Protocols for Secure Communication in Wireless Sensor Networks

146 Chapter 5. Multipath Communication
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Common nodes on two tree paths
Figure 5.8: Distribution of common nodes between tree paths
pairs of tree paths. As expected, due to the physical separation of tree paths,
the number of tree pairs that share a large number of nodes decreases rapidly.
5.3.3 Traffic Overhead
Tree paths are not designed to minimize the number of hops necessary to deliver
a message. They are intended to provide for disjoint and spatially separated
paths. In addition, the use of tree paths in conjunction with a geographical
addressing scheme leads to failures in delivering messages. Thus, the intended
gain in security must be paid for in terms of a higher hop count and a delivery
rate below one.
Let L be the average length (hop distance) of shortest paths in the network.
Using a tree path, a message is routed first to the root, and then on to the target.
If all messages are routed through the root, this yields a path length of approximately
Lm ≈ 2L. However, a few messages do not need to go through the root
as the source and the destination node are both contained in the same subtree.
These connections lower the average tree path length by a small amount. From
the same simulation experiment as above we also obtain the distribution of the
path lengths. Figure 5.9 shows the distribution of the shortest path lengths, i.e.
the shortest connections between source and destination with no tree involved.
The average path length is 6.9. Figure 5.10 illustrates the distribution of the
tree path lengths; this yields an average of 11.9, i.e. Lm ≈ 1.72L. This experiment
shows that the expected path length using tree routing is well above the
best achievable path length but would still be acceptable in many cases.