II-A. Interpretation of Results

The resulting solution from CPLEX gives us two important things:

• The minimum possible value of e, which is the absolute minimum distortion that we can achieve in theory.
• The scaled weights, x, which will accomplish this minimum distortion.

The x values that are returned will most likely not be integral. To remedy this, we use them to calculate probability values in vector p. We define these as such:

Let X be the sum of all x_i values in our vector x.

p_i = x_i / X     for all indices i.

We can use these p values as follows:

Let's say we want to create some encoding of our alphabet A,G,C,T so that each symbol maps to a binary string of some length N. We now know with what probability the rows 0001, 0010, etc. should occur in our encoding.

So if p₁ = 0.15, then for any slice in our encoding scheme, there is a 15% chance that for a specific bit, A, G, and C will have 0 while T has 1, and so on.

With these probability values, we can experiment with randomly generated encoding schemes of any length to observe how long we need our encoded strings to be before we approach the minimum possible distortion.