We first define a matrix M5 =
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
1 1 2 5 4 3 − − − − − − − − − − − − − − −
2 4 5 6 4 1 7 6 1 5 7 − − − − − − − − − −
3 3 2 7 5 6 3 2 7 4 1 − − − − − − − − − −
4 6 5 4 6 3 4 2 1 4 2 1 4 5 1 3 5 2 6 5 2
5 1 2 7 1 2 5 3 7 5 3 6 2 7 6 2 4 3 7 4 3
where “−” means the corresponding item is not used. By Lemma 2.5, H40 is a subgraph of H5. Thus one way to color H5 is to extend a coloring of H40and this leads to Algorithm OP T 5. See Figure 7 for an illustration of this algorithm.
Algorithm 4 OP T 5 (As Executed At Every Vertex)
1: if c ≤ 5 then
2: vertices (c, s) get the color M5(c, s);
3: else // c ≥ 6.
4: let b = b 8·s
5·2b c2cc;
5: vertex (c, s) gets the color A(|c|4, |s|3)◦Q
j<bpj; // use the |`|3 = 2 case in OP T 8.
6: end if
Theorem 3.4. Algorithm OP T 5 is distributed, takes constant number of steps, and pro-duces an optimal distance-two 7-coloring for H5.
Proof. It is obvious that OP T 8 is distributed and takes constant time. Let f be the coloring produced by OP T 5. We now verify that f is a distance-two coloring. Suppose
y y
Figure 7: The distance-two 7-coloring for H5 produced by OP T 5; all the vertices are colored by using A (along with permutations p0, p1, . . . , p6) except that those highlighted are colored by using M5.
(c, s) and (c0, s0) are two distinct vertices that are of distance at most 2. If at least one of (c, s) and (c0, s0) is highlighted (see Figure 7), then f (c, s) 6= f (c0, s0) can be verified by a brute-force checking. If both of (c, s) and (c0, s0) are not highlighted, then OP T 5 performs in the same way as the |`|3 = 2 case of OP T 8; hence f (c, s) 6= f (c0, s0) by Theorem 3.2.
From the above, f is a distance-two coloring. It is obvious that f uses 7 colors. Thus by Theorem 2.3, OP T 5 is optimal and we have this theorem.
4 The leader election problem
The leader election problem is to select a leader (from the sensors in a cluster) to perform certain tasks on each cluster. Because sensor networks contain many sensed data of the local environment, leader election can be used to combine or aggregate the data into meaningful information. More precisely, leader election has applications to coordination and data fusion, the latter is also called data aggregation and can be used to reduce the number of data to be communicated between the sensor node and the actor so that to
avoid information overload. Leaders paly the most important role of each cluster. Thus an efficient process for the election of a cluster leader (or data aggregator node) is essential.
In [13], the authors mentioned that they use the uniform leader election for radio networks protocol in [15] (abbreviated as ULERNP) to select a leader for each cluster.
Unfortunately, we find that this is incorrect. In ULERNP, the network has to be a single-hop network (i.e., every two nodes can communicate directly). Therefore to use ULERNP to select a leader for each cluster in the virtual infrastructure G`, the nodes in each cluster have to form a complete graph; however, it is usually impossible that every two nodes in a cluster can communicate directly. Furthermore, when the nodes are very dense, ULERNP usually produces dramatic communication overhead.
In [8], a hybrid approach that combines the energy conservation with the simplicity was introduced. This approach is based on four selection parameters: (1) the available energy, (2) the number of neighbouring sensor nodes, (3) the distance from the current group leader, and (4) the level of trust; for details, please refer to [8]. This approach can be used in leader election for G` and H`. However, nodes may produce a lot of communication overhead since G` and H` are usually multi-hop networks. For other leader election protocols, please see [10, 16].
Before closing this section, we propose an idea of how to perform leader election in a multi-hop network like G` and H`. We will only consider the parameter (1) and the distance from the candidate node to the other nodes in the cluster (the leader should be easy accessed from the other nodes). If more than one node can be selected, we randomly select one of them as the leader.
5 The concluding remarks
In this thesis, we propose a virtual infrastructure called H` and an distance-two col-oring algorithm for H`. Our virtual infrastructure H` provides a coarse-grained location
to the sensors in a network and allows geographic routing. Our distance-two coloring algorithm can be used to assign the frequency channels (or colors) in a fully distributed manner and our algorithm uses fewer channels than the previous work [13]. In the fu-ture, we intend to determine an appropriate way for the leader election problem, because choosing the right leader can help enhancing the network lifetime and can make routing more easier. In real world applications, the environment may have obstruction in it. Thus it is also challenging to find a virtual infrastructure for such an environment.
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