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Computing Height-Optimal Tangles Faster

  • Oksana FirmanEmail author
  • Philipp Kindermann
  • Alexander Ravsky
  • Alexander Wolff
  • Johannes Zink
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11904)

Abstract

We study the following combinatorial problem. Given a set of n y-monotone wires, a tangle determines the order of the wires on a number of horizontal layers such that the orders of the wires on any two consecutive layers differ only in swaps of neighboring wires. Given a multiset L of swaps (that is, unordered pairs of numbers between 1 and n) and an initial order of the wires, a tangle realizes L if each pair of wires changes its order exactly as many times as specified by L. The aim is to find a tangle that realizes L using the smallest number of layers. We show that this problem is NP-hard, and we give an algorithm that computes an optimal tangle for n wires and a given list L of swaps in \(O((2|L|/n^2+1)^{n^2/2} \cdot \varphi ^n \cdot n)\) time, where \(\varphi \approx 1.618\) is the golden ratio. We can treat lists where every swap occurs at most once in \(O(n!\varphi ^n)\) time. We implemented the algorithm for the general case and compared it to an existing algorithm. Finally, we discuss feasibility for lists with a simple structure.

Notes

Acknowledgments

We thank Thomas C. van Dijk for stimulating discussions and the anonymous reviewers for helpful comments.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut für InformatikUniversität WürzburgWürzburgGermany
  2. 2.National Academy of Sciences of UkraineLvivUkraine

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