A Generalized Balanced Pair Algorithm

TL;DR

This paper introduces a generalized version of the balanced pair algorithm that works in reducible cases, terminates more frequently, and discusses factors affecting its termination with illustrative examples.

Contribution

It extends the balanced pair algorithm to reducible cases and analyzes conditions influencing its termination, providing new insights and conjectures.

Findings

The generalized algorithm terminates more often than the standard version.

Examples illustrate the algorithm's behavior in reducible cases.

Discussion of features leading to non-termination and related conjectures.

Abstract

We present here a more general version of the balanced pair algorithm. This version works in the reducible case and terminates more often than the standard algorithm. We present examples to illustrate this point. Lastly, we discuss the features which lead to balanced pair algorithms not terminating and state several conjectures.