GD 2017 September 25, 2017
Debajyoti Mondal
Department of Computer Science
University of Saskatchewan, Canada
Therese Biedl
Cheriton School of Computer Science
University of Waterloo, Canada
2GD 2017 September 25, 2017
G A planar drawing
on a 4×4 grid.
[Krug and Wagner 2008, Biedl 2014]
NP-hard for
• Arbitrary planar graphs
• Planar graphs with bounded pathwidth
• Outerplanar graphs
[M et al. 2008, Biedl 2014]
Polynomial time for
• Planar 3-trees
• Planar graphs with both bounded
treewidth and bounded face-degrees
3GD 2017 September 25, 2017
T
[Alam et al. 2008, M et al 2011, Biedl 2015]
Minimizing one dimension
• Upward drawings (rooted trees)
• Strictly-upward drawings (rooted trees)
• Non-upward drawings, height (unrooted trees)
Minimizing
Height
Minimizing
Width
4GD 2017 September 25, 2017
[Bhatt and Cosmadakis 1987, Gregori 1989, Brunner
and Matzeder 2010, Bachmaier and Matzeder 2013]
NP-hard for
• Drawing ordered/unordered trees on a k-grid, k ∈ {4,
6, 8}, with unit edge length
[Supowit and Reingold 1982]
NP-hard for
• Ordered trees - under constraints such as isomorphic
subtrees must be drawn identically, left and right child
must be placed on the left and right of their parent
T
5GD 2017 September 25, 2017
Straight-line drawings of Trees?
Upward straight-line drawings of rooted Trees?
Strictly upward straight-line drawings of rooted Trees?
Strictly upward straight-line drawings of ordered rooted Trees?
T
6GD 2017 September 25, 2017
T
r
r
a
b c
d
e
f
g
a c b
f
g
Does T admit a strictly-
upward drawing
on the given grid?
d
e
A strictly-upward drawing of T
• Straight-line planar drawing
• Every child is drawn strictly below to its parent
• The ordering of the children can be chosen
A given grid
7GD 2017 September 25, 2017
A reduction from Numerical 3-Dimensional Matching (N3DM)
• Instance: Positive integers ri , gi , bi , where 1 ≤ i ≤ k, and an integer B such
that Σi (ri + bi + gi) = k · B.
• Question: Do there exist permutations π and π/ of {1, . . . , k} such that
rπ(i) + bi + gπ/(i) = B for all 1 ≤ i ≤ k?
ri
524
bi
577
gi
327
ri
254
bi
577
gi
723
Σi (ri + bi + gi) = k · B
42 = 3 · 14
+
+
+
+
+
+
=
=
=
14
14
14
8GD 2017 September 25, 2017
Remains NP-hard under the following restrictions:
• gis are huge ∈ O(k4c)
• bis are odd and large ∈ O(k2c)
• ris are small ∈ O(kc)
A reduction from Numerical 3-Dimensional Matching (N3DM)
• Instance: Positive integers ri , gi , bi , where 1 ≤ i ≤ k, and an integer B such
that Σi (ri + bi + gi) = k · B.
• Question: Do there exist permutations π and π/ of {1, . . . , k} such that
rπ(i) + bi + gπ/(i) = B for all 1 ≤ i ≤ k?
9GD 2017 September 25, 2017
ri
524
bi
577
gi
327
B+2
Supporting paths
10GD 2017 September 25, 2017
ri
524
bi
577
gi
327
Spinal path
GD 2017
ri
524
bi
577
gi
327
11 September 25, 2017
Add wall vertices
GD 2017
ri
524
bi
577
gi
327
12 September 25, 2017
GD 2017
ri
524
bi
577
gi
327
13 September 25, 2017
GD 2017
ri
524
bi
577
gi
327
14 September 25, 2017
I has an affirmative solution if and only if
T admits a drawing on a (B+4)×(2k+3) grid
I
T
GD 2017 15 September 25, 2017
ri
524
bi
577
gi
327
ri
254
bi
577
gi
723
+
+
+
+
+
+
=
=
=
14
14
14
B
B+4
2k+3
GD 2017 16 September 25, 2017
ri
524
bi
577
gi
327
ri
254
bi
577
gi
723
+
+
+
+
+
+
=
=
=
14
14
14
B
B+4
2k+3
GD 2017 17 September 25, 2017
ri
524
bi
577
gi
327
ri
254
bi
577
gi
723
+
+
+
+
+
+
=
=
=
14
14
14
B
B+4
2k+3
GD 2017 18 September 25, 2017
ri
524
bi
577
gi
327
ri
254
bi
577
gi
723
+
+
+
+
+
+
=
=
=
14
14
14
B
B+4
2k+3
All the rows (except the topmost row)
must be completely used up by the nodes of the tree.
GD 2017 19 September 25, 2017
ri
254
bi
577
gi
723
+
+
+
+
+
+
=
=
=
14
14
14
B
B+4
2k+3
v1
The bottommost layer contains
• two vertices from the supporting paths and
• Wall vertices of the bottommost wall parent.
GD 2017 20 September 25, 2017
ri
254
bi
577
gi
723
+
+
+
+
+
+
=
=
=
14
14
14
B
B+4
2k+3
v1
The next wall parent (w3)
• can have at most two children on l8 and
• all the remaining children lie consecutively on l7
w4
w3
w2
w1
l8
l6
l7
l9
l1
GD 2017 21 September 25, 2017
ri
254
bi
577
gi
723
+
+
+
+
+
+
=
=
=
14
14
14
B
B+4
2k+3
v1
w3
w2
w1
The next wall parent (w3)
• can have at most two children on l8 and
• all the remaining children lie consecutively on l7
l8
l6
l7
l9
GD 2017 22 September 25, 2017
ri
254
bi
577
gi
723
+
+
+
+
+
+
=
=
=
14
14
14
B
B+4
2k+3
v1
???
w3
w2
w1
The next wall parent (w3)
• can have at most two children on l8 and
• all the remaining children lie consecutively on l7
l8
l6
l7
l9
GD 2017 23 September 25, 2017
ri
254
bi
577
gi
723
+
+
+
+
+
+
=
=
=
14
14
14
B
B+4
2k+3
v1
w3
w2
w1
The next wall parent (w3)
• can have at most two children on l8 and
• all the remaining children lie consecutively on l7
l8
l6
l7
l9
GD 2017 24 September 25, 2017
ri
254
bi
577
gi
723
+
+
+
+
+
+
=
=
=
14
14
14
B
B+4
2k+3
v1
w3
w2
w1
The remaining space is too large for two green stars.
l8
l6
l7
l9
GD 2017 25 September 25, 2017
ri
254
bi
577
gi
723
+
+
+
+
+
+
=
=
=
14
14
14
B
B+4
2k+3
v1
w3
w2
w1
The remaining space is too large for two green stars,
and placing two red stars would violate planarity.
l8
l6
l7
l9
GD 2017 26 September 25, 2017
ri
254
bi
577
gi
723
+
+
+
+
+
+
=
=
=
14
14
14
B
B+4
2k+3
v1
w3
w2
w1
The remaining space is too large for two green stars,
and placing two red stars would violate planarity.
l8
l6
l7
l9
GD 2017 27 September 25, 2017
ri
254
bi
577
gi
723
+
+
+
+
+
+
=
=
=
14
14
14
B
B+4
2k+3
v1
w3
w2
w1
The remaining space is too large for two green stars,
and placing two red stars would violate planarity.
Since all grid points must be used up, there can be exactly
one green and one red star defining a triple that sum to B.
l8
l6
l7
l9
GD 2017 September 25, 201728
Straight-line drawings of Trees?
Upward straight-line drawings of rooted Trees?
Strictly upward straight-line drawings of rooted Trees?
Strictly upward straight-line drawings of ordered rooted Trees?
ri
254
bi
577
gi
723
+
+
+
+
+
+
=
=
=
14
14
14
B
ri
524
bi
577
gi
327