Abstract
We study the crossing-minimization problem in a layered graph drawing of planar-embedded rooted trees whose leaves have a given total order on the first layer, which adheres to the embedding of each individual tree. The task is then to permute the vertices on the other layers (respecting the given tree embeddings) in order to minimize the number of crossings. While this problem is known to be NP-hard for multiple trees even on just two layers, we describe a dynamic program running in polynomial time for the restricted case of two trees. If there are more than two trees, we restrict the number of layers to three, which allows for a reduction to a shortest-path problem. This way, we achieve XP-time in the number of trees.
J. Katheder is supported by DFG grant Ka 812-18/2 and
J. Zink is supported by DFG grant Wo 758/11-1.
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Notes
- 1.
XP is a parameterized running-time class and an XP-algorithm has a running time in \(\mathcal {O}(|I|^{f(k)})\), where |I| is the size of the instance, f a computable function, and k the parameter. Note that every FPT-algorithm is an XP-algorithm but not vice versa.
- 2.
This is a generalization of the positions introduced in Sect. 3 where all positions were relative to (the given embedding of) \(T_1\).
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Acknowledgments
We thank the organizers of the workshop GNV 2022 in Heiligkreuztal for the fruitful atmosphere where some of the ideas of this paper arose. We also thank the anonymous reviewers for their helpful feedback.
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Katheder, J., Kobourov, S.G., Kuckuk, A., Pfister, M., Zink, J. (2024). Simultaneous Drawing of Layered Trees. In: Uehara, R., Yamanaka, K., Yen, HC. (eds) WALCOM: Algorithms and Computation. WALCOM 2024. Lecture Notes in Computer Science, vol 14549. Springer, Singapore. https://doi.org/10.1007/978-981-97-0566-5_5
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