Protocols for Secure Communication in Wireless Sensor Networks

5.3. Properties of Tree Paths 143
area would yield an intersection.
When the deployment area is a square, we can roughly approximate the
probability with which an intersection between two trees occurs in the following
way. The “grey” zone occurs only on the side of line AB on which C is
located. The closer C is to one node, the smaller the grey zone on this node’s
side will be, but it will be larger on the other’s (this can be seen in Figure 5.3,
where the shaded area close to B is much smaller than that close to A). In the extreme
case, the grey zone will fill out the whole upper area almost completely.
The other extreme is C being close to the line AB, and placed between A and B;
in this case, the grey zone will be rather small in total. For a rough estimation,
we may assume that one half of the upper area will be grey. In total, this means
that in about one quarter of all pairs of tree paths, there will be an intersection.
Figure 5.4: Paths crossing each other without an intersection node
This estimate is based on an ideal, continuous model. A real WSN is a
discrete system, communication paths are not straight lines, and their density
is limited. This leads to a number of differences to the ideal model:
• Paths that intersect in the model do not necessarily intersect in a real deployment.
An example is shown in Figure 5.4. Note that if a Gabriel
graph would be used, which is not necessary for tree routing, a situation
as in this example cannot occur and the paths will definitely intersect.
• In the real world, it is likely that paths intersect in a location close to the
communication endpoints since paths are closer to each other and, since
there are only finitely many nodes to choose from, it is more likely that
they have one or more nodes in common.
• An intersection need not be restricted to one common node. It is possible
that two paths have a segment in common, especially close to the
endpoints. An example is shown in figure 5.6.
These considerations lead us to the conclusion that the fraction of tree paths
that intersect should be roughly above one-quarter, but below one-half. In order
to validate this assessment, we performed a simulation experiment. Simulation
runs were executed for ten different networks, 30 spanning tree pairs for each