Ordering in Sequence Spaces: An Overview

• Peter Vanroose
Chapter

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.

Keywords

Transition Matrix Sequence Space Large Eigenvalue Optimal Rate State Diagram
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

1. [1]
R. Ahiswede, J.-P. Ye and Z. Zhang, “Creating order in sequence spaces with simple machines”. Information and Computation, 89 (1), 1990, 47 – 94.
2. [2]
R. Ahiswede and Z. Zhang, “Contributions to a theory of ordering for sequence spaces”. Problems of Control and Information theory, 18 (4), 1989, 197 – 221.
3. [3]
H. D. L. Hollmann and P. Vanroose, “Entropy reduction, ordering in sequence spaces, and semigroups of nonnegative matrices”, Preprint 95–092, SFB 343 “Diskrete Strukturen in der Mathematik”, Universität Bielefeld, 1995.Google Scholar
4. [4]
U. Tamm, “The influence of memory on creating order”. Preprint 96–031, SFB 343 “Diskrete Strukturen in der Mathematik”, Universität Bielefeld, 1996.Google Scholar
5. [5]
U. Tamm, “Ballot sequences in creating order”. Preprint, SFB 343 “Diskrete Strukturen in der Mathematik”, Universität Bielefeld, 1998.Google Scholar
6. [6]
P. Vanroose, Een ordeningsresultaat voor de situatie (0, 2, 1, T+) (in Dutch). PhD supplement, Katholieke Universiteit Leuven, 1989.Google Scholar
7. [7]
Jian-Ping Ye, Towards a Theory of Ordering in Sequence Spaces. PhD thesis, Fakultät für Mathematik der Universität Bielefeld, 1988.Google Scholar