Reconstructing Strings from Substrings

Abstract

We consider an interactive approach to DNA sequencing by hybridization, where we are permitted to ask questions of the form "is s a substring of the unknown sequence S?", where s is a specific query string. We are not told where s occurs in S, nor how many times it occurs, just whether or not s a substring of S. Our goal is to determine the exact contents of S using as few queries as possible. Through interaction, far fewer queries are necessary than using conventional fixed sequencing by hybridization (SBH) sequencing chips. We provide tight bounds on the complexity of reconstructing unknown strings from substring queries. Our lower bound, which holds even for a stronger model that returns the number of occurrence of s as a substring of S, relies on interesting arguments based on de Bruijn sequences. We also demonstrate that subsequence queries are significantly more powerful than substring queries, matching the information theoretic lower bound. Finally, in certain applications, something may already be known about the unknown string, and hence it can be determined faster than an arbitrary string. We show that building an optimal decision tree is NP-complete, then give an approximation algorithm that gives trees within a constant multiplicative factor of optimal.

Restricting SBH Ambiguity via Restriction Enzymes

“…Indeed, [19] proves the problem of generating an SBH-consistent sequence of a given length is NP-complete. Thus we employ a search-based algorithm to find a suitable sequence.…”

Efficient Reconstruction of RC-Equivalent Strings

“…the information theoretic lower bound on the number of queries necessary for exact reconstruction. We also give a simple algorithm for arbitrary alphabets, adapted from a paper by Skiena and Sundaram [13], where the length of the queries is not bounded, using O(n log |Σ|) queries.…”

“…We adapt a simple algorithm from [13], originally developed for the classic case (where queries would answer no if the subsequence only appears in the reverse complement of the string).…”

Optimal Probing Patterns for Sequencing by Hybridization

“…Thus, if, for example, we wish to reconstruct at least 0.9 fraction of the sequences of length n, then n must be less than roughly 2 k [2,8,20,23]. Several methods for overcoming this limitation of SBH were proposed: interactive protocols [10,17,25,28], using location information [1,4,5,7,11,23], using a known homologous string [18,19,27], and using restriction enzymes [24,26].…”