PQ-Tree Algorithm [#!BL76!#]

A PQ-Tree is a rooted, ordered tree. We will use a PQ-tree with the elements of U as leaves, and internal nodes of two types: P-nodes and Q-nodes. A P-node whose sub-nodes are $T_{1},\ldots,T_{k}$ for $k \geq 2$ represents k subsets of U (the leaf sets of $T_{1},\ldots,T_{k}$), each of which is known to be a consecutive block of elements, but with the order of the blocks unknown. A Q-node whose sub-nodes are $T_{1},\ldots,T_{k}$ for $k \geq 3$ represents that the k blocks corresponding to the leaf set of $T_{1},\ldots,T_{k}$ 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 P-nodes and Q-nodes we must allow the following legal transformations (see figure 9.5 1 - 2).

1.

Reordering the sub-nodes of some P-node arbitrarily

2.

Reversing the order of the sub-nodes of some Q-node

 

  

Figure 9.3: PQ-tree node types: we use circles and bars to denote P-nodes and Q-nodes, respectively.

  

Figure 9.4: Frontier of a PQ-tree

  

Figure 9.5: Permitted transformations of a PQ-tree

  

Figure 9.6: Permitted transformations of a PQ-tree

  Two PQ-trees 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 $T \equiv T'$.  



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 PQ-tree representing $\Pi(\varphi)$.



PQ-Tree Algorithm for Unique Probe DNA Mapping:

1.

Initialize the tree as a root P-node with all elements of U as sub-nodes (leaves).

2.

For $i = 1,\ldots, n$ : 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 PQ-tree 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 bottom-up 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 PQ-tree based algorithm

Figure 9.7 shows an example of application of the PQ-Tree algorithm for unique probe DNA mapping. The problem with using PQ-trees 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 PQ-tree¹ 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.