## 2 Materials and Methods  
### Problem Statement and Solution Overview  
**Given:**  
* Ad hoc or sensor network topology (node coordinates; line properties between nodes) and the set of active lines. The set of active lines can consist of either all lines or some subset of lines, which can be based, for example, on routes between nodes exchanging information or a network minimal spanning tree. * Each line of the ad hoc or sensor network uses one equal (in terms of bandwidth) orthogonal channel resource. The examples are as follows: time slots of equal length, frequency channels of equal bandwidth, etc. * Each line can use a different type of signal. For example, given a specific frequency band of the same width, one line can use QAM-16 modulation, and the other line can use QPSK modulation. The type of signal defines the bit error curve. * Each line has a power budget for other lines' interference. If there is no budget for other lines' interference, then only one line can transmit using one specific orthogonal channel resource. In this case, no simultaneous use of shared channel resource is allowed. * Maximal acceptable bit error rate for each line. * Each line has its own transmission power to satisfy the maximal bit error rate requirement under the given signal propagation loss, receiver thermal noise power, and power budget for interference from other lines. Lines with different distances between the transmitter and receiver have different transmission powers, hence interference to other lines. **Find:**  
* The minimal quantity of orthogonal channel resources to allow active lines transmission with a bit error rate below the allowed threshold. * The resource allocation for each specific line. The proposed solution consists of two phases. **First phase:** Find line sets with mutual interference within given bounds (mutually acceptable interfering line sets). **Second phase:** Find the minimal set cover of the mutually acceptable interfering line sets. We set the definitions to describe the algorithm to find mutually acceptable interfering line sets in the table (Table 1). The ideas of mutually acceptable interfering line sets and pairwise mutually acceptable lines are illustrated in Figures 1 and 2. Nodes are depicted with circles; lines between nodes represent communications lines; multicolored bars associated with adjacent lines represent interference budgets of these lines. Interference budget bar colors are the same as the colors of lines that produce this interference. \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbols** & **Descriptions** \\ \hline \(T\) & A tree representing sets of lines with mutual interference \\  & lower than acceptable threshold. \\ \hline \(L\) & All lines set. \\ \hline \(\mu\) & An empty line. \\ \hline \(n\) & The node of set tree of layer. \\ \hline \(n\in T\) & \\ \hline \(n=\left\{l,n_{previous}\right\}\) & \(l\in L\) is the line of node; \(n_{previous}\) is the previous node. \\ \hline \(v=\left\{\mu,v\right\}\) & The line set tree root node; the only node which contains an empty line \(\mu\). \\ \hline \(l(n)\) & A line of node \(n\). \\ \hline \(A_{l}\) & The pairwise mutually acceptable line set for the line \\  & \(l\in L\). \(l\) can transmit simultaneously with any single line \\  & from \(A_{l}\): any pair of lines \(\left\{l,\forall x\in A_{l}\right\}\) can use the same \\  & orthogonal channel resource. \\ \hline \(A(l)\) & A function (algorithm) to obtain the pairwise mutually \\  & acceptable line set. \\ \hline \(A(\mu)\to L\) & \\ \hline \(A(l\
eq\mu)\to A_{l}\subseteq\left\{L\setminus l\right\}\) & \\ \hline \(D_{n}\) & A set of the lines taken from the set of nodes constituting \\  & the path to the tree root from node \(n\). \\ \hline \(D_{v}=\left\{\mu\right\}\) & \\ \hline \(D_{n\
eq v}=\left\{l(n),\ldots,\mu\right\}\) & \\ \hline \(C_{n}\) & Candidate successor set of lines for the node \(n\). \\ \hline \(C(n)\) & A function (algorithm) to obtain \(C_{n}\).