Michael A' Bekos1

1
PCI 2005
PCI2005 10TH PANHELLENIC CONFERENCE ON INFORMATICS
Boundary Labeling of Optimal Total Leader Length

• Michael A. Bekos1
• Michael Kaufmann2
• Katerina Potika1
• Antonios Symvonis1

1 National Technical University of Athens 2
University of Tubingen
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Outline
PCI 2005

• Problem definition
• Previous results
• Our algorithm
• Conclusions and open problems

3
Problem Definition
PCI 2005

• In a boundary labeling
• Labels are attached on the boundary of a
  rectangle R.
• Each site is connected to its label by
  non-intersecting polygonal lines (leaders).
• Problem
• Motivation

• Problem Definition
• Sites
• Leaders
• Labels

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Sites
PCI 2005

• In a boundary labeling, sites should lie in
  general positions, i.e.
• No three sites lie on a line.
• No two sites share the same x or y coordinate.

• Problem Definition
• Sites
• General position

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Leaders
PCI 2005

• Different Leader Types
• Type-s Type-po Type-opo

• Problem Definition
• Leaders
• type-s
• type-po
• type-opo
• Track Routing Area

Track Routing Area
Type-s leaders
Type-po leaders
Type-opo leaders
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Labels
PCI 2005

• Different Label Types
• Size Uniform / Non-Uniform.
• Sides N / S / W / E
• Ports Fixed / Sliding.
• Position Predefined or Fixed / Sliding.

• Problem Definition
• Labels
• Size
• Sides
• Ports
• Position

N
E
W
S
(a)
(b)
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Objectives
PCI 2005

• Under a labeling model, we aim at
• Legal solution
• Non-overlapping labels.
• Non-intersecting leaders.
• Legal solution with minimum total leader length
  (TLLM).
• Legal solution with minimum number of bends (BM).

• Problem Definition
• Objectives
• Legal solution
• Minimize Total Leader Length
• Minimize number of bends.

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Examples
PCI 2005

• Problem Definition
• Examples
• Minimize Total Leader Length
• Minimize number of bends.

Minimizing number of bends
Minimizing total leader length
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NP-Completeness
PCI 2005

• A negative result
• Non-Uniform labels.
• 2 opposite sides. Partition
• Legal solution.
• Simplifying assumptions
• Uniform labels.
• Labels in predefined positions.

• Previous Results
• A NP-Complete Problem

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Previous results on TLLM problem
PCI 2005

• Type-opo leaders
• 1 side O(nlogn) Simple Algorithm
• 2 opposite sides O(n2) Dynamic Programming

• Previous Results
• Minimum Total Leader Length
• Uniform Labels
• Predefined position labels
• Sliding or fixed label ports

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Previous results on BM problem
PCI 2005

• Type-opo leaders
• 1 side O(n2) Dynamic Programming

• Previous Results
• Minimum number of bends
• Variable size Labels
• Sliding labels
• Sliding ports

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Our Problem
PCI 2005

• Description
• Leaders of type-opo.
• Uniform Labels in predefined position.
• 4 allowed sides.
• Fixed / Sliding ports.
• Minimize total leader length.

• Our Algorithm
• Labeling Model Specification

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Description of our Algorithm
PCI 2005

• Algorithm
• Complete weighted bipartite graph
• G (P U L, E, w)
• w(pi, lj) Manhattan length of the shortest
  leader connecting site pi with
• label lj
• Vaidya's algorithm gt Obtain labeling.
• Resolve crossings.

• Our Algorithm
• Labeling Model
• Type-opo leaders
• 4-Side
• Uniform labels
• Fixed positions
• Fixed / Sliding ports
• Minimize total leader length

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Leader Orientation
PCI 2005

• Definition
• Leader c is oriented towards corner A of R if
  its labels port and corner A are on the same
  half-plane

• Our Algorithm
• Labeling Model
• Type-opo leaders
• 4-Side
• Uniform labels
• Fixed positions
• Fixed / Sliding ports
• Minimize total leader length

ci
pi
cj
pj
A
A

• Leader ci is oriented towards corner A.
• Leader cj is oriented away from corner A.

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Resolve Crossings (1)
PCI 2005

• Cases that can not occur

• Our Algorithm
• Labeling Model
• Type-opo leaders
• 4-Side
• Uniform labels
• Fixed positions
• Fixed / Sliding ports
• Minimize total leader length

ci
ci
pi
pi
Reroute(ci, cj)
pj
pj
cj
cj
(b)
(a)
ci
cj
pi
pj
(c)

• Labels associated with two crossing leaders ci,
  cj are located at two adjacent sides of R.

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Resolve Crossings (2)
PCI 2005

• Cases that can not occur

• Our Algorithm
• Labeling Model
• Type-opo leaders
• 4-Side
• Uniform labels
• Fixed positions
• Fixed / Sliding ports
• Minimize total leader length

ci
pi
pi
Reroute(ci, cj)
pj
pj
cj
ci
cj
A
A
(a)
(b)
ci
pi
pi
Reroute(ci, cj)
pj
pj
cj
ci
cj
A
A
(c)
(d)

• Two crossing leaders ci, cj are oriented towards
  the same corner.

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Resolve Crossings (3)
PCI 2005

• How to reroute two crossing leaders

• Our Algorithm
• Labeling Model
• Type-opo leaders
• 4-Side
• Uniform labels
• Fixed positions
• Fixed / Sliding ports
• Minimize total leader length

ci
pi
pi
ci
Reroute(ci, cj)
pj
pj
cj
cj
A
A

• If we keep the position of the ports unchanged,
  then the total leader length remains unchanged,
  after the rerouting.

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Resolve Crossings (4)
PCI 2005

• How to resolve all crossings

• Our Algorithm
• Labeling Model
• Type-opo leaders
• 4-Side
• Uniform labels
• Fixed positions
• Fixed / Sliding ports
• Minimize total leader length

B
B
pk
pk
pi
pi
Reroute(ci, c)
ci
pj
pj
c
p
p
ci
cj
ck
cj
ck
c
A
A

• The leftmost site, which participates in a
  crossing, moves to the right.
• In a left to right pass, all left-to-right
  crossings are eliminated.

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Resolve Crossings (5)
PCI 2005

• How to resolve all crossings

• Our Algorithm
• Labeling Model
• Type-opo leaders
• 4-Side
• Uniform labels
• Fixed positions
• Fixed / Sliding ports
• Minimize total leader length

Reroute(ci, c)
ci
pi
pi
c
p
p
ci
c
A
A
D
D

• In the left to right pass, no right-to-left
  crossing can be introduced.
• We can independently resolve all crossings in two
  passes.

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On Time Complexity
PCI 2005

• Resolve crossings
• Visibility query
• Return the first line segment to the right of
  p(xp,yp,) that is intersected by line yyp
• Line segments O(log2n)
• Semi-infinite rays O(logn)

pk

• Our Algorithm
• Labeling Model
• Type-opo leaders
• 4-Side
• Uniform labels
• Fixed positions
• Fixed / Sliding ports
• Minimize total leader length

pi
pj
c
p
ci
cj
ck
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On Time Complexity
PCI 2005

• Total time complexity
• Vaidya's algorithm O(n2log3n)
• Resolve Crossings O(nlogn)
• gt O(n2log3n)

• Our Algorithm
• Labeling Model
• Type-opo leaders
• 4-Side
• Uniform labels
• Fixed positions
• Fixed / Sliding ports
• Minimize total leader length

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Open Problems
PCI 2005

• Mixed Model

• Open Problems
• Mixed Model
• Optimization
• Floating Points
• Complexity

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Open Problems
PCI 2005

• Optimization
• Several parameters
• Leaders, label size, sides, ports,
  label position.
• Several open optimization problems
• Leader length minimization.
• Bend minimization.
• Floating Points

• Open Problems
• Mixed Model
• Optimization
• Floating Points
• Complexity

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Open Problems
PCI 2005

• Complexity
• Pseudopolynomial algorithms
• O(nH)
• O(nH2)
• What is the computational complexity of these
  problems?

• Open Problems
• Mixed Model
• Optimization
• Floating Points
• Complexity

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On Boundary Labeling.
PCI 2005

• Thank you.