# Lightweight Path Recovery in IPv6 Internet-of-Things Systems
## 3 System Model and Problem Statement  
### Network Model  
We assume that in the multihop RPL wireless sensor network, \(n\) nodes form a set \(N\) that are fixed in a square region with side length \(L\). Among them, \(N_{1}\), located in the rectangular area's center, is a sink node (i.e., gateway node or base station) with adequate computing resource and energy, which collects data from other nodes and is responsible for maintaining the network. The remaining \(n-1\) resource-constrained sensor nodes denoted as \(N_{2}\), \(N_{3}\),..., \(N_{n}\) are randomly scattered in this square region, as shown in Figure 1. During the initialization of the network, a sink node will assign a unique identifier to every sensor node, represented by a two-byte ID. All nodes have a fixed and limited transmission range \(TL\). In each collection cycle, the probability \(P_{active}\) indicates that a sensor node is active, and the probability \(P_{fault}\) means that a sensor node temporary fails. Whether individual sensor nodes are active or not is a mutually independent event, and an active sensor node sends packets as well as receives packets from others themselves. The failure of each sensor node is also independent of each other, and when a sensor node fails, it is unable to perform the activities of sending, forwarding, and receiving packets. In this network model, point-to-point traffic (between devices inside the sensor network), point-to-multipoint traffic (from the sink node to a subset of sensor nodes), and multipoint-to-point traffic (from nodes inside the sensor network towards the sink node) are supported. Taking point-to-point traffic as an example, the shortest path from sensor node \(N_{i}\) to sink node and then from sink node to sensor node \(N_{j}\) without passing through the faulty node constitutes the routing path as shown in Figure 2. Where we assume that the upstream routing from sensor node \(N_{i}\) to sink node and the downstream routing from sink node to sensor node \(N_{i}\) are mirrored, i.e., the reverse path of the path from sensor node \(N_{i}\) to sink node is the path from sink node to the sensor node \(N_{i}\). As mentioned in several studies [26; 27; 28], it is assumed in RPL that the links are symmetric and that both \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & **Year** & **Overhead** & **Key Idea** & **Limitation** \\ \hline \({}_{c}\)Path\({}_{ST}\)[19] & 2019 & 8B & Exploits compression-aware techniques and path correlations & Not suitable for networks with inactive nodes \\ \hline RTI [18] & 2019 & 12B & Exploits packet tracing and local probing to reconstruct the route path & (1) Sensitive to nodes fails (2) High storage overhead \\ \hline PAT [17] & 2017 & 11B & Exploits path compression and correlations & (1) Initialization phase to reconstruct the network topologies are needed (2) High storage overhead \\ \hline iPath [12] & 2016 & 6B & Exploits path correlations & Not suitable for networks with inactive nodes \\ \hline CSPR [13] & 2016 & 8B & Exploits compression-aware technique & (1) Not suitable for dynamic networks (2) Not suitable for networks with inactive nodes \\ \hline Pathfinder [24] & 2015 & 9B & Exploits path correlations and inconsistency of packets & Not suitable for sparse networks \\ \hline INS- & 2015 & 11B & Exploits path correlations & (1) Not suitable for networks with inactive nodes (2) High storage overhead \\ \hline PathZip [14] & 2014 & 8B & Exploits path compression with topology-aware and geometry-assistant techniques & (1) Sensitive to nodes fails (2) High storage overhead \\ \hline MNT [16] & 2012 & 6B & Exploits path correlations & Not suitable for networks with inactive nodes \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Literature Review on Path Reconstruction Algorithms. directions of the links are connected. Then, the paths consisting of symmetric links have the reasonableness of symmetric assumption.