PC Trees and circularones arrangements

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PC Trees and circular-ones arrangements

• Speaker David Bañeres
• Authors Wen-Lian Hsu
• Ross M. McConnell

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Index

• Introduction
• Booth and Luekers algorithm
• Hsus approach
• PC-Tree structure
• Conclusions

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Index

• Introduction
• Booth and Luekers algorithm
• Hsus approach
• PC-Tree structure
• Conclusions

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Introduction

• What is a Interval graph?
• Its the intersection graph of intervals on a line

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Introduction

• Interval graphs are used for many problems
• Molecular biology (1950s ? work with genetic
  material)
• Scheduling jobs
• Maximum clique
• Minimum coloring
• But we need algorithms for recognizing and
  constructing interval graphs.
• Booth and Lueker found an algorithm with linear
  cost time (1976)

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Index

• Introduction
• Booth and Luekers algorithm
• Hsus approach
• PC-Tree structure
• Conclusions

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Booth and Luekers algorithm

• Booth and Luekers algorithm
• This algorithm represents graphs with a 0-1
  matrix
• It tries to check if the matrix has the
  consecutive-ones property
• It the matrix has this property ? we can find a
  representation with intervals in linear time
• Consecutive-ones property
• If the columns can be ordered , so that, the ones
  are consecutive in every row.

Consecutive-ones property
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Booth and Luekers algorithm

• Booth and Lueker developed a representation

PQ-TREE
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Booth and Luekers algorithm

• This algorithm for constructing PQ-Trees is
  difficult to program
• Other authors tried to solve this problem
• P.N. Klein, J.H. Reif, An efficient parallel
  algorithm for planarity, J. Comput. System Sci 19
  (1988) 190-246
• W. L Hsu, A simple test for the consecutive ones
  property, J. Algorithms 42 (2002) 1-16

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Index

• Introduction
• Booth and Luekers algorithm
• Hsus approach
• PC-Tree structure
• Conclusions

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Hsus approach

• Check for circular-ones property
• If the columns can be ordered such that in every
  row, either the zeros are consecutive or the ones
  are consecutive.
• Test circular-ones property reduces in linear
  time to test for the consecutive-ones property.

Consecutive-ones property
Circular-ones property
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Hsus approach

• How to check for circular-ones property?

PC-TREE
P
C
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Index

• Introduction
• Booth and Luekers algorithm
• Hsus approach
• PC-Tree structure
• Conclusions

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Definitions
1
1
1
1
1
0
0
1
0
0
0
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Constructing the PC-Tree

• The Initial PC-Tree is a P node that is adjacent
  to all leaves
• At each row
• Search the Terminal path and modify nodes so that
  all ones lie on one side of the path
• Split each node on the path into two nodes, one
  adjacent to the edges to full leaves and one
  adjacent to the edges to empty leaves
• Delete the edges of the path and replace them
  with a new C node x whose cyclic order preserves
  the order of the nodes on this path
• Contract all edges from x to C-node neighbors,
  and any node that has only two neighbors

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Example
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Example
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Example
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Example
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Example
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Example
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Example
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Example
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Example
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Example
G
H
A
B
C
E
D
F
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How to build a PQ-Tree from a PC-Tree

• Add a new column x to the matrix that has all
  zeros
• Compute PC-Tree
• Pick-up the tree at x
• The child of x is the PQ-Tree
• Substitute all C nodes for Q nodes

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Example
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Example
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Example
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Example
31
Example
32
Example
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Example
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Example
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Example
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Example
G
H
A
B
C
E
D
F
X
X
0
0
0
0
0
0
0
0
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Example
A
E
G
D
C
X
B
F
H
PC-Tree
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Example
E
B
C
D
F
G
H
A
Circular-ones property
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Example
E
B
C
D
F
G
H
A
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Example
E
B
C
D
F
G
H
A
consecutive-ones property
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Data structure for PC-Tree
P
Twin Arcs
Double linked list of arcs
Q
(x,y)
P node
ptr 1rt neighbor
ptr parent edge
ptr 2nd neighbor

• We have a root tree internally for efficiency
• C nodes dont have explicit representation

ptr twin arc
bit Is y parent of x?
Ptr y (if y is a P-node)
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Conclusions

• The PC-Tree is a good structure for checking the
  circular-ones property.
• We can use the PC-Trees for creating PQ-Trees
• The cost of the algorithm for building a PC-Tree
  is linear

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References

• Wen-Lian Hsu, Ross M. McConnell, PC trees and
  circular-ones arrangements, Theoretical Computer
  Science 296 99-116, 2003