PC trees and Circular One Arrangements

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PC trees and Circular One Arrangements

• Wen-Lian Hsu
• Ross M. McConnell

Presented by Noa Bar-Yosef
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Outline

• Motivation
• PQ Trees to solve consecutive-ones problem
• PC Trees to solve circular-ones problem
• Algorithm to build a PC tree
• Turning a PC tree into a PQ tree
• Data Structure for representing the PC tree and
  algorithm time analysis
• Summary

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Interval Graphs
c
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Motivation

• Molecular Biology
• 1950s Benzer showed that the intersection graph
  of a large number of fragments of genetic
  material was an interval graph.
• Other Applications
• Scheduling jobs
• Maximum cliques
• Minimum coloring

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A Clique Matrix
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Recognizing Interval Graphs

• 1976 - Booth and Leuker
• If a consecutive-ones ordering is found, it is
  easy to find a representation of the graph with
  intervals in linear time.

A
C
A
D
B
E
C
B
D
E
Consecutive-Ones
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A matrix WITHOUT a consecutive-ones property
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Outline

• Motivation
• PQ Trees to solve consecutive-ones problem
• PC Trees to solve circular-ones problem
• Algorithm to build a PC tree
• Turning a PC tree into a PQ tree
• Data Structure for representing the PC tree and
  algorithm time analysis
• Summary

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PQ-Trees
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Order of Children Node are Reversed
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Order of children of a P node are permuted
arbitrarily
P
P
Q
Q
1
13
11
12
2
3
P
6
7
4
8
5
9
10
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PQ-Trees

• Booth and Leukers observation
• testing for the circular-ones property reduces in
  linear time to testing for the consecutive-ones
  property.

Circular Ones
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A matrix WITHOUT a circular-ones property
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PQ-Trees

• Notoriously difficult to program!
• Extensive Literature on the subject try to solve
  the complexity of the algorithm.
• Why not use the circular-ones property?
• Solve the consecutive-ones problem and reduce it
  to the circular-ones problem.

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Outline

• Motivation
• PQ Trees to solve consecutive-ones problem
• PC Trees to solve circular-ones problem
• Algorithm to build a PC tree
• Turning a PC tree into a PQ tree
• Data Structure for representing the PC tree and
  algorithm time analysis
• Summary

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PC - Trees
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Reverse the leaves on one side of an edge
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Reverse leaves of forest if a P -Node would be
removed
4
9
5
10
p
6
11
7
c
1
8
3
2
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An Unrooted Set Family

• Let an unrooted set family be a set family F on
  domain V with the following properties
• The singleton subsets of V are members of F
• Whenever X?F, then so X V-X.
• Whenever X,Y?F, X-Y, Y-X, Y?X and (X?Y)
  V-(X?Y) are all non-empty, then X?Y, X?Y, and X?Y
  are also in F

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Theorem of Representation

• Theorem For an unrooted set family, there is an
  unrooted tree whose nodes are labeled prime and
  degenerate, and whose leaves are the singletons
  subsets of V. X?F iff it is the union of the leaf
  set of a subtree that results from the deletion
  of an edge, or the union of leaf sets of some of
  the subtrees that result from the deletion of a
  degenerate node.

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Circular Module
X B,C is uniform in Row II
Y B,C is a circular module
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The circular modules are an unrooted set family!

• Unrooted Set Family
• The singleton subsets of V are members of F.
• Whenever X?F, then so X V-X.
• Whenever X,Y?F, X-Y, Y-X, Y?X and (X?Y)
  V-(X?Y) are all non-empty, then X?Y, X?Y, and X?Y
  are also in F

• Circular modules

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Circular Modules Edge and P Modules

• A circular module X is either an edge module,
  which means the leaves on one side of an edge

X
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Circular Modules Edge and P Modules

• Or a P module, which means a set that is not an
  edge module but the union of leaves in a subset
  of the trees formed when a P node is removed.

X
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Outline

• Motivation
• PQ Trees to solve consecutive-ones problem
• PC Trees to solve circular-ones problem
• Algorithm to build a PC tree
• Turning a PC tree into a PQ tree
• Data Structure for representing the PC tree
• Algorithm time Analysis

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Constructing the PC-Trees

• On induction on the number of rows of a matrix.
• The ith step of the algorithm modifies the PC
  tree so that it is correct for the submatrix
  consisting of the first i rows of the matrix.

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Constructing the PC-tree Cont.

• When adding a row, no new circular modules are
  created. But older ones may become defunct
  because a constraint was added.

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Order preserving Contraction
f
a
g
e
b
z
c
d
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Leaves and Edges
Step i
0 Empty leaf 1 Full leaf
Terminal Edge
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Construction Algorithm Illustration
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Construction Algorithm

• The initial PC tree is a P node that is adjacent
  to all leaves, which allows all (n-1)! circular
  orderings.
• At each row
• Find the terminal path, and then perform flips of
  C nodes and modify the cyclic order of edges
  incident to P nodes so that all 1s 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 2 neighbors.

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PC Tree Example - Initially
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PC Tree Example Row I
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PC Tree Example Row II
E0
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PC Tree Example Row III
D0
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PC Tree Example Row IV
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PC Tree Example Row V
G0
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PC Tree Example Row VI
C0
F1
H1
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PC Tree Example Row VII
A1
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PC Tree Example Last Row
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Circular Ones Matrix
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Correctness of Algorithm

• By induction
• i1 a tree that represents all possible circular
  permutations.
• i
• Ignore effects of contraction
• T be the PC tree before the ith row, T after.
• An edge set of T is in the set family represented
  by T iff it continues to be a circular module.
• A P set of T continues to be represented in T
  iff t is a circular module in the first i rows.
• The cyclic order of all C nodes must be correct.

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Correctness of Algorithm Illustrated
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Outline

• Motivation
• PQ Trees to solve consecutive-ones problem
• PC Trees to solve circular-ones problem
• Algorithm to build a PC tree
• Turning a PC tree into a PQ tree
• Data Structure for representing the PC tree and
  algorithm time analysis
• Summary

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Solving the Consecutive-Ones Problem

• Add the zero vector as a new column of the matrix
• Compute the PC-tree for the new matrix
• Pick it up by the leaf corresponding to the new
  column to root it
• The subtree rooted at its child is the PQ tree
  for the original matrix.
• Think of cutting up the vertical cylinder
  at X!

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Computing Consecutive Ones Matrix
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Consecutive Ones - Initially
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Consecutive Ones Row I
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Consecutive Ones Row II
E0
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Consecutive Ones Example
?
2
1
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Consecutive Ones Last Row
A1
D1
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Circular Ones Matrix
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Creating the Consecutive One Matrix
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Consecutive Ones Matrix
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PQ Tree
D1
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PQ Tree and Matrix
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Outline

• Motivation
• PQ Trees to solve consecutive-ones problem
• PC Trees to solve circular-ones problem
• Algorithm to build a PC tree
• Turning a PC tree into a PQ tree
• Data Structure for representing the PC tree and
  algorithm time analysis
• Summary

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Data Structure for PC tree representation
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Full-Partial Labeling Algorithm
Partial
Full
1
1
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Finding the Terminal Path Efficiently - 1
Key Observation
Apex the least common ancestor of partial nodes
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Finding the Terminal Path Efficiently - 1
0
1
0
1
p
0
1
0
f
p
1
p
1
f
p
0
p
1
p
0
1
0
1
0
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Finding the terminal Path Efficiently - 3

• If X is a node on the terminal path other than
  the apex, the parent of X is also on the terminal
  path.
• Finding Xs parent
• P node O(1)
• C node All full neighbors are consecutive! Find
  the neighbor that isnt full! Cost proportional
  to number of full neighbors.

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Performing the update step on the terminal path
efficiently
O(pk)
Assigning Parent bits O(p) Contraction O(1)
O(p)
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Linear Time Bound

• Amortized Analysis
• F(M,i) 2ui Ci ?x?Pi(deg(x) 1)

F(M,0) T(m)
1 Credit for each C-node
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Outline

• Motivation
• PQ Trees to solve consecutive-ones problem
• PC Trees to solve circular-ones problem
• Algorithm to build a PC tree
• Turning a PC tree into a PQ tree
• Data Structure for representing the PC tree and
  algorithm time analysis
• Summary

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Summary

• PC tree solves a more general problem than that
  of the PQ tree in a very simple matter.
• The PQ tree can be obtained from the PC tree
• The linear time bound analysis is similar
• Perhaps the problem with the PQ-tree is that it
  is rooted???

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References

• Algorithmic Graph Theory and Perfect Graphs,
  Martin C. Golumbic
• PC Trees vs. PQ Trees, Wen-Lian Hsu
• PC trees and Circular-One Arrangements, Wen-Lian
  Hsu, Ross M. McConnell
• PQ Trees, PC Trees, and Planar Graphs, Wen-Lian
  Hsu, Ross M. McConnell
• And thank you to http//www.lsi.upc.edu/dbaneres/
  for his examples

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