Unique Probe Mapping

An STS (Sequence Tagged Site) probe or STS discriminator is a filter that can uniquely determine whether or not a specific short sequence of DNA appears along any given (longer) sequence. The filter can identify the existence of the short sequence but provides no information as to its location. Using a short sequence of 200 - 300 bases ensures that the probability of an error in recognition is relatively low. Running a number of STS probes against numerous clones results in a matrix cell M_i,j with the entry 1 (0) representing a positive (negative) result of probe j against clone i,Figure 10.1 gives an example of ordered clones and corresponding STS probes. Figure 10.2 presents the resulting STS matrix.

  

Figure 10.1: Ordered clones and several STS probes

  

Figure 10.2: Resulting STS matrix. 1 in line i column j denotes that clone i contains probe j.

Problem 10.1   The unique probes mapping problem.
INPUT: A set of elements U (probes) and a collection of subsets $\varphi = \{S_{1},S_{2},\ldots,S_{n}\}, \forall \imath : S_{\imath} \subseteq U$
QUESTION: Find the set $\Pi(\varphi)$ of all permutations over U along which every $S_{\imath}$ is continuous.

Problem 10.1 is equivalent to the problem of rearranging the columns of the STS result matrix so that all 1's in the rows of the matrix are continuous. This attribute is also known as the consecutive 1's property. The problem of finding the set of permutations $\Pi(\varphi)$ is a well known problem in computer science. A linear time algorithm for solving this problem was presented in 1976 by Booth and Lueker [1]. Clearly, an explicit representation of the collection of all the resulting permutations may require much more than linear space. Therefore, a linear time algorithm requires a linear space representation of this collection. This linear representation is achieved by PQ-trees which are described in the following section.