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.

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We present a new, practical algorithm to resolve the experimental data in restriction site analysis, which is a common technique for mapping DNA. Specifically, we assert that multiple digestions with a single restriction enzyme can provide sufficient information to identify the positions of the restriction sites with high probability. The motivation for the new approach comes from combinatorial results on the number of mutually homeometric sets in one dimension, where two sets of n points are homeometric if the multiset of n(n-1)/2 distances they determine are the same. Since experimental data contain errors, we propose algorithms for reconstructing sets from noisy interpoint distances, including the possibility of missing fragments. We analyse the performance of these algorithms under a reasonable probability distribution, establishing a relative error limit of r = theta(1/n2) beyond which our technique becomes infeasible. Through simulations, we establish that our technique is robust enough to reconstruct data with relative errors of up to 7.0% in the measured fragment lengths for typical problems, which appears sufficient for certain biological applications.

We present some experiences with the problem of multiple genome comparison, analogous to multiple sequence alignment in sequence comparison, under the inversion and transposition distance metrics, given a fixed phylogeny. We first describe a heuristic for the case in which phylogeny is a star on three vertices and then use this to approximate the multiple genome comparison problem via local search.

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