Abstract
A tanglegram is a pair of trees on the same set of leaves with matching leaves in the two trees joined by an edge. Tanglegrams are widely used in biology – to compare evolutionary histories of host and parasite species and to analyze genes of species in the same geographical area. We consider optimizations problems in tanglegram drawings. We show a linear time algorithm to decide if a tanglegram admits a planar embedding by a reduction to the planar graph drawing problem. This problem was considered by Fernau, Kauffman and Poths. (FSTTCS 2005). Our reduction method provides a simpler proof and helps to solve a conjecture they posed, showing a fixed-parameter tractable algorithm for minimizing the number of crossings over all d-ary trees.
For the case where one tree is fixed, we show an O(n logn) algorithm to determine the drawing of the second tree that minimizes the number of crossings. This improves the bound from earlier methods. We introduce a new optimization criterion using Spearman’s footrule optimization and give an O(n 2) algorithm.
We also show integer programming formulations to quickly obtain tanglegram drawings that minimize the two optimization measures discussed. We prove lower bounds on the maximum gap between the optimal solution and the heuristic of Dwyer and Schreiber (Austral. Symp. on Info. Vis. 2004) to minimize crossings.
This research was partially supported by NSF grants SEI-BIO 0513910, SEI-SBE 0513660, CCF-0515378, and IIS-0803564.
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Venkatachalam, B., Apple, J., St. John, K., Gusfield, D. (2009). Untangling Tanglegrams: Comparing Trees by Their Drawings. In: Măndoiu, I., Narasimhan, G., Zhang, Y. (eds) Bioinformatics Research and Applications. ISBRA 2009. Lecture Notes in Computer Science(), vol 5542. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01551-9_10
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