## 3 Network Initialization  
In our work, since the nodes are static, all nodes know their own locations before network initialization. In the initialization stage of the network, each node sends its own location information to its one hop neighbors. Meanwhile, each node also receives all the location information from all its one hop neighbors. When nodes acquire their neighbor location information, they compute the distance between themselves and the sink, and the distance between every one hop neighbor and the sink. Then, each node compares those distances and concludes which node is closer than itself to the sink, then it counts the number of one hop neighbors closer to the sink in Euclidean distance between nodes, and this number is its out-degree number. These nodes are then included in the out-degree set. The rest of the nodes that are not in the out-degree set constitute the in-degree set. Obviously, nodes in the in-degree set are farther away from the sink. After the computation and comparison are finished, each node has acquired out-degree and in-degree sets of their own. There are two extreme cases that could happen for the out-degree set or the in-degree sets of some nodes. Figure 1 shows the illustration of this for the cases where the out-degree of a node is zero and in-degree of a node is zero.  
1. The in-degree of some node is zero. From Figure 1, we can see that \(Z\) is a node whose in-degree is zero and it is located at the boundary of the obstacle convexity. We can conclude that this node has no forwarding task. Therefore, in this paper, we do not take into consideration such kinds of nodes.
2. The out-degree of some node is zero. In Figure 1, \(A\) is a node whose out-degree is zero. This kind of nodes is extremely special and their number is exiguous, but we cannot neglect the existence of such nodes.  
The existence of a zero out-degree node can have a fatal influence on the routing of wireless sensor networks. Thus, we present a separate discussion of these nodes. From Figure 1, we can see that nodes whose out-degree is zero are always located at the extreme points of the concave boundary of obstacles. When some nodes discover their out-degree is zero, they send a packet to all neighbors within one hop and the packet includes its location, unique \(ID\) and connectivity (out-degree and in-degree). Once the neighbor nodes receive the packet, they would prune off one degree from their own out-degree set. Figure 2 shows the concave region when an obstacle exists in a sensor field. From Figure 2, we can see that the node \(A\) has two neighbors (node \(B\) and \(C\)) that have only one forwarding link to node \(A\). If they receive a packet from \(A\), they prune off one degree from their out-degree set. Meanwhile, their out-degree sets would also change to zero. Nodes \(B\) and \(C\) would reduplicate the operation that node \(A\) did once: send a packet to its neighbor, and so on, we would obtain a concave region from the composition of the nodes whose out-degree is zero. In Figure 2, the  
Figure 1: Illustration for the node out-degree is zero and in-degree is zero cases in sensor networks with obstacles.  
region formed by white dots (out-degree is zero) is the concave region. Based on the out-degree, we can divide the nodes into two categories, zero out-degree nodes and non-zero out-degree nodes.