## 2 Materials and Methods
### Random Tree Pruning Based Search
Using QPSK modulation, the target SINR can be calculated with the reverse of the following equation: \(bitErrorRateQPSK=erfc(\sqrt{SINR})/2\). For the \(8.375\times 10^{-7}\) bit error rate the target SINR is about 12 dB. Given that the SNR is equal to the 30 dB, we have about 18 dB for the other lines' interference. We have performed a grid search to estimate the good parameters for pruning distributions and the GA: distribution scale factor: \(\{10,100,1000\}\); Pareto \(\alpha\) parameter: \(\{0.001,1000\}\); GA mutation parameter: \(\{0.1,0.2\}\). In total, we have tested 30 permutations with the number of nodes from 10 to 70 with step 10 and 30 random topologies for each test. The batch size is 128. The scale factor for pruning is equal to 1. The number of iterations after the best found solution has not improved is 2 times larger than the iteration when improvement has happened. The minimal number of iterations is 20. The AWGN mutual interference assumption is used. The SNR is 30 dB. Figure 6: Random network topologies. The cost value is the ratio of the minimal quantity of orthogonal resources to the number of lines. The lower the cost value, the better; a cost equal to 1 is the worst one. Figure 7 shows the cost values averaged over 30 random topologies. Figure 8 shows the place of algorithm parameters set in the sorted sequence of cost values; the less place, the better. Table 3 shows the sum of sorted ranks over all the number of nodes. We have chosen pruning with uniform distribution: scale factors: \(\{1,2,4\}\); batch size: 128; mutation probability: 0.1; minimal number of iterations: 20; number of iterations to stop after the solution has improved: 2 times larger than the last improvement iterations; AWGN mutual interference assumption is used; SNR: 30 dB. We have performed experiments on 30 random network topologies for each quantity of nodes from 10 through 250 with a step of 10. Averaged over 30 random topologies, results are presented in Figures 9 and 10. Results without pruning are called ultimate. Experiments without pruning are realized for up to 60 nodes, for GA up to 100 nodes, and for random search the whole range. Figure 8: Rankings of different settings of the algorithm. Figure 7: The cost for different settings of the algorithm. Figure 10: (**a**) Mean space complexity. (**b**) Mean time complexity. (**c**) Mean number of iterations. (**d**) Mean number of lines. Figure 9: Cost (the ratio of minimal quantity of orthogonal resources to the number of lines). ## 4 Discussion

The grid search was conducted to choose the settings of the proposed algorithm. The results of the grid search (Table 3, Figures 7 and 8) reveal the following. The genetic algorithm search performs better than the random search algorithm. Lower mutation probability, mP = 0.1 in our case, provides better results than mP = 0.2 and mP = 1 (random search is equivalent to the genetic algorithm with mP = 1). The uniform distribution for pruning gives better results than the linear and Pareto ones. The more the linear and Pareto distribution resembles the uniform one, the better the results. In all cases, we obtain monotonically decreasing cost function values as the number of nodes grows, except for the random search with extremely skewed Pareto distribution, where we have a convex cost function. We have performed experiments on 30 random network topologies for each quantity of nodes from 10 to 250 with step 10 under the AWGN mutual interference assumption. Examples of random topologies consisting of 10, 50, 100, and 250 nodes are shown in Figure 6. The results are presented in Figures 9 and 10. We can see the following dependence for all algorithm settings: the more nodes in the network, the lower the ratio of minimal orthogonal resource quantity to the active lines quantity. Without pruning the tree of mutually acceptable interfering line sets, we obtain the best results.