Recovery of cyclic words by their subwords

TL;DR

This paper investigates the problem of reconstructing cyclic binary words from their scattered subwords, establishing bounds on the subword length needed for unique reconstruction and providing counterexamples.

Contribution

It demonstrates that cyclic binary words can be reconstructed from scattered subwords of length approximately three-fourths of the word length, and shows the existence of non-unique cases at slightly smaller lengths.

Findings

Reconstruction possible with subwords of length 3/4 n + 4

Existence of two cyclic words with identical subwords of length 3/4 n - 1.5

Bounds on subword length for unique reconstruction

Abstract

A problem of reconstructing words from their subwords involves determining the minimum amount of information needed, such as multisets of scattered subwords of a specific length or the frequency of scattered subwords from a given set, in order to uniquely identify a word. In this paper we show that a cyclic word on a binary alphabet can be reconstructed by its scattered subwords of length $\frac34n+4$, and for each $n$ one can find two cyclic words of length $n$ which have the same set of scattered subwords of length $\frac34n-\frac32$.