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 BuchinMaike Buchin
Jaroslaw ByrkaMartin NöllenburgYoshio 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)
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Comparing pairs of trees
• Comparing trees– Visually
• Applications– Software
visualization– Hierarchical
clustering– Phylogenetic trees
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4 inter-tree crossings5 inter-tree crossings3 inter-tree crossings
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
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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 Comparison
of Hierarchically Organized Data
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Previous work (cont’d)
• 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 c≈1024
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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 * under widely accepted conjectures
binary TL
complete binary TL
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2-approximation algorithm
• Simple recursive approach• Try each of 4 combinations, and recurse
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Initial algorithm
• Algorithm:– Try each of the 4
combinations– Count crossings– Return the best
one
• Can’t count all crossings!
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?
?
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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… ?
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Need to remember more
• Sometimes we can…
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Problematic situation:
Good situation
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Use labels
• To preserve this knowledge
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Initial layout
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Use labels
• Using labels, we can count more crossings
Drawing Complete Binary Tanglegrams
Problematic situation only if labels are
equal(indeterminate
crossing)
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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)
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• Mistakes from indeterminate crossings– We cannot count them
• How many can we have?
• We show that #IND copt
• Therefore calg 2 copt
Approximation factor
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# crossings in optimal drawing
# crossings in algorithm drawing
# indeterminate crossings
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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
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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