## 3 Trail-Based Search Strategy
### Search Phase  
The trail-based search strategy consists of two phases: search phase and chase phase. The basic process of the search phase is the same as ERS. The source generates a search packet with a preset time-to-live (TTL) integer, and then broadcasts the packet to its neighbor nodes. Meanwhile, the source initiates a waiting timer \(T_{w}\) to wait for the acknowledgment from the actor. The nodes that receive the search packet continue to broadcast this packet with a decrease in TTL value. The forwarding stops when \(TTL=0\). If the actor locates within the transmission range of any sensor node that broadcasts the search packet, it will receive this packet, and the search attempt is successful. Then the actor sends an acknowledgment (ACK) packet back to the source. The waiting timer \(T_{w}\) should be set long enough to ensure the ACK packet can be received by the source before it terminates. If no ACK packet is received when \(T_{w}\) terminates, the source will initiate another search attempt with a larger TTL value and search for a larger search area. Different from ERS, TS searches not only the actor, but also its footprints. We define the _host_ as the node that receives the search packet and obtains a footprint with \(\alpha_{h}\leq\varepsilon\). If the host decides to broadcast the search packet, it will record its footprint into the packet. The footprint recorded in the packet is denoted as \(\theta_{p}\). Generally, the actor may leave a long trail in the network which increases the success rate of search attempt in TS. Nevertheless, it also results in search redundancy that multiple nodes discover the footprints simultaneously in the search phase. To solve this problem, we propose the definition of the _effective footprint_ as follows. **Definition 1** (Effective Footprint). : _Given a host that receives the search packet with \(\theta_{p}\) and \(TTL=0\), its footprint \(\theta_{h}\) is an effective footprint when it satisfies \(\alpha_{h}<\varepsilon\) and \(\alpha_{h}<\alpha_{p}\) simultaneously._  
In the search phase, the host that obtains an effective footprint should transmit an ACK packet back to the source and initiate the chase phase. The definition of the effective footprint is derived based on the following observation. If the footprint is discovered within the search area, there will be two scenarios. The first one is that the actor is moving inside the search area as shown in Figure 1(a). In this case, the actor will be discovered in the search phase, thus there is no need to chase the actor along the trail. Another scenario is that the actor is moving away from the source as shown in Figure 1(b), such that only the footprint can be discovered in the search phase. In this case, the host should transmit an ACK packet back to the source and trigger the chase phase. As shown in Figure 1(b), the actor is ensured to leave footprints on the edge of the search area (the host that receives the search packet with \(TTL=0\)). By comparing \(\alpha_{h}\) and \(\alpha_{p}\), the host can estimate in which direction the actor moves. If \(\alpha_{h}<\alpha_{p}\), which means that the actor is moving out of the search area, the host will initiate the chase phase. The working flow of the search phase is summarized in Algorithm 1. Figure 2: The scenarios that the packet discovers the footprints.