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PQTree Algorithm [#!BL76!#]
A PQTree is a rooted, ordered tree. We will use a PQtree
with the elements of U as leaves, and internal nodes of two
types: Pnodes and Qnodes.
A Pnode whose subnodes are
for
represents k subsets of U (the leaf sets of
), each of which is known to be a consecutive
block of elements, but with the order of the blocks unknown. A
Qnode whose subnodes are
for
represents that the k blocks corresponding to the leaf set of
are known to appear in this order, up to a
complete reversal (see figure 9.3).
It is therefore clear that in order to have these meanings of the
Pnodes and Qnodes we must allow the following legal
transformations (see figure 9.5
1  2).
 1.
 Reordering the subnodes of some Pnode arbitrarily
 2.
 Reversing the order of the subnodes of some Qnode
Figure 9.3:
PQtree node types: we
use circles and bars to denote Pnodes and Qnodes, respectively.

Figure 9.4:
Frontier of a PQtree

Figure 9.5:
Permitted
transformations of a PQtree

Figure 9.6:
Permitted
transformations of a PQtree

Two PQtrees T and T' are said to be
equivalent if there exists a set of legal transformations
leading from one tree to the other. In such a case, we write
.
Therefore, the problem of permuting the probes in order to achieve
the consecutive 1's property of the STS matrix is equivalent to
finding a PQtree representing
.
PQTree Algorithm for Unique Probe DNA Mapping:
 1.
 Initialize the tree as a root Pnode with all elements of U as subnodes
(leaves).
 2.
 For
: Reduce (T,S_{i})
After Stage i, T is induced, i.e. modified, so such that
consistent(T) contains
only permutations in which S_{i} is continuous.
Reduce(T,S_{i})
 1.
 Color all S_{i} leaves.
 2.
 Apply transformations to replace T with an equivalent
PQtree along whose frontier all of the colored leaves are
consecutive.
 3.
 Identify the deepest node
Root(T,S_{i}) whose subtree
spans all colored leaves.
 4.
 Apply replacement rules presented in figure 9.6: Traverse the nodes of this subtree, working
bottomup till reaching
Root(T,S_{i}) (``root'' or the figure means root of the subtree) and apply the appropriate rule upon visiting a node. For an empty P node, and
for full or empty Q node, there is no change needed. If there is no matching pattern, return the null tree (meaning
there is no correct arrangement).
 5.
 Remove the coloring (all the information in S_{i} is now included in
the structure of the tree).
Figure 9.7:
Example of PQtree
based algorithm

Figure 9.7 shows an example of application
of the PQTree algorithm for unique probe DNA mapping.
The problem with using PQtrees for solving the unique mapping
problem is that the algorithm does not support noise:
Unfortunately, due to "real life" measurement errors the input
matrix usually has either extra or missing 1's entries. In such
case, the resulting PQtree^{1} will not produce the best
(minimum error) solution available, but rather an arbitrary
solution depending on the clone order chosen. Since all data is
obtained by experiments and errors are not uncommon, this
deficiency deters one from using the algorithm.
Next: Solving the Unique Mapping
Up: DNA Physical Mapping
Previous: Unique Probe Mapping
Peer Itsik
20010109