On the Word Fragment Length for Unambiguous Reconstruction of a Periodic Word from a Complete Multiset of Fragments of Fixed Length

Abstract

We consider the problem of reconstructing a word from a multiset of its fragments of fixed length. Words consist of symbols from a finite alphabet. The word to be reconstructed is assumed to be periodic or contain a periodic word as a subword. It is shown that a periodic word with a period p can be reconstructed from a multiset of its fragments of length k, where k geqslant leftlfloor {frac{{16}}{7}sqrt p } rightrfloor + 5. For a word consisting of a q-periodic prefix repeated m times and a p-periodic suffix repeated l times, if l geqslant m{{q}^{{leftlfloor {frac{{16}}{7}sqrt P } rightrfloor + 5}}}, then the estimate becomes k geqslant leftlfloor {frac{{16}}{7}sqrt P } rightrfloor + 5, where P = {text{max(}}p,~q{text{)}}.