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 10.3). 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 10.5 1 - 2).

1.

Reordering the sub-nodes of some P-node arbitrarily

2.

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

 

is the set of all leaves, read in a left-to-right order. As demonstrated in figure 10.4

  

Figure 10.3: PQ-tree node types: we use circles and bars to denote P-nodes and Q-nodes, respectively.

  

Figure 10.4: Frontier of a PQ-tree

  

Figure 10.5: Permitted transformations of a PQ-tree

  

Figure 10.6: Permitted transformations of a PQ-tree

 

if there exists a set of legal transformations leading from one tree to the other. In such a case, we write $T \equiv T'$.  

Theorem 10.2   Booth-Lueker 1976 [1]

1.

For every $U,\varphi$ there exists a PQ-tree T s.t. $Consistent(T) = \Pi(\varphi)$

2.

For every given PQ-tree T, there exists $U,\varphi$ s.t. $Consistent(T) = \Pi(\varphi)$

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)

The procedure reduce (T,S_i) returns a tree for any permutation in consistent(T) 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 10.6 on this subtree, working bottom-up till reaching Root(T,S_i)

  

Figure 10.7: Example of PQ-tree based algorithm

Figure 10.7 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-tree^10.1 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.