# Routing Strategies for Isochronal-Evolution Random Matching Network
## 4 Two Routing Strategies
### Planning Algorithm for the LHPMD  
An ordered tree for planning LHPMD (which is abbreviated to LHPMD-OT) is shown in Figure 3. The node of an LHPMD-OT also represents the vertex of an IERMN. The \(\lambda\)-th level of an LHPMD-OT is denoted by \(LHPMD-OT(\lambda)\); the number of nodes in \(LHPMD-OT(\lambda)\) is denoted by \(|LHPMD-OT(\lambda)|\); the \(\xi\)-th node from the left in \(LHPMD-OT(\lambda)\) is denoted by \(LHPMD-OT(\lambda,\xi)\); and the root of the LHPMD-OT is denoted by \(LHPMD-OT(0,1)\). The spatial length of the path from the vertex that  
Figure 2: Binary Search Tree for planning the LDPMH. The dotted rectangles indicate the levels in the tree. The solid rectangles indicate the nodes in the tree. is represented by \(LHPMD-OT(0,1)\) to each vertex that is represented by the node in \(LHPMD-OT(\lambda)\) is \(\lambda\) hops. Each node of an LHPMD-OT is designated by a 4-tuple  
\[\left\langle Me,Parent,Level,Number\right\rangle \tag{13}\]  
where \(Me\) is the number of the vertex that is represented by the node, \(Parent\) is the number of the vertex that is represented by the parent of the node, \(Level\) is the level in which the node exists, and \(Number\) shows the position of the node in its parent's children. In this paper, the 4-tuple formula of \(LHPMD-OT(\lambda,\xi)\) is set as \(\left\langle\mu,\rho,\lambda,i\right\rangle\). The planning algorithm for the LHPMD based on building an LHPMD-OT is shown in Algorithm 3. ``` Input:\(X\), source vertex \(v_{i}\), destination vertex \(v_{j}\), planning slot \(k\) Output: the LHPMD from \(v_{i}\) to \(v_{j}\)
1. Set \(i\) as the root of the LHPMD-OT and create \(Z=\left\{v_{i}\right\}\), the set of nodes of the LHPMD-OT whose value is \(j\). 2.for\(\lambda=0\) to \(T\)do
3. Construct a new level \(LHPMD-OT(\lambda+1)\)
4.for\(\xi=1\) to\(\left|LHPMD-OT(\lambda)\right|\)do
5.for\(\tau=k+\lambda+\iota-1\) to \(\Gamma\)do
6. The 4-tuple formula of \(LHPMD-OT(\lambda+1,\tau)\) is set as \(\left\langle x_{\tau,\mu},\mu,\lambda+1,\tau-k-\lambda-\iota+2\right\rangle\), because it is the \(\tau\)-th child of \(LHPMD-OT(\lambda,\xi)\) whose 4-tuple formula is \(\left\langle\mu,\rho,\lambda,i\right\rangle\). 7.if\(x_{\tau\mu}=j\)do
8.\(Z\gets Z\cup\left\{LHPMD-OT(\lambda+1,\tau)\right\}\)
9.endif
10.endfor
11.endfor
12.if\(Z\
eq\varnothing\)then
13. Select the nodes with the lowest values of \(Level\) from among \(Z\)
14. Select the node with the lowest value of \(Number\) from among the nodes selected by Step 13
15. Backtrack from the selected node by Step 14 to the root \(i\) to collect the nodes that have been traveled through
16. Arrange the collected nodes in reverse order to get a sequence that will be transformed into the LHPMD
17.break
18.endif
19.endfor

Figure 3: Ordered tree for planning the LHPMD. The dotted rectangles indicate levels in the tree. The solid rectangles indicate nodes in the tree. The level of an LHPMD-OT represents the spatial length of the path from the vertex, which is represented by the root, to each vertex, which is represented by the node in it. Hence, a lower level will lead to a path with a smaller spatial length. The children of a node in an LHPMD-OT represent the vertices that are connected by the vertex represented by the node in the future. Thus, a node with a smaller value of \(Number\) means that the vertex represented by it will be connected to the vertex represented by its parent earlier. Therefore, by backtracking from the node with the lowest values of \(Level\) and \(Number\) in \(Z\) to the root, the LHPMD can be obtained, according to Steps 13-16. ## 5 Simulation Experiments and Analyses

### Simulation Design

Since the vertices of an IERMN are able to control packets and make routing decisions autonomously, the vertices were modeled as agents by the method of Agent Based Modeling and Simulation (ABMS). Then, we conducted traffic dynamics simulations on the System-of-Systems Effectiveness Analysis Simulation (SEAS) platform to compare the two proposed routing strategies. The SEAS is an ABMS platform [35].