Download - Drawing (Complete) Binary Tanglegrams Hardness, Approximation, Fixed-Parameter Tractability Utrecht U, NL TU Eindhoven, NL Karlsruhe U, DE Tokio Inst

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

Drawing (Complete) Binary Tanglegrams 2

Tanglegram:• 2 trees• leaves matched 1-to-1


Application example

• Phylogenetic trees

Pocket gopher drawings from The Animal Diversity Web (http://animaldiversity.org)


Comparing pairs of trees

• Comparing trees– Visually

• Applications– Software

visualization– Hierarchical

clustering– Phylogenetic trees


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


Related work

• 2-sided crossing minimization problem

– Introduced by Sugiyama et al.

• Several differences– Arbitrary degree– Any ordering allowed


Previous work

• Holten and Van Wijk (’08)– Visual Comparison

of Hierarchically Organized Data


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


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


2-approximation algorithm

• Simple recursive approach• Try each of 4 combinations, and recurse

Drawing Complete Binary Tanglegrams


Initial algorithm

• Algorithm:– Try each of the 4

combinations– Count crossings– Return the best

one

• Can’t count all crossings!


?

?


Types of crossings

• Lower-level– Created by recursive calls– Nothing to do about them

• Current-level– Can be avoided at this level

• What about… ?



Need to remember more

• Sometimes we can…


Problematic situation:

Good situation


Use labels

• To preserve this knowledge


Initial layout


Use labels

• Using labels, we can count more crossings


Problematic situation only if labels are

equal(indeterminate

crossing)


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)



• Mistakes from indeterminate crossings– We cannot count them

• How many can we have?

• We show that #IND copt

• Therefore calg 2 copt

Approximation factor


# crossings in optimal drawing

# crossings in algorithm drawing

# indeterminate crossings


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



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

Download - Drawing (Complete) Binary Tanglegrams Hardness, Approximation, Fixed-Parameter Tractability Utrecht U, NL TU Eindhoven, NL Karlsruhe U, DE Tokio Inst

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