Ordering in Sequence Spaces: An Overview

Abstract

“Creating order” is maybe one of the most important human activities. In its simplest form, ordering is just “sorting”, which is a mathematically well understood problem. However, in real life we are often facing practical limitations which inhibit complete sorting. These limitations can be either knowledge (information) restrictions —we don’t know the future, we forget the past— or manipulation restrictions —we don’t want to carry objects too far—.

A mathematical theory of ordering (with constraints) in sequence spaces was first presented in [7] and [1]. In their setup, an algorithm is sought which “orders” any sequence of length n, i.e., which transforms the sequence $\overrightarrow x$ into the sequence $\overrightarrow y$ (of the same length and with the same symbols in it), such that the number of possible resulting sequences $\overrightarrow y$ is as small as possible. In this sense ordering is a generalization of sorting $\overrightarrow x$, as this would yield the absolute minimal number of sequences $\overrightarrow y$.

However, the model imposes extra restrictions on the ordering algorithm: a window of size ß moves over the sequence, and the algorithm is only allowed to interchange the symbols within the window; moreover, at any time the algorithm cannot examine the sequence except for π “past” and ø “future” symbols.

This simple setup leads to several nice nontrivial mathematical problems, several of which are still unsolved.