Positional sequencing by hybridization.

Sequencing by hybridization (SBH) is a promising alternative to the classical DNA sequencing approaches. However, the resolving power of SBH is rather low: with 64kb sequencing chips, unknown DNA fragments only as long as 200 bp can be reconstructed in a single SBH experiment. To improve the resolving power of SBH, positional SBH (PSBH) has recently been suggested; this allows (with additional experimental work) approximate positions of every l-tuple in a target DNA fragment to be measured. We study the positional Eulerian path problem motivated by PSBH. The input to the positional eulerian path problem is an Eulerian graph G(V, E) in which every edge has an associated range of integers and the problem is to find an Eulerian path e1,...,e/E/ in G such that the range of ei contains i. We show that the positional Eulerian path problem is NP-complete even when the maximum out-degree (in-degree) of any vertex in the graph is 2. On a positive note we present polynomial algorithms to solve a special case of PSBH (bounded PSBH), where the range of the allowed positions for any edge is bounded by a constant (it corresponds to accurate experimental measurements of positions in PSBH). Moreover, if the positions of every l-tuple in an unknown DNA fragment of length n are measured with O(log n) error, then our algorithm runs in polynomial time. We also present an estimate of the resolving power of PSBH for a more realistic case when positions are measured with theta (n) error.