complexity results for threedimensional orthogonal graph drawing

1
complexity results for three-dimensional
orthogonal graph drawing

• maurizio patrignani
• third university of rome
• graph drawing
• dagstuhl 05191-2005

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three-dimensional orthogonal GD

• nodes are (distinct) points in 3d space
• edges are composed by sequences of axis-parallel
  segments

• only degree six graphs admit such drawings

3
what we know (1)

• volume is ?(n3/2)
• rosenberg. three-dimensional vlsi a case study.
  j acm 1983
• volume is ?(n3/2)
• eades, stirk, and whitesides, the techniques of
  komolgorov and bardzin for three-dimensional
  orthogonal graph drawings. ipl 96
• up to 16 bends per edge in time
• eades, symvonis, and whitesides,
  three-dimensional orthogonal graph drawing
  algorithms. discr. appl. math 2000
• up to 7 bends per edge in time

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what we know (2)

• if only three bends per edges are allowed
• eades, symvonis, and whitesides,
  three-dimensional orthogonal graph drawing
  algorithms. discr. appl. math 2000
• linear time complexity in O(n3) volume
• papakostas and tollis. algorithms for incremental
  orthogonal graph drawing in three-dimensions.
  jgaa 1999
• linear time complexity in O(n3) volume
• other algorithms
• biedl. heuristics for 3d orthogonal graph
  drawing. twente workshop 1995
• 14 bends per edge in linear time and O(n2) volume
• closson, gartshore, johansen, and wismath. fully
  dynamic 3-dimensional orthogonal graph drawing.
  jgaa 2000
• 6 bends per edge in O(n2) volume and linear time,
  but insertions/deletions in O(1) time
• wood. an algorithm for three-dimensional
  orthogonal graph drawing. gd 1998
• 4 bends per edge in O(n3) time, but less than
  7m/3 bends in total
• di battista, patrignani, and vargiu. a splitpush
  approach to 3d orthogonal drawing. jgaa 2000
• no bound given

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plenty of drawings
papakostas and tollis 1999
eades, stirk, and whitesides 1996
eades, symvonis, and whitesides 2000
di battista, patrignani, and vargiu 2000
eades, symvonis, and whitesides 2000
biedl 1995
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what we would like to know

• two very difficult problems
• what happens if a maximum of two bends per edge
  is allowed?
• can we extend to 3d the topology-shape-metrics
  approach?

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2-bend drawing problem

• does a (degree six) graph always admit a 3d
  orthogonal drawing with at most 2 bends per edge?
• a positive answer could provide an algorithm of
  unprecedented effectiveness
• a negative answer was conjectured
• eades, symvonis, and whitesides. two algorithms
  for three dimensional orthogonal graph drawing.
  gd96, 1997
• but the K7 graph that was thought to require 3
  bends turned out to admit a 2-bend drawing
• wood. on higher dimensional orthogonal graph
  drawing. cats97
• problem 46 of the open problem project
• demaine, mitchell, and orourke

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topology-shape-metrics approach in 2d
V1,2,3,4,5,6 E(1,4),(1,5),(1,6),
(2,4),(2,5),(2,6), (3,4),(3,5),(3,6)
6
1
2
5
3
4
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topology-shape-metrics approach in 3d
V1,2,3,4,5,6 E(1,4),(1,5),(1,6),
(2,4),(2,5),(2,6), (3,4),(3,5),(3,6)
6
1
2
5
3
4
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simple and not simple shape graphs
not simple shape graph (always intersects)
simple shape graph (admitting non-intersecting
metrics)
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characterization of simple shapes

• known results
• characterization for cycles
• di battista, liotta, lubiw, and whitesides.
  orthogonal drawings of cycles in 3d space, gd00,
  2001
• characterization for paths (with additional
  constraints)
• di battista, liotta, lubiw, and whitesides.
  embedding problems for paths with direction
  constrained edges. theor. comp. sci., 2002
• proof that the characterization for cycles is not
  easy to extend to simple graphs (theta graphs)
• di giacomo, liotta, and patrignani. a note on 3d
  orthogonal drawings with direction constrained
  edges. ipl, 2004
• characterizing simple shapes is an open problem
• problem 20 of brandenburg, eppstein, goodrich,
  kobourov, liotta, and mutzel. selected open
  problems in graph drawing. gd 2003

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two open problems

• existence of a 2-bend drawing
• characterization of simple shapes

can complexity considerations give us some
insight?
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what we show

• given a 6-degree graph we prove that
• statement 1 simplicity testing is NP-hard
• if you fix edge shapes (with a maximum of 2
  bends per edge) finding the metrics corresponding
  to a non intersecting drawing is NP-hard
• statement 2 2-bend routing is NP-hard
• if you fix node positions finding a routing
  without intersections with a maximum of two bends
  per edge is NP-hard

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consequences of statement 1(simplicity testing
is NP-hard)

• any characterization of simple orthogonal shapes
  involves a hard computation
• even if we were able to find simple orthogonal
  shapes the compaction step would be NP-hard
• questions
• are there classes of graphs such that the
  compaction step is polynomial?
• are there families of shape graphs such that each
  graph is represented and the metrics can always
  be computed in polynomial time?

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consequences of statement 2(2-bend routing is
NP-hard)

• yet another problem where two bends per edge
  implies NP-hardness
• two bends per edge fixed shape ? NP-hardness
• two bends per edge fixed positions ?
  NP-hardness
• two bends per edge diagonal layout ?
  NP-hardness
• wood. minimising the number of bends and volume
  in 3d orthogonal graph drawings with a diagonal
  vertex layout. algorithmica, 2004
• question
• what is the problem of finding a 2-bend drawing
  of a graph?

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how we prove the statements

• reductions from the 3sat problem
• instance a set of clauses c1, c2, , cm each
  containing three literals from a set of boolean
  variables v1, v2, , vn
• question can truth values be assigned to the
  variables so that each clause contains at lest
  one true literal?

example of 3sat instance (v1 ? v3 ? v4) ? (v1
? v2 ? v5) ? (v2 ? v3 ? v5)
c3
c1
c2
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the 3sat reduction framework
variable gadgets
joint gadgets
clause gadgets
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variable gadget
true variable
false variable
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variable gadget propagating truth values
false variable
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joint-gadget
T
F
T
F
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joint-gadget
T
F
T
F
F
T
T
F
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clause gadget
from the joint gadget
from the variable gadget
from the joint gadget
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all literals false ? intersecting clause gadget
F
T
F
T
F
F
T
T
F
F
F
F
F
T
F
T
F
T
F
T
T
T
T
T
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variable gadget
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variable gadget propagating truth values
to clause gadget c1
variable gadget
to clause gadget c2
to clause gadget c3
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joint gadget
from the variable gadget
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joint gadget
from the variable gadget
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joint gadget
to the clause gadget
from the variable gadget
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clause gadget
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conclusions

• simplicity testing is NP-hard
• 2-bend routing is NP-hard
• open problems
• classes of graphs for which simplicity testing is
  polynomial?
• classes of shapes for which simplicity testing is
  polynomial?
• complexity of finding 2-bend drawings?

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questions?
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