Computing Height-Optimal Tangles Faster

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 time, where is the golden ratio. We can treat lists where every swap occurs at most once in 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.