On the Number of Subsequences in the Nonbinary Deletion Channel

TL;DR

This paper investigates the number of subsequences resulting from non-binary strings after deletions, providing bounds and characterizations for strings with maximum subsequences based on run-length.

Contribution

It introduces improved bounds and characterizes non-binary strings with maximum subsequences after deletions, with polynomial-time computability.

Findings

Characterized a family of r-run non-binary strings with maximum subsequences.

Provided bounds on the number of subsequences for r-run non-binary strings.

Showed the maximum number of subsequences can be computed in polynomial time.

Abstract

In the deletion channel, an important problem is to determine the number of subsequences derived from a string $U$ of length $n$ when subjected to $t$ deletions. It is well-known that the number of subsequences in the setting exhibits a strong dependence on the number of runs in the string $U$, where a run is defined as a maximal substring of identical characters. In this paper we study the number of subsequences of a non-binary string in this scenario, and propose some improved bounds on the number of subsequences of $r$-run non-binary strings. Specifically, we characterize a family of $r$-run non-binary strings with the maximum number of subsequences under any $t$ deletions, and show that this number can be computed in polynomial time.