String Reconstruction from Substring Compositions

Abstract

Motivated by mass-spectrometry protein sequencing, we consider a simply-stated problem of reconstructing a string from the multiset of its substring compositions. We show that all strings of length 7, one less than a prime, or one less than twice a prime, can be reconstructed uniquely up to reversal. For all other lengths we show that reconstruction is not always possible and provide sometimes-tight bounds on the largest number of strings with given substring compositions. The lower bounds are derived by combinatorial arguments and the upper bounds by algebraic considerations that precisely characterize the set of strings with the same substring compositions in terms of the factorization of bivariate polynomials. The problem can be viewed as a combinatorial simplification of the turnpike problem, and its solution may shed light on this long-standing problem as well. Using well known results on transience of multi-dimensional random walks, we also provide a reconstruction algorithm that reconstructs random strings over alphabets of size ≥ 4 in optimal near-quadratic time. * UCSD.

Reconstruction and Error-Correction Codes for Polymer-Based Data Storage

“…We answer both questions affirmatively by describing a coding scheme that allows for unique reconstruction and correction of a single deletion-insertion mass error. Encoding is performed by interleaving symmetric strings with Catalantype paths, while decoding is accomplished through a modification of the backtracking decoding algorithm described in [7]. Our work extends the existing literature in coded string reconstruction [9], [10].…”

“…In an earlier line of work, the authors of [7] introduced the problem of binary string reconstruction from its substring composition multiset to address the issue of MS/MS readout analysis. The substring composition multiset of a binary string is obtained by writing out all substrings of the string of all possible length and then representing each substring by its composition.…”

“…Note that composition multisets ignore information about the actual order of the bits and may hence be seen as only capturing the information about the "mass" or "weight" of the string. The problem addressed in [7] was to determine for which string lengths may one guarantee unique reconstruction from an error-free composition multiset up to string reversal. The main results of [7,Theorem 17,18,20] asserts that binary strings of length ≤ 7, one less than a prime, or one less than twice a prime are uniquely reconstructable up to reversal.…”

“…For our line of work, we also rely on the two modeling assumptions described in [7]: Assumption 1. We can infer the composition of a polymer substring from its mass.…”

“…Unlike the work in [7] which has solely focused on the problem of unique string reconstruction, we view the problem from a coding-theoretic perspective and ask the following:…”