## 2 Materials and Methods
### Random Tree Pruning Based Search
## 5 Conclusions

In this paper, we have proposed the parallel heuristic algorithm for the NP-hard problem of global minimal orthogonal channel resource allocation and grouping of concurrently transmitting lines for ad hoc and sensor networks. Given the set of active lines, bit error rate requirements, and the model to estimate mutual interference, we can, in a parallel manner, find the minimal quantity of orthogonal resources for the lines. Simultaneously, we discover which lines can concurrently transmit together using one common orthogonal resource. This is achieved with the developed approach of incrementally creating the tree with mutually acceptable interfering line sets using pairwise line limitations, and, using the minimal set cover algorithm, we obtain the smallest set of concurrently transmitting line sets. To combat the combinatorial explosion of tree size with the increase in active line quantity, we use tree pruning. Afterwards, we use the genetic algorithm to optimize the tree pruning information to find the pruned tree with the sets that give the smallest minimal set cover. The tree pruning lowers the worst case exponential complexity of full tree construction to the polynomial complexity. We conducted a grid search to find the best parameters of the proposed technique. Using the parameters found, we conducted experiments that show the minimal quantity of orthogonal resources, space complexity, and time complexity of the proposed technique for random network topologies with up to 250 nodes and the mean number of lines up to 2000. The cases without tree pruning, with random search pruning, and with genetic algorithm pruning are considered. Genetic algorithm pruning achieves the best result, but has relatively high time complexity. Contrary, random pruning gives suboptimal results with substantially lower time complexity. Due to the parallel nature and the combinatorial explosion of line sets mitigation, the proposed algorithm can be used for the optimal orthogonal channel resource allocation for large scale ad hoc and sensor networks. An additional benefit of the algorithm lies in its usability to create training sets for machine learning algorithms for the real-time distributed channel resource allocation and to optimize the NOMA groups of 6G cellular networks. **Author Contributions:** Conceptualization, V.I; methodology, V.I. ; software, V.I. ; validation, V.I. ; investigation, V.I. ; data curation, V.I. ; writing--original draft preparation, V.I. and M.T. ; writing--review and editing, V.I. and M.T. ; visualization, V.I. ; supervision, M.T. ; project administration, M.T. ; funding acquisition, M.T. All authors have read and agreed to the published version of the manuscript. This research has received no external funding. The authors thank [PERSON] for his valuable insights, [PERSON] for his substantial suggestions, and [PERSON] for her consulting and inspiration. The authors declare no conflicts of interest. ## Abbreviations

The following abbreviations are used in this manuscript: \begin{tabular}{l l}
6G & Sixth-generation technology for wireless communications \\ AI & Artificial Intelligence \\ AWGN & Additive white Gaussian noise \\ CFT & Caprara, Fischetti, and Toth algorithm for the minimal set cover search \\ DSP & Digital signal processing \\ EHLO & Elephant herding lion optimizer \\ GA & Genetic optimization algorithm \\ GBD & Generalized Benders Decomposition \\ MFCM & Multiple Fuzzy C-means Clustering algorithms \\ MIMO & Multiple Input-Multiple Output \\ MPA & Marine predator optimization algorithm \\ NOMA & Non-orthogonal multiple access \\ OFDMA & Orthogonal frequency division multiple access \\ \end{tabular} 

## References

* (1) [PERSON] _Ad-Hoc Networks: Fundamental Properties and Network Topologies_; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2006. * (2) [PERSON]; [PERSON] _Protocols and Architectures for Wireless Sensor Networks_; John Wiley & Sons: Hoboken, NJ, USA, 2007.