Synchronization Strings: Highly Efficient Deterministic Constructions over Small Alphabets

Abstract

Synchronization strings are recently introduced by Haeupler and Shahrasbi [15] 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. For all such applications, synchronization strings are desired to be over alphabets that are as small as possible, since a larger alphabet size corresponds to more redundant information added.Haeupler and Shahrasbi [15] showed that for any parameter ε > 0, synchronization strings of arbitrary length exist over an alphabet whose size depends only on ε. Specifically, [15] obtained an alphabet size of O(ε −4 ), which left an open question on where the minimal size of such alphabets lies between Ω(ε −1 ) and O(ε −4 ). In this work, we partially bridge this gap by providing an improved lower bound of Ω ε −3/2 , and an improved upper bound of O ε −2 . We also provide fast explicit constructions of synchronization strings over small alphabets.Further, along the lines of previous work on similar combinatorial objects, we study the extremal question of the smallest possible alphabet size over which synchronization strings can exist for some constant ε < 1. We show that one can construct ε-synchronization strings over alphabets of size four while no such string exists over binary alphabets. This reduces the extremal question to whether synchronization strings exist over ternary alphabets.

Synchronization strings: explicit constructions, local decoding, and applications

“…The first boosting procedure, given in Section 4.4, leads to a deterministic linear time synchronization string construction. We remark that concurrently and independently Cheng, Li, and Wu obtained a deterministic O(n log 2 log n) time synchronization string construction [8].…”

Near-linear time insertion-deletion codes and (1+ ε )-approximating edit distance via indexing

“…We will also make use of the following construction of synchronization strings that is developed in [15,23]. Haeupler et al [24] suggest a construction of list-decodable insertion-deletion codes by indexing the codewords of a list-recoverable code with symbols of a synchronization string.…”

“…Having such lists, the receiver can use the list-recovery function of C to obtain an L-list-decoding for C . The encoding complexity follows from the fact that synchronization strings be constructed in linear time [15,23], the decoding complexity follows from Theorem 7.1, and the alphabet of C is trivially Σ × Σ s × Σ I as it is obtained by indexing codewords of C with the ε s -synchronization string and the ε I -indexing sequence.…”

“…Theorem 1.3 from[15]). For any ε ∈ (0, 1) and integer n, one can construct an ε-synchronization string of length n over an alphabet of size O ε −3 in linear time.…”

Deterministic Document Exchange Protocols, and Almost Optimal Binary Codes for Edit Errors

“…al [12], [11], [5] constructed explicit codes that can correct 1 − ε fraction of edit errors with rate Ω(ε 5 ) and alphabet size poly(1/ε); and codes that can correct 1 − 2 t+1 − ε fraction of errors with rate (ε/t) poly(1/ε) for a fixed alphabet size t ≥ 2. Another line of work by Haeupler et al [14], [15], [7] introduced and constructed a combinatorial object called synchronization string, which can be used to transform standard error correcting codes into codes for edit errors by increasing the alphabet size. Via this transformation, [14] achieved explicit codes that can correct δ fraction of edit errors with rate 1 − δ − ε and alphabet size exponential in 1 ε , which approaches the singleton bound.…”