Sequencing by Hybridization

Standard oligo chips can, at least theoretically, be used for sequencing. Let us prepare an oligo chip that contains all possible sequences of length k. These sequences are called k-mers. Practical values of k are 8-10. If we expose this chip to a solution containing some target DNA, the results will show which k-mers occur in the target sequence. This gives rise to the definition of the   k-spectrum of a sequence T as the multi-set of all its substrings of length k. We would now like to reconstruct this sequence.

Problem 12.1   Reconstructing a sequence from hybridization data
INPUT: A multi-set S of k-mers
QUESTION: Is S the spectrum of a sequence T? If yes, reconstruct this sequence.

It should be noted that we assume that if a k-mer appears several times in the target DNA, the hybridization experiment will report its multiplicity (this is why we require the input to be a multi-set and not simply a set). To date, this requirement is impractical. For instance, for k=3:

$$\begin{eqnarray*}{} \centering T & = & ATGCAGGTCCAG \\ S & = & \{ATG, AGG, CAG, GCA, GGT, GTC, TCC, TGC, CCA, CAG\} \end{eqnarray*}$$