr) + n segments
v1 v2
v3
v4v5
v6 v7
v8
Incremental construction in canonical orderwhile maintaining nice drawings of the subtrees
v1 v2
v3
Better Upper Bound for TriangulationsIdea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees
Draw G with at most leaf (Tl) + leaf (Tr) + n segments
v1 v2
v3
v4v5
v6 v7
v8
v1 v2
v3
v1 v2
v3v4
Incremental construction in canonical orderwhile maintaining nice drawings of the subtrees
Better Upper Bound for TriangulationsIdea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees
Draw G with at most leaf (Tl) + leaf (Tr) + n segments
v1 v2
v3
v4v5
v6 v7
v8
v1 v2
v3
v4
v1 v2
v3
v4
v5
Incremental construction in canonical orderwhile maintaining nice drawings of the subtrees
Better Upper Bound for TriangulationsIdea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees
Draw G with at most leaf (Tl) + leaf (Tr) + n segments
v1 v2
v3
v4v5
v6 v7
v8
v1 v2
v3
v4
v5
Incremental construction in canonical orderwhile maintaining nice drawings of the subtrees
v1 v2
v3
v4
v5
v6
Better Upper Bound for TriangulationsIdea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees
Draw G with at most leaf (Tl) + leaf (Tr) + n segments
v1 v2
v3
v4v5
v6 v7
v8
v1 v2
v3
v4
v5
Incremental construction in canonical orderwhile maintaining nice drawings of the subtrees
v1
v6
v2
v3
v4
v5
v6 v7
Better Upper Bound for TriangulationsIdea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees
Draw G with at most leaf (Tl) + leaf (Tr) + n segments
v1 v2
v3
v4v5
v6 v7
v8
v1 v2
v3
v4
v5
Incremental construction in canonical orderwhile maintaining nice drawings of the subtrees
v6v7
v1 v2
v3
v4
v5
v6
v7
v8
leaf (Tl) + leaf (Tr) + 3 segments
Better Upper Bound for TriangulationsIdea: Nice Drawings of Trees + Decomposition of a Triangulation into Trees
Draw G with at most leaf (Tl) + leaf (Tr) + n segments
v1 v2
v3
v4v5
v6 v7
v8
v1 v2
v3
v4
v5
Incremental construction in canonical orderwhile maintaining nice drawings of the subtrees
v6v7
v1 v2
v3
v4
v5
v6
v7
v8
leaf (Tl) + leaf (Tr) + 3 segments+ at most (n-3) segments
28CCCG 2014
August 11, 2014
Why Does this Work?
v1 v2
v3
v4v5
v6 v7
v8
v1 v2
v3
v4
v5
v6v7
v8
Question 1. What makes it possible to maintain nice drawings of the subtrees?
- We can always create a new segment satisfying the ‘divergence’ property.
A triangulation G and A drawing of G with at most leaf (Tl) + leaf (Tr) + n segments
v
p q
29CCCG 2014
August 11, 2014
Why Does this Work?
v1 v2
v3
v4v5
v6 v7
v8
v1 v2
v3
v4
v5
v6v7
v8
A triangulation G and A drawing of G with at most leaf (Tl) + leaf (Tr) + n segments
v
Question 1. What makes it possible to maintain nice drawings of the subtrees?
- We can always create a new segment satisfying the ‘divergence’ property.
p qp
q
Why Does this Work?
v1 v2
v3
v4v5
v6 v7
v8
v1 v2
v3
v4
v5
v6v7
v8
A triangulation G and A drawing of G with at most leaf (Tl) + leaf (Tr) + n segments
v
p qp q
vQuestion 2. Why the drawing of the edges in Tm does not create any edge crossing?
- The slopes of the l-edges incident to the outerface are smaller than the slope of edge (v, p).- The slopes of the r-edges incident to the outerface are larger than the slope of edge (v, q).
31
Final Upper Bounds
CCCG 2014
August 11, 2014
Graph Class Lower Bounds Upper Bounds References
Trees | {v: deg(v) is odd} | / 2 | {v: deg(v) is odd} | / 2 Dujmović, Eppstein,
Suderman and Wood, CGTA 2007Maximal Outerplanar n n
Plane 2-Trees and 3-Trees 2n 2n Samee, Alam, Adnan
and Rahman, GD 2008
3-Connected Cubic Plane Graphs n/2 n/2
Mondal, Nishat, Biswas, and Rahman,
JOCO 2010
3-Connected Plane Graphs 2n 5n/2 (= 2.50n)
Dujmović, Eppstein, Suderman and Wood,
CGTA 2007
Triangulations 2n leaf (Tl) + leaf (Tr) + n <= 2.33n This Presentation
4-Conneted Triangulations 2n leaf (Tl) + leaf (Tr) + n <= 2.25n This Presentation
Combine the upper bounds on the number of leaves
[Bonichon, Saëc and Mosbah, ICALP 2002]
[Zhang and He, DCG 2005]
32
Future Research
CCCG 2014
August 11, 2014
Tight Bounds: What is the smallest constant c such that every n vertex planar graph admits a (cn)-segment drawing? Can we improve the bound in the variable embedding setting?
Generalization: Does the upper bound of 7n/3 segments hold also for 3-connected planar graphs?
Optimization: Is there a polynomial-time algorithm for computing minimum-segment drawings of triangulations, or simpler classes of graphs such as plane 3-trees or outerplanar graphs?
Thank You..
v1 v2
v3
v4v5
v6 v7
v8
v1 v2
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v5
v6v7
v8
a
b
cd h
f
g
ij
k
le
Tight Bounds?
Generalization?
Optimization?
