1. 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

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

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

4. Drawing (Complete) Binary Tanglegrams 4 Outline of this talk Introduction 2-approximation algorithm –Algorithm –Approximation factor Conclusions

5. Drawing (Complete) Binary Tanglegrams 5 Comparing pairs of trees Comparing trees –Visually Applications –Software visualization –Hierarchical clustering –Phylogenetic trees

6. 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

7. Drawing (Complete) Binary Tanglegrams 7 Related work 2-sided crossing minimization problem –Introduced by Sugiyama et al. Several differences –Arbitrary degree –Any ordering allowed

8. Drawing (Complete) Binary Tanglegrams 8 Previous work Holten and Van Wijk (’08) –Visual Comparison of Hierarchically Organized Data

9. 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

10. 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

11. Drawing (Complete) Binary Tanglegrams 11 2-approximation algorithm Simple recursive approach Try each of 4 combinations, and recurse Drawing Complete Binary Tanglegrams

12. 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 ? ?

13. 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

14. Drawing (Complete) Binary Tanglegrams 14 Need to remember more Sometimes we can… Drawing Complete Binary Tanglegrams Problematic situation: Good situation

15. Drawing (Complete) Binary Tanglegrams 15 Use labels To preserve this knowledge Drawing Complete Binary Tanglegrams Initial layout

16. 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)

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. Drawing (Complete) Binary Tanglegrams 23