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Drawing (Complete) Binary Tanglegrams

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

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Drawing (Complete) Binary Tanglegrams

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

  2. Tanglegram: • 2 trees • leaves matched 1-to-1 Drawing (Complete) Binary Tanglegrams

  3. Application example • Phylogenetic trees Pocket gopher drawings from The Animal Diversity Web (http://animaldiversity.org) Drawing (Complete) Binary Tanglegrams grandis castanops bursarius neglectus

  4. Comparing pairs of trees • Comparing trees • Visually • Applications • Software visualization • Hierarchical clustering • Phylogenetic trees Drawing (Complete) Binary Tanglegrams

  5. Input: 2 trees: S, T With leaves in 1-to-1 correspondence Output: plane drawings of S and T Minimizing # inter-tree crossings Problem statement:TL (Tanglegram Layout) S T 6 inter-tree crossings 4 inter-tree crossings 5 inter-tree crossings 3 inter-tree crossings Drawing (Complete) Binary Tanglegrams

  6. Related work • 2-sided crossing minimization problem • Introduced by Sugiyama et al. • Several differences • Arbitrary degree • Any ordering allowed Drawing (Complete) Binary Tanglegrams

  7. Previous work • Holten and Van Wijk (’08) • Visual Comparisonof Hierarchically Organized Data Drawing (Complete) Binary Tanglegrams

  8. 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 Drawing (Complete) Binary Tanglegrams

  9. 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 Drawing (Complete) Binary Tanglegrams

  10. 2-approximation algorithm • Simple recursive approach • Try each of 4 combinations, and recurse Drawing (Complete) Binary Tanglegrams Drawing Complete Binary Tanglegrams

  11. Initial algorithm ? • Algorithm: • Try each of the 4 combinations • Count crossings • Return the best one • Can’t count all crossings! ? Drawing (Complete) Binary Tanglegrams Drawing Complete Binary Tanglegrams

  12. 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 Drawing Complete Binary Tanglegrams

  13. Need to remember more Problematic situation: • Sometimes we can… Good situation  Drawing (Complete) Binary Tanglegrams Drawing Complete Binary Tanglegrams

  14. Use labels • To preserve this knowledge Initial layout Drawing (Complete) Binary Tanglegrams Drawing Complete Binary Tanglegrams

  15. Use labels • Using labels, we can count more crossings Problematic situation only if labels are equal (indeterminate crossing) Drawing (Complete) Binary Tanglegrams Drawing Complete Binary Tanglegrams

  16. 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 Drawing Complete Binary Tanglegrams

  17. 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 Drawing Complete Binary Tanglegrams

  18. Approximation factor (2) • Obs: Indeterminate crossings used to be “good” • Upperbound #INDby # of these crossing • Use that trees are complete • We know exactly how many edges each subtree has Drawing (Complete) Binary Tanglegrams Drawing Complete Binary Tanglegrams

  19. Conclusions • Studied binary TL / completebinary TL • binary TL has no constant factor apx. • completebinary TL remains NP-hard • completebinary TL has simple FPT algorithm • 2-approximation algorithm for completebinary TL • In practice, useful for non-complete trees too Drawing (Complete) Binary Tanglegrams

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