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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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Drawing (Complete) Binary Tanglegrams 2 Tanglegram: 2 trees leaves matched 1-to-1
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Drawing (Complete) Binary Tanglegrams 3 Application example Phylogenetic trees castanops neglectus bursarius grandis Pocket gopher drawings from The Animal Diversity Web (http://animaldiversity.org)
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Drawing (Complete) Binary Tanglegrams 4 Outline of this talk Introduction 2-approximation algorithm –Algorithm –Approximation factor Conclusions
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Drawing (Complete) Binary Tanglegrams 5 Comparing pairs of trees Comparing trees –Visually Applications –Software visualization –Hierarchical clustering –Phylogenetic trees
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Drawing (Complete) Binary Tanglegrams 6 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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Drawing (Complete) Binary Tanglegrams 7 Related work 2-sided crossing minimization problem –Introduced by Sugiyama et al. Several differences –Arbitrary degree –Any ordering allowed
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Drawing (Complete) Binary Tanglegrams 8 Previous work Holten and Van Wijk (’08) –Visual Comparison of Hierarchically Organized Data
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Drawing (Complete) Binary Tanglegrams 9 Previous work (cont’d) Dwyer and Schreiber (’04) –2.5D drawings of stacked trees –One sided (binary) version, O(n 2 log n) time. Fernau, Kaufmann and Poths (’05) –TL is NP-hard –1 (binary) tree fixed: O(n log 2 n) time. –FPT algorithm O*(c k ), for c≈1024
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Drawing (Complete) Binary Tanglegrams 10 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(4 k n 2 )-time FPT algorithm * under widely accepted conjectures binary TL complete binary TL
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Drawing (Complete) Binary Tanglegrams 11 2-approximation algorithm Simple recursive approach Try each of 4 combinations, and recurse Drawing Complete Binary Tanglegrams
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Drawing (Complete) Binary Tanglegrams 12 Initial algorithm Algorithm: –Try each of the 4 combinations –Count crossings –Return the best one Can’t count all crossings! Drawing Complete Binary Tanglegrams ? ?
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Drawing (Complete) Binary Tanglegrams 13 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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Drawing (Complete) Binary Tanglegrams 14 Need to remember more Sometimes we can… Drawing Complete Binary Tanglegrams Problematic situation: Good situation
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Drawing (Complete) Binary Tanglegrams 15 Use labels To preserve this knowledge Drawing Complete Binary Tanglegrams Initial layout
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Drawing (Complete) Binary Tanglegrams 16 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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Drawing (Complete) Binary Tanglegrams 17 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(n 3 ) Drawing Complete Binary Tanglegrams
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Drawing (Complete) Binary Tanglegrams 18 Mistakes from indeterminate crossings –We cannot count them How many can we have? We show that #IND c opt Therefore c alg 2 c opt Approximation factor Drawing Complete Binary Tanglegrams # crossings in optimal drawing # crossings in algorithm drawing # indeterminate crossings
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Drawing (Complete) Binary Tanglegrams 19 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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Drawing (Complete) Binary Tanglegrams 20 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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Drawing (Complete) Binary Tanglegrams 21 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(n 1+2log_d(d!) ) time – factor 1+(d choose 2) Drawing Complete Binary Tanglegrams
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Drawing (Complete) Binary Tanglegrams 22 Ribosomal DNA sequencing rDNA: genotypic identification procedure What’s the difference between these: Involves the amplification of a phylogenetically informative target, such as the small-subunit (16S) rRNA gene
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Drawing (Complete) Binary Tanglegrams 23
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