Drawing (Complete) Binary Tanglegrams

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Drawing (Complete) Binary Tanglegrams
Hardness, Approximation, Fixed-Parameter
Tractability
Utrecht U, NL TU Eindhoven, NL Karlsruhe U,
DE Tokio Inst. Tech., JP
Kevin Buchin Maike Buchin Jaroslaw Byrka Martin
Nöllenburg Yoshio Okamoto Rodrigo I.
Silveira Alexander Wolff
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• Tanglegram
• 2 trees
• leaves matched 1-to-1

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Application example

• Phylogenetic trees

Pocket gopher drawings from The Animal Diversity
Web (http//animaldiversity.org)
grandis
castanops
bursarius
neglectus
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Outline of this talk

• Introduction
• 2-approximation algorithm
• Algorithm
• Approximation factor
• Conclusions

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Comparing pairs of trees

• Comparing trees
• Visually
• Applications
• Software visualization
• Hierarchical clustering
• Phylogenetic trees

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Problem statement TL (Tanglegram Layout)

• Input 2 trees S, T
• With leaves in 1-to-1 correspondence

• Output plane drawings of S and T
• Minimizing inter-tree crossings

S
T
6 inter-tree crossings
4 inter-tree crossings
5 inter-tree crossings
3 inter-tree crossings
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Related work

• 2-sided crossing minimization problem
• Introduced by Sugiyama et al.
• Several differences
• Arbitrary degree
• Any ordering allowed

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Previous work

• Holten and Van Wijk (08)
• Visual Comparisonof Hierarchically Organized
  Data

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Previous work (contd)

• Dwyer and Schreiber (04)
• 2.5D drawings of stacked trees
• One sided (binary) version, O(n2 log n) time.
• Fernau, Kaufmann and Poths (05)
• TL is NP-hard
• 1 (binary) tree fixed O(n log2 n) time.
• FPT algorithm O(ck), for c1024

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Our results

• We study 2 versions of TL
• We show
• binary TL is NP-hard to approximate within any
  constant
• complete binary TL is NP-hard
• complete binary TL has 2-APX algorithm
• complete binary TL has O(4kn2)-time FPT algorithm

binary TL
complete binary TL
under widely accepted conjectures
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2-approximation algorithm

• Simple recursive approach

• Try each of 4 combinations, and recurse

Drawing Complete Binary Tanglegrams
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Initial algorithm
?

• Algorithm
• Try each of the 4 combinations
• Count crossings
• Return the best one
• Cant count all crossings!

?
Drawing Complete Binary Tanglegrams
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Types of crossings

• Lower-level
• Created by recursive calls
• Nothing to do about them
• Current-level
• Can be avoided at this level
• What about ?

Drawing Complete Binary Tanglegrams
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Need to remember more
Problematic situation

• Sometimes we can

Good situation ?
Drawing Complete Binary Tanglegrams
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Use labels

• To preserve this knowledge

Initial layout
Drawing Complete Binary Tanglegrams
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Use labels

• Using labels, we can count more crossings

Problematic situation only if labels are
equal (indeterminate crossing)
Drawing Complete Binary Tanglegrams
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Algorithm

• For each way of arranging the subtrees
• Assign labels to some leaves
• Solve recursively
• gives lower-level crossings
• Compute current-level crossings
• Return best of 4 combinations
• Running time T(n)?8T(n/2) O(n)O(n3)

Drawing Complete Binary Tanglegrams
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Approximation factor

• Mistakes from indeterminate crossings
• We cannot count them
• How many can we have?
• We show that IND ? copt
• Therefore calg ? 2 copt

indeterminate crossings
crossings in optimal drawing
crossings in algorithm drawing
Drawing Complete Binary Tanglegrams
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Approximation factor (2)

• Obs Indeterminate crossings used to be good
• Upperbound IND by of these crossing
• Use that trees are complete
• We know exactly how many edges each subtree has

Drawing Complete Binary Tanglegrams
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Conclusions

• Studied binary TL / complete binary TL
• binary TL has no constant factor apx.
• complete binary TL remains NP-hard
• complete binary TL has simple FPT algorithm
• 2-approximation algorithm for complete binary TL
• In practice, useful for non-complete trees too

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Other remarks

• The factor 2 is tight
• For non-complete trees
• In theory, no guarantee
• In practice, experiments show good results
• Average factor well below 2
• Generalization to d-ary trees
• O(n12log_d(d!)) time
• factor 1(d choose 2)

Drawing Complete Binary Tanglegrams
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Ribosomal DNA sequencing

• rDNA  genotypic identification procedure
• Whats the difference between these
• Involves the amplification of a phylogenetically
  informative target, such as the small-subunit
  (16S) rRNA gene

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