Modules in Robinson Spaces

TL;DR

This paper studies the structure of modules in Robinson spaces and introduces an efficient divide-and-conquer algorithm to recognize Robinson spaces in optimal quadratic time.

Contribution

It characterizes modules in Robinson spaces and presents a practical algorithm for their recognition with optimal complexity.

Findings

Modules in Robinson spaces have a specific structure that can be exploited.

The proposed algorithm recognizes Robinson spaces in O(n^2) time.

The copoint partition aids in the divide-and-conquer recognition process.

Abstract

A Robinson space is a dissimilarity space $(X,d)$ (i.e., a set $X$ of size $n$ and a dissimilarity $d$ on $X$) for which there exists a total order $<$ on $X$ such that $x<y<z$ implies that $d(x,z)\ge \max\{ d(x,y), d(y,z)\}$. Recognizing if a dissimilarity space is Robinson has numerous applications in seriation and classification. An mmodule of $(X,d)$ (generalizing the notion of a module in graph theory) is a subset $M$ of $X$ which is not distinguishable from the outside of $M$, i.e., the distance from any point of $X\setminus M$ to all points of $M$ is the same. If $p$ is any point of $X$, then $\{ p\}$ and the maximal by inclusion mmodules of $(X,d)$ not containing $p$ define a partition of $X$, called the copoint partition. In this paper, we investigate the structure of mmodules in Robinson spaces and use it and the copoint partition to design a simple and practical divide-and-conquer algorithm for recognition of Robinson spaces in optimal $O(n^2)$ time.