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