Synchronization Strings: Efficient and Fast Deterministic Constructions over Small Alphabets

Abstract: Synchronization strings are recently introduced by Haeupler and Shahrasbi [HS17a] in the study of codes for correcting insertion and deletion errors (insdel codes). A synchronization string is an encoding of the indices of the symbols in a string, and together with an appropriate decoding algorithm it can transform insertion and deletion errors into standard symbol erasures and corruptions. This reduces the problem of constructing insdel codes to the problem of constructing standard error correcting codes, which is much better understood. Besides this, synchronization strings are also useful in other applications such as synchronization sequences and interactive coding schemes. Amazingly, [HS17a] showed that for any error parameter $\varepsilon>0$, synchronization strings of arbitrary length exist over an alphabet whose size depends only on $\varepsilon$. Specifically, they obtained an alphabet size of $O(\varepsilon^{-4})$, as well as a randomized construction that runs in expected time $O(n^5)$. However, it remains an interesting question to find deterministic and more efficient constructions. In this paper, we improve the construction in [HS17a] in three aspects: we achieve a smaller alphabet size, a deterministic construction, and a faster algorithm. Along the way we introduce a new combinatorial object, and establish a new connection between synchronization strings and insdel codes --- such codes can be used in a simple way to construct synchronization strings. This new connection complements the connection found in [HS17a], and may be of independent interest. In an independent work [HS17c], Haeupler and Shahrasbi also give deterministic constructions of synchronization strings over arbitrary length (or even infinite length). Their constructions can achieve linear construction time, but have alphabet size $\varepsilon^{-O(1)}$, which may be larger than ours.