Weighted Parsimony

In this version of the problem the price of a change is not constant. Instead, denote by C^c_ij the cost of the character c changing from state i to state j. The problem is still to minimize the total cost of the tree given the topology and the leaf labels.

Problem 9.5   Weighted Small Parsimony.
INPUT:

• The topology of a rooted phylogenetic tree with leaves having labels in A_c.
• The costs C^c_ij for $i,j \in A_c$.

QUESTION:

1.

What is the minimum possible cost for this topology?

2.

What is the optimal labeling of the internal nodes?

We will present an algorithm by Sankoff [12] which is a generalization^9.2 of the Fitch algorithm. Sankoff's algorithm:

Step 1: We will compute, for each node v and each state t a quantity S_t(v) which is the minimum cost of the subtree whose root is v given v_c = t. The order of calculation will be, as in step 1 of Fitch, postorder: For each leaf v:

$$S^c_t(v) = \left\{ \begin{array}{ll} 0 & v_c = t \\ \infty & \rm {otherwise} \end{array} \right.$$ (9.2)

For an internal node v, with subnodes u and w, it is easy to see that:

$$S^c_t(v) = \min_i\left\{C^c_{ti} + S^c_i(u)\right\} + \min_j\left\{C^c_{tj} + S^c_j(w)\right\}$$ (9.3)

The minimum total cost of a tree with root r is:

$$S(T) = \sum_{c=1}^m \min_t S^c_t(r)$$ (9.4)

Step 2: Based on the numbers S^c_t(v) calculated in step 1, we will now determine the optimal values for each character c in the internal nodes. We will work in preorder this time:
For the root node r, we will choose $r_c = \arg\min_t S^c_t(r)$.
For any other node v, with parent node u,

$$v_c = \arg\min_t(C^c_{u_ct} + S^c_t(v))$$

Complexity: For every node we do O(k) work in each step, meaning $O(n \cdot k)$ per character. The algorithm should be performed once for each character, with a total complexity of $O(m \cdot n \cdot k)$.