Protocols for Secure Communication in Wireless Sensor Networks

5.2. Routing on Spanning Trees 139
Source
Root of subtree
Target
Figure 5.2: Target address contained in the convex hull of the cover area, but not in the cover
area itself
Here, we propose to use the convex hull of a set of regions to represent
the sum of the regions. Convex hulls have some desirable properties. First, a
convex hull can be represented by very few points compared to the point set
it represents. Second, if the point set is extended, the new convex hull can be
solely computed from the existing points on the hull and the new point set. No
information about the points contained in the convex hull is required. A third
property is efficient computability using Graham’s algorithm (cf. [44]), which
has O(nlogn) time complexity in the number of nodes.
Using convex hulls for representing the area covered by a set of nodes in
a subtree has the disadvantage that the convex hull may contain areas that are
in fact not covered by any nodes in the subtree. This means that although an
address, i.e. a geographic location, may be purportedly reachable through a
subtree, since it is contained within the convex hull covered by the subtree,
there is in fact no node in the subtree that actually covers this location. Hence,
such a message will be routed through that subtree without ever reaching a node
that actually covers the addressed location.
Such a situation is depicted in Figure 5.2. The boxes illustrate the cover
areas of the nodes, the union of these boxes is the total cover area of the subtree.