## 4 The Proposed Method: SLP-E
### Protocol Execution Phase
#### 4.2.2 Scenario-2
* _Max Delay in Phase-4_: The maximum length covered in this phase is \(90^{\circ}\): Figure 4:2 (11)
* _Max Delay in Phase-5_: The maximum distance we have is \(R\), and the remaining path to be traversed by packets in SPR after _phase-4_ is \(R-\frac{2R}{3}\); therefore, we have: Figure 4:3 (12)

The total number of hops \(H_{t_{2}}\) is

Figure 4:4

Figure 4 shows the overall routing protocols' maximum delay, expressed in terms of hops. These are the hops that the packets travel over to reach the BS. When compared to the shortest path routing (SPR), SLP-R [37], PRLPRW [34], SSR [28], and PSSLP [16] schemes, the proposed scheme has the largest delay. Furthermore, the proposed scheme's delay graph is nearly straight, demonstrating that it balances the number of hops a packet must take to reach the BS at any location of the source node in the network as compared to the existing protocols. Hence, the balanced NLT and safety period. ## 7 SLP-E's Complexity Analysis

In this section, we provide a comprehensive analysis of the algorithm's complexity, specifically focusing on the number of message transmissions needed to successfully convey the packet from the source to the base station (BS). The proposed algorithm begins by initializing the network through a flooding message, which is continually rebroadcasted at each node until it reaches the network's boundaries. In the case of a network consisting of \(N\) nodes, the initialization phase involves the transmission of messages in the order of \(O(N)\). This characteristic holds true for all other existing SLP techniques. During the operational phase of SLP-E, each source node transmits a packet to the base station (BS), and a node that receives this message forwards the packet to its neighbor that lies in the routing path. This entails message transmission of order \(O(1)\) at each node per hop. The total number of hops the message traverses in the network depends on the distance between the source node and the BS and the routing protocol design. The message transmission complexity for the routing phase depends on the average number of hops the packets take to reach the BS. This analysis is given in Section 6.1 and Equations (8) and (13). Therefore, the total number of message transactions in the network of the proposed scheme is given by

Figure 4:5 Figure 4:6

## 8 Results and Discussion

In this section, we go over in great depth the simulation parameters as well as the assessments of the simulation outcomes. To assess our technique's performance, we first contrast it with shortest path routing (SPR) and then with other similar approaches (i.e., SLP techniques based on random walks). Comparisons are made using the SPR (no privacy strategy), SLP-R [37], SRR [28], PRLPRW [34], and PSSLP [16] schemes. The performance indicators used to assess the efficiency of the suggested remedy are described in Section 6. ### Simulation Settings

The following simulation settings and parameters are taken into account. We take into account a circular deployment architecture with a network radius of 1050 units, as shown in Figure 4:7 Based on the factor \(r_{s}\times\sqrt{2}/2\), we take into account an _offset_ distance of 50 units between two nodes. _Offset_ is defined as the spacing between two adjacent nodes in the network. Except for the boundary nodes, this option guarantees that each node has eight neighbors. Using this number, we calculate the radio range of the sensor to be 72 units, the total number of nodes in the network to be 1460, and the total number of annular rings to be 21. We consider two groups (scenarios) where one group's nodes are within a distance of \(R/4\) in hops with reference to the BS, and otherwise for the other group. \(R/4\) units are chosen as the threshold value. Here, we consider a single source node and change its location within the network using BS as a point of comparison. We run 100 trials and send 1000 packets to the base station for each position of the source node.