Computing Height-Optimal Tangles Faster [article]

Oksana Firman, Philipp Kindermann, Alexander Ravsky, Alexander Wolff, Johannes Zink
<span title="2019-09-05">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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
more &raquo; ... 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·φ^n · n) time, where φ≈ 1.618 is the golden ratio. We can treat lists where every swap occurs at most once in O(n!φ^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.
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