# Prefix Block-Interchanges on Binary and Ternary Strings

## TL;DR

This paper studies prefix block-interchanges on binary and ternary strings, providing bounds, algorithms, and classifications to optimize string grouping and sorting operations relevant to genome rearrangement problems.

## Contribution

It introduces new upper bounds, algorithms, and classifications for prefix block-interchanges on binary and ternary strings, improving understanding of genome rearrangement operations.

## Key findings

- Linear-time algorithm for grouping binary normalized strings.
- Polynomial-time algorithm for grouping ternary normalized strings.
- Classification of ternary strings based on prefix block-interchange operations.

## Abstract

The genome rearrangement problem computes the minimum number of operations that are required to sort all elements of a permutation. A block-interchange operation exchanges two blocks of a permutation which are not necessarily adjacent and in a prefix block-interchange, one block is always the prefix of that permutation. In this paper, we focus on applying prefix block-interchanges on binary and ternary strings. We present upper bounds to group and sort a given binary/ternary string. We also provide upper bounds for a different version of the block-interchange operation which we refer to as the `restricted prefix block-interchange'. We observe that our obtained upper bound for restricted prefix block-interchange operations on binary strings is better than that of other genome rearrangement operations to group fully normalized binary strings. Consequently, we provide a linear-time algorithm to solve the problem of grouping binary normalized strings by restricted prefix block-interchanges. We also provide a polynomial time algorithm to group normalized ternary strings by prefix block-interchange operations. Finally, we provide a classification for ternary strings based on the required number of prefix block-interchange operations.