We performed experiments on each of our datasets as follows. It was noted in II.A that we can calculate probability values from the x values returned by CPLEX. Each of these p values is the probability of selecting the associated cross section in a random selection. Using these probabilities, we build encoding schemes one cross section at a time. After each cross section was added, we measured the current distortion against the distance vector used for that data set.
Calculating the actual distortion of an encoding scheme against a distance vector was solvable as a simple 2-variable convergence problem. We then graphed distortion at each distance on a range of distances from 1 to 5000. Keep in mind that on each graph, the encoding scheme used at any distance N is the same as the encoding scheme at N-1 plus one extra cross section, randomly selected from our weighted distribution.The general trend in these graphs is that they approach the minimum distortion rapidly at first, and then level out before actually reaching the minimal distortion. It is easy to determine at what length further cross sections have little benefit in reducing distortion (this is usually around 1000 bits). Due to the random nature of the experiment, there are some slight rises in distortion throughout the lengthening process, but the overall trend still applies.