A simple and optimal algorithm for strict circular seriation

TL;DR

This paper introduces a simple, efficient $O(n \,\log n)$ algorithm for strict circular seriation, improving upon previous methods, and provides verification and characterization of circular Robinson dissimilarities.

Contribution

It presents a new simple $O(n \,\log n)$ algorithm for strict circular seriation and characterizes circular Robinson dissimilarities as pre-circular Robinson dissimilarities.

Findings

The new algorithm runs in $O(n \,\log n)$ time.

The algorithm can verify compatibility in $O(n^2)$ time.

Circular Robinson dissimilarities are exactly the pre-circular Robinson dissimilarities.

Abstract

Recently, Armstrong, Guzm\'an, and Sing Long (2021), presented an optimal $O(n^2)$ time algorithm for strict circular seriation (called also the recognition of strict quasi-circular Robinson spaces). In this paper, we give a very simple $O(n\log n)$ time algorithm for computing a compatible circular order for strict circular seriation. When the input space is not known to be strict quasi-circular Robinson, our algorithm is complemented by an $O(n^2)$ time verification of compatibility of the returned order. This algorithm also works for recognition of other types of strict circular Robinson spaces known in the literature. We also prove that the circular Robinson dissimilarities (which are defined by the existence of compatible orders on one of the two arcs between each pair of points) are exactly the pre-circular Robinson dissimilarities (which are defined by a four-point condition).