Synchronization Strings: Efficient and Fast Deterministic Constructions over Small Alphabets

Synchronization strings are recently introduced by Haeupler and Shahrasb [11] in the study of codes for correcting insertion and deletion errors. 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 codes for insertion and deletion errors 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, Haeupler and Shahrasb [11] showed that for any error parameter ε > 0, synchronization strings of arbitrary length exist over an alphabet whose size depends only on ε. Specifically, [11] obtained an alphabet size of O(ε−4), as well as a randomized algorithm to construct such a synchronization string with length n in expected time O(n). However, it remains an interesting question to find deterministic and more efficient constructions. In this paper, we improve the construction in [11] in three aspects: we achieve a smaller alphabet size, a deterministic construction, and a faster algorithm to construct synchronization strings. Along the way we introduce a new combinatorial object, and establish a new connection between synchronization strings and codes for insertion and deletion errors — such codes can be used in a simple way to construct synchronization strings. This new connection complements the connection found in [11], and may be of independent interest. ∗[email protected]. Department of Computer Science, Johns Hopkins University. Supported by NSF Grant CCF-1617713. †[email protected]. Department of Computer Science, Johns Hopkins University. Supported by NSF Grant CCF-1617713. ‡[email protected]. Department of Computer Science, Johns Hopkins University. 1 ar X iv :1 71 0. 07 35 6v 1 [ cs .I T ] 1 9 O ct 2 01 7