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 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.