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on the complexity of orthogonal compaction . maurizio patrignani univ. rome III. circuit schematics. entity relationship diagrams . industrial plants. integrated circuits. network topologies. data flow diagrams. orthogonal drawings. 6. 1. 5. 2. 3. 6. 1. 5. 2. 3. 6. 4. 5. - PowerPoint PPT Presentation
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on the complexity of orthogonal compaction
maurizio patrignaniuniv. rome III
industrial
plants
data flow diagramsnetwork topologies
integrated circuits
circuit schematicsentity relationship
diagrams
orthogonal drawings
topology-shape-metrics approach
V={1,2,3,4,5,6}E={(1,4),(1,5),(1,6), (2,4),(2,5),(2,6), (3,4),(3,5),(3,6)}
46
1 25
3
4
6
1 25
3
4
planarization
orthogonalization
compaction
61
25
3
the compaction step
input: an orthogonal representation or shape
output: an orthogonal grid drawing
without loss of generality we will consider only graphs without bends
2
3/2
3/2
/2
/2
/2
/2/2
/2/2
/2
3/2
3/2 3/2
3/2 a(f) · - 2
/2/2
a(f) · + 2
a(f) = number of vertices of face f
1)
2)
= 2
minimizing total edge length
minimizing area
minimizing maximum edge length
state of the art
orthogonal compaction wrt areawas mentioned as open problem(G. Vijayan and A. Widgerson)
linear time compaction heuristic based on rectangularization(R. Tamassia)
optimal compaction wrt total edge length by means of ILP + branch & bound or branch & cut techniques(G. W. Klau and P. Mutzel)
polynomial time compaction heuristic based on turn-regularization(S. Bridgeman, G. Di Battista, W. Didimo, G. Liotta, R. Tamassia, and L. Vismara)
1985
1987
1998
1998
formulating a decision problem
x2 x4 x1 x2 x3 x1 x2 x3 x4 x3
problem: satisfiability (SAT)instance: a set of clauses, each containing
literals from a set of boolean variables
question: can truth values be assigned to the variables so that each clause contains at least one true literal?
problem: orthogonal area compactioninstance: an orthogonal representation H
and a value kquestion: can an orthogonal drawing of H
be found such that its area is less or equal to k?
variable set ={x1 , x2 , x3 , x4}
reduction
compacted as much as possible
not compacted as much as possible
local and global properties
SAT
instance
compacted drawing
SAT
solution
sliding rectangles gadget
n timesr r r r l l l l r r r r
n timesr r r r l l l l r r r r
n timesr r r r l l l l r r r r
1 2 3 n...
transferable path properties
r r r l l l l r
r l l l l r r r
removing
inserting
a global property made local
a variant of the sliding rectangles gadget
an exponential number of orthogonal drawings with the minimum area
( parenthesis
parenthesis )
different
“shapes”...
... all sharing
the same
orthogonal
shape
NP-hardness proof
x3
falsex2
truex1
falsex4
truex5
true
clause 1
clause 2
clause 3
clause 4
clause gadget
xi is falsexi is true
xi does not occur in the clause
xi occurs in theclause with a positiveliteral
xi occurs in the clause witha negativeliteral
? ? ? ?
one is missing!
clause gadget example
variable set ={x1 , x2 , x3} x1 x2clause
truefalse
true
true truefalse
falsetrue
false
x1 x2
x1 x2
but we have only five “A”-shaped structures!
an example
x2 x4 x1 x2 x3 x1 x2 x3 x4 x3 clause 2 clause 3 clause 4clause 1
x1
falsex3
falsex2
truex4
true
clau
se 1
clau
se 2
clau
se 3
clau
se 4
NP-completeness
property: the compaction problem with respect to area is NP-hard
property: the compaction problem with respect to area is in NP
theorem: the compaction problem with respect to area is NP-complete
compaction with respect to total edge length
corollary: the compaction problem with respect to total edge length is NP-complete
compaction with respect to maximum edge length
corollary: the compaction problem with respect to maximum edge length is NP-complete
approximability considerations
does not admit a polinomial-time approximation scheme (not in PTAS)
3
3
conclusions• we have shown that the compaction
problem with respect to area, total edge length, or maximum edge length is NP-complete
• we have shown that the three problems are not in PTAS
• it is possible to modify the constructions so to have biconnected orthogonal representations
• does an orthogonal representation consisting in a single cycle retain the complexity of the three general problems?
• how many classes (rectangular, turn-regular, ...) of orthogonal representations admit a polynomial solution?
open problems