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Orthogonal Compaction Using Additional Bends Michael Jnger 1 , Petra Mutzel 2 , Christiane Spisla 2 1 University of Cologne, 2 TU Dortmund University January 2018, Aussois Overview 1 Problem Definition 2 Combinatorial Algorithm 3 Experimental

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Orthogonal Compaction Using Additional Bends

Michael Jünger1, Petra Mutzel2, Christiane Spisla2

1University of Cologne, 2TU Dortmund University

January 2018, Aussois

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Overview

1 Problem Definition 2 Combinatorial Algorithm 3 Experimental Evaluation

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Orthogonal Graph Drawing

Planar orthogonal grid drawings

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Orthogonal Graph Drawing

Planar orthogonal grid drawings

  • vertices on gridpoints
  • horizontal and vertical edges

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Orthogonal Graph Drawing

Planar orthogonal grid drawings

  • vertices on gridpoints
  • horizontal and vertical edges

Aesthetic criteria:

  • few bends
  • short edges
  • small area

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Orthogonal Graph Drawing

Compaction of orthogonal drawings

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Orthogonal Graph Drawing

Compaction of orthogonal drawings

  • coordinate / length assignment

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Orthogonal Graph Drawing

Compaction of orthogonal drawings

  • coordinate / length assignment
  • improve geometric properties

(edge length, area)

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Orthogonal Graph Drawing

Compaction of orthogonal drawings

  • coordinate / length assignment
  • improve geometric properties

(edge length, area)

  • preserve orthogonal shape

(bends, angles)

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Orthogonal Compaction

  • rthogonal compaction problem (OCP)

Given an orthogonal drawing Γ of a graph G = (V ,E), compute another drawing Γ′ of G with the same orthogonal shape minimizing the total edge length.

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Orthogonal Compaction

  • rthogonal compaction problem (OCP)

Given an orthogonal drawing Γ of a graph G = (V ,E), compute another drawing Γ′ of G with the same orthogonal shape minimizing the total edge length.

  • NP-hard problem (Patrignani, 1999)

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Orthogonal Compaction

  • rthogonal compaction problem (OCP)

Given an orthogonal drawing Γ of a graph G = (V ,E), compute another drawing Γ′ of G with the same orthogonal shape minimizing the total edge length.

  • NP-hard problem (Patrignani, 1999)
  • ILP formulation (Klau and Mutzel, 1999)

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Orthogonal Compaction

  • rthogonal compaction problem (OCP)

Given an orthogonal drawing Γ of a graph G = (V ,E), compute another drawing Γ′ of G with the same orthogonal shape minimizing the total edge length.

  • NP-hard problem (Patrignani, 1999)
  • ILP formulation (Klau and Mutzel, 1999)
  • special cases solvable in polynomial time (Bridgeman et al., 2000)

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Orthogonal Compaction

  • rthogonal compaction problem (OCP)

Given an orthogonal drawing Γ of a graph G = (V ,E), compute another drawing Γ′ of G with the same orthogonal shape minimizing the total edge length.

  • NP-hard problem (Patrignani, 1999)
  • ILP formulation (Klau and Mutzel, 1999)
  • special cases solvable in polynomial time (Bridgeman et al., 2000)
  • heuristics in practice:

alternating one-dimensional compaction algorithms

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Orthogonal Compaction

  • ne-dimensional orthogonal compaction problem (1DIM OCP)

Given an orthogonal drawing Γ of a graph G = (V ,E), compute another drawing Γ′ of G with the same orthogonal shape minimizing the total vertical(horizontal) edge length subject to fixed x(y)-coordinates.

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Orthogonal Compaction

  • ne-dimensional orthogonal compaction problem (1DIM OCP)

Given an orthogonal drawing Γ of a graph G = (V ,E), compute another drawing Γ′ of G with the same orthogonal shape minimizing the total vertical(horizontal) edge length subject to fixed x(y)-coordinates.

  • solvable in polynomial time

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Orthogonal Compaction

  • ne-dimensional orthogonal compaction problem (1DIM OCP)

Given an orthogonal drawing Γ of a graph G = (V ,E), compute another drawing Γ′ of G with the same orthogonal shape minimizing the total vertical(horizontal) edge length subject to fixed x(y)-coordinates.

  • solvable in polynomial time
  • flow-based method, longest path method, ... e.g. Tamassia (1987)

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Orthogonal Compaction

  • ne-dimensional orthogonal compaction problem (1DIM OCP)

Given an orthogonal drawing Γ of a graph G = (V ,E), compute another drawing Γ′ of G with the same orthogonal shape minimizing the total vertical(horizontal) edge length subject to fixed x(y)-coordinates.

  • solvable in polynomial time
  • flow-based method, longest path method, ... e.g. Tamassia (1987)
  • drawings still improvable

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Orthogonal Compaction

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Orthogonal Compaction

→ Trade-off between bends and edge length

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Orthogonal Compaction

  • ne-dimensional monotone flexible edge orthogonal compaction

problem with fixed vertex star geometry (1DIM Fled-Five OCP)

Given an orthogonal drawing Γ of a graph G = (V ,E), compute another drawing Γ′ of G that minimizes the total vertical(horizontal) edge length subject to fixed x(y)-coordinates and in which all horizontal(vertical) edge segments are redrawn x(y)-monotonically and the vertex star geometries are maintained.

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Orthogonal Compaction

  • ne-dimensional monotone flexible edge orthogonal compaction

problem with fixed vertex star geometry (1DIM Fled-Five OCP)

Given an orthogonal drawing Γ of a graph G = (V ,E), compute another drawing Γ′ of G that minimizes the total vertical(horizontal) edge length subject to fixed x(y)-coordinates and in which all horizontal(vertical) edge segments are redrawn x(y)-monotonically and the vertex star geometries are maintained.

  • change the orthogonal shape of edge

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Orthogonal Compaction

  • ne-dimensional monotone flexible edge orthogonal compaction

problem with fixed vertex star geometry (1DIM Fled-Five OCP)

Given an orthogonal drawing Γ of a graph G = (V ,E), compute another drawing Γ′ of G that minimizes the total vertical(horizontal) edge length subject to fixed x(y)-coordinates and in which all horizontal(vertical) edge segments are redrawn x(y)-monotonically and the vertex star geometries are maintained.

  • change the orthogonal shape of edge
  • maintain angles around vertices

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Orthogonal Compaction

What is allowed:

  • add/remove vertical segments to horizontal edges

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Orthogonal Compaction

What is allowed:

  • add/remove vertical segments to horizontal edges

What is not allowed:

  • “turnarounds” of horizontal edges

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Orthogonal Compaction

What is allowed:

  • add/remove vertical segments to horizontal edges

What is not allowed:

  • “turnarounds” of horizontal edges
  • change x-coordinates of vertices

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Flow-based Compaction

How to solve orthogonal compaction problems?

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Flow-based Compaction

How to solve orthogonal compaction problems? Transformation to a network flow problem.

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Flow-based Compaction

How to solve orthogonal compaction problems? Transformation to a network flow problem. Minimum cost flow in network N ↔ optimal solution for the vertical OCP

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Flow-based Compaction

How to solve orthogonal compaction problems? Transformation to a network flow problem. Minimum cost flow in network N ↔ optimal solution for the vertical OCP modification Minimum cost flow in network N′ ↔ optimal solution for the vertical Fled-Five OCP

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Classical Flow Network

Given: drawing Γ of graph G

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Classical Flow Network

Given: drawing Γ of graph G replace bends

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Classical Flow Network

Given: drawing Γ of graph G add vertical dissecting edges

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Classical Flow Network

Given: drawing Γ of graph G Construct network N:

  • ∀ face f ∈ G :

node nf with demand d(nf ) = 0

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Classical Flow Network

Given: drawing Γ of graph G Construct network N:

  • ∀ face f ∈ G :

node nf with demand d(nf ) = 0

  • ∀ vert. edge e ∈ G :

arc ae with lower bound lb(ae) = 1 upper bound ub(ae) = ∞

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Classical Flow Network

Given: drawing Γ of graph G Construct network N:

  • ∀ face f ∈ G :

node nf with demand d(nf ) = 0

  • ∀ vert. edge e ∈ G :

arc ae with lower bound lb(ae) = 1 upper bound ub(ae) = ∞ cost(ae) = 1 (original)

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Classical Flow Network

Given: drawing Γ of graph G Construct network N:

  • ∀ face f ∈ G :

node nf with demand d(nf ) = 0

  • ∀ vert. edge e ∈ G :

arc ae with lower bound lb(ae) = 1 upper bound ub(ae) = ∞ cost(ae) = 1 (original) cost(ae) = 0 (dummy)

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Classical Flow Network

  • flow on ae corresponds to length
  • f e

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Classical Flow Network

  • flow on ae corresponds to length
  • f e
  • flow conservation: faces are

drawn properly

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Classical Flow Network

  • flow on ae corresponds to length
  • f e
  • flow conservation: faces are

drawn properly

  • capacities: minimum edge length

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Classical Flow Network

  • flow on ae corresponds to length
  • f e
  • flow conservation: faces are

drawn properly

  • capacities: minimum edge length
  • minimum cost flow ⇒ minimal

vertical length assignment

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Classical Flow Network

  • flow on ae corresponds to length
  • f e
  • flow conservation: faces are

drawn properly

  • capacities: minimum edge length
  • minimum cost flow ⇒ minimal

vertical length assignment

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Modified Flow Network

Given: drawing Γ of graph G replace bends

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Modified Flow Network

Given: drawing Γ of graph G add bend vertices

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Modified Flow Network

Given: drawing Γ of graph G add vertical dissecting edges

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Modified Flow Network

Given: drawing Γ of graph G Construct network N:

  • as before

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Modified Flow Network

Given: drawing Γ of graph G Construct network N:

  • as before
  • ∀ bend vertices v ∈ G :

arcs a↑

v, a↓ v

with lower bound lb(a↑/↓

v ) = 0

upper bound ub(a↑/↓

v ) = ∞

cost(a↑/↓

v ) = 1

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Modified Flow Network

Given: drawing Γ of graph G Construct network N:

  • as before
  • ∀ bend vertices v ∈ G :

arcs a↑

v, a↓ v

with lower bound lb(a↑/↓

v ) = 0

upper bound ub(a↑/↓

v ) = ∞

cost(a↑/↓

v ) = 1

  • ∀ middle segment e ∈ G :

lower bound lb(ae) = 0

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Modified Flow Network

  • flow on a↑/↓

v

corresponds to downward/upward edge segment in the edge that is split by v

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Modified Flow Network

2 2 2 2 2 2

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Modified Flow Network

2 2 2 2 2 2

vertical edge length before: 19 vertical edge length after: 15

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Experimental Evaluation

Comparison of the Fled-Five approach against the traditional orthogonal compaction

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Experimental Evaluation

Comparison of the Fled-Five approach against the traditional orthogonal compaction Input: bend minimal orthogonal drawings

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Experimental Evaluation

Comparison of the Fled-Five approach against the traditional orthogonal compaction Input: bend minimal orthogonal drawings

  • FF = Fled-Five OCP algorithm, repeatedly in x- and y-direction
  • TRAD = traditional OCP algorithm, repeatedly in x- and y-direction

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Experimental Evaluation

Total edge length

1000 2000 3000 4000 5000 6000 100 200 300 400 500

total edge length

nodes

TRAD FF 13

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Experimental Evaluation

Drawing area

2000 4000 6000 8000 10000 12000 14000 100 200 300 400 500

area

nodes

TRAD FF 14

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Experimental Evaluation

Number of bends

50 100 150 200 250 300 350 100 200 300 400 500

bends

nodes

TRAD FF 15

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Conclusion

Introduction of the Fled-Five OCP

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Conclusion

Introduction of the Fled-Five OCP

  • relaxation of the classical OCP

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Conclusion

Introduction of the Fled-Five OCP

  • relaxation of the classical OCP
  • polynomial-time algorithm

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Conclusion

Introduction of the Fled-Five OCP

  • relaxation of the classical OCP
  • polynomial-time algorithm
  • better geometric results at the expense of bends

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Conclusion

Introduction of the Fled-Five OCP

  • relaxation of the classical OCP
  • polynomial-time algorithm
  • better geometric results at the expense of bends
  • no control over the number of bends

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Conclusion

Introduction of the Fled-Five OCP

  • relaxation of the classical OCP
  • polynomial-time algorithm
  • better geometric results at the expense of bends
  • no control over the number of bends

Thank you!

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References I

Bridgeman, S. S., Di Battista, G., Didimo, W., Liotta, G., Tamassia, R., and Vismara, L. (2000). Turn-regularity and optimal area drawings of orthogonal

  • representations. Comput. Geom., 16(1):53–93.

Klau, G. W. and Mutzel, P. (1999). Optimal compaction of orthogonal grid

  • drawings. In Cornuéjols, G., Burkard, R. E., and Woeginger, G. J., editors,

Integer Programming and Combinatorial Optimization, 7th International IPCO Conference, 1999, Proceedings, volume 1610 of LNCS, pages 304–319. Springer. Patrignani, M. (1999). On the complexity of orthogonal compaction. In Algorithms and Data Structures, 6th International Workshop, WADS ’99, Proceedings, volume 1663 of LNCS, pages 56–61. Springer. Tamassia, R. (1987). On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput., 16(3):421–444.

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