The former hierarchical multicast schemes only consider the factor of geographical position, which means the shortest path way using the least number of links. While constructing the hierarchical multicast tree, the system often chooses the geographical central node as the cluster core or near the core. Hence it can save the transmission distance [8]. However, in sensor grid, the data quantities of different nodes are much different [10]. Usually 80% of the data often is centralized in 20% nodes; naturally these important nodes should be paid more attention to. Generally speaking, the more data the nodes have, the more data transmission will happen from the nodes [11]. If the data scale is the only factor we consider, to choose the node with larger data quantity as root or near the root would undoubtedly improve the efficiency of the data transmission.

As a result, the system should consider not only the space factor, but also the data quantity as the factor [9]. The two factors are independent with each other and related with each other. In other words, their relationship is game and balance. We try to set a group of functions in order to draw an elaborate balance between them in our to-be-presented algorithm. The basic idea goes through the whole process of constructing the hierarchical multicast tree. The space factor and data factor are two factors independent with each other, which have meaning and formation respectively; both of them tend to maximize their result. Namely the two factors game with each other. On the other hand, the two factors also co-exist in a system, common working, mutual interaction and constraint. Namely they balance with each other. We must synthetically consider the space and data factors while constructing the multicast tree.

After summarizing the context of the algorithms, this subsection discusses the concrete implementation of the algorithms [12]. The motivation of this paper is to design a multicast scheme in m-D Sensor grid that can achieve not only shorter multicast delay and less resource consumption, but also the efficient data transmission.

The network is partitioned into clusters in terms of some regular Sensor grid area. After group members are initially scattered into different clusters, a tree is built to connect the cluster members within each other. The connection among different clusters is done through hooking the tree roots [13].

To construct such an architecture, a set of novel algorithms based on the m-D Sensor grid are presented:

Cluster formation algorithm that divides the group members into different clusters in terms of static delay distance;

After checking the relative documents, this paper is one of the pioneer to use the multicast architecture for grid computing field [14], which—according to the idea from qualitative to quantitative—builds mathematic model, finds game and balance relationship, resolves linear parameter, and accurately improves the efficiency of transmission of sensor grid.

Each cluster will have a spatial center node as the space core. The space core can be the root of the tree in the cluster. The following theorem provides the sufficient and necessary conditions to select a spatial core in each cluster that is optimal in terms of the minimum sum of static delay distances to all the other cluster members.

Theorem 1 Let U be the cluster member that occupies the node (u₀, . . ., u_j, . . ., u_m−1) in an m-D grid and n > j,n < j and n = j be the number of cluster members with the j-th coordinates larger than (right nodes of j-th row), less than (left nodes of j-th row), and equal to u_j (the nodes just on j-th row) respectively. Then U is the spatial center node if and only if the following inequalities hold simultaneously: (2) | n < j − n > j | ≤ n = j ,   j = 0 ,   1 ,   … , m − 1

Proof:(⇒):Suppose U = (u₀, . . ., u_j, . . ., u_m−1) is a spatial center node, and then to any member U′ in the sensor grid, there exists f(U) ≤ f(U′). To achieve (2), we firstly considers a node U′ = (u₀, . . ., u_j+1, . . ., u_m−1) and its multicast static delay distance f(U′). Given any member U_i = (u_i,0, . . ., u_i,j, . . ., u_i,m−1) and u_j ≤ u_i,j, the distance from U_i to the end host U is one unit longer than the distance from U_i to the node U′. Similarly, it can be seen that to any member U_i = (u_i,0, . . ., u_i,j, . . ., u_i,m−1) and u_i,j ≤ u_j the distance from U_i to the end host U is one unit shorter than the distance from U_i to U′. There exist (n_>uj + n_=uj) members whose j − th coordinates are larger than or equal to u_j, and n_<uj cluster members whose j − th coordinates are less than U_j, then it can be concluded that 0 ≤ f ( u ′ ) − f ( u ) = ∑ j = 0 n ′   ( d ( u ′ ,   u j ) − d ( u ,   u j ) ) = n > U j + n = U j − n < U j → n < U j − n > U j ≤ n = U j. By comparing f(u₀, . . ., u_j−1, . . ., u_m−1) with f(U) in the same way as above, the inequality of 2 can be achieved.

(⇐): It is easy to demonstrate that if (2) is violated, and then U cannot be the spatial center nodes. Assume n_>uj − n_<uj > n_=uj, then n_>uj > n_<uj + n_=uj. Similarly, this paper firstly considers a node U′ = (u₀, . . ., u_j+1, . . ., u_m−1) and its multicast static delay distance f(U′). Given any member U_i = (u_i,0, . . ., u_i,j, . . ., u_i,m−1) and u_j ≤ u_i,j the distance from U_i to the end host U is one unit longer than the distance from U_i to U′. Similarly, it can be seen that to any member U_i = (u_i,0, . . ., u_i,j, . . ., u_i,m−1) and u_i,j ≤ u_j the distance from U_i to the node U is one unit shorter than the distance from U_i to U′. f ( u ) − f ( u ′ ) = ∑ i = 0 n ′   ( d ( u ,   u i ) − d ( u ′ ,   u i ) ) = n < u j + n = u j − n > u j > 0 → f ( u ) > f ( u ′ ). Therefore the distance from U to these end hosts is larger than some other end hosts, which is a desired contradiction.

The physical meaning of the theory is obvious. Firstly, we process on X axis. For example N₌₂ = 4, namely there are 4 nodes just on of second row: (2, 6), (2, 4), (2, 2), (2, 1); N_<2 = 2, namely there are 2 nodes in the left of second row: (1, 3), (1, 1); N_>2 = 4, namely there are 4 nodes in the right of second row (3, 5), (3, 1), (5, 5), (5, 2), so |n_<2 − n_>2| ≤ n₌₂. Thus N₌₂ is satisfied coordinates on X axis. On other hand, N₌₃ = 2, including (3, 5), (3, 1); N_<3 = 6, including (2, 6), (2, 4), (2, 2), (2, 2), (1, 3), (1, 1); N_>3 = 2, including (5, 5), (5, 1), so |n_<3 − n_>3| ≥ n₌₃. Thus N = 3 is not satisfied coordinates. In the same way, we can do it again on Y axis. Then we can find the (2, 2) is the space central node, namely the space core of the cluster.

At the beginning of this discussion, it can be presumed that the system establishes a multicast tree to transfer data packet, which choose the space core as the root and organize the architecture according to the space weight vector [21]. It is anticipated that the tree should maximize the sharing of link utilization within the clusters so that the rest of the links may be used for other traffic [22]. Our approach is to connect all the members according to (1) the branch on the tree between two adjacent members is the shortest path in the cluster, (2) the total number of links on the tree should also be minimized. Before discussing the algorithm, it is necessary to define the following terminologies (using a 2-D cluster as the model):

Shortest path area nodes (SPAN): For any two nodes (x₀, y₀) and (x₁, y₁), let X_min = min{x₀, x₁}, X_max = max{x₀, x₁}, Y_min = min{y₀, y₁} and Y_max = max{y₀, y₁}. They uniquely define a rectangle area [x₀, y₀] × [x₁, y₁]. Each node (x, y) in [x₀, y₀] × [x₁, y₁], which is on one of the shortest paths between (x₀, y₀) and (x₁, y₁), so it is called the shortest path area nodes (SPAN) between (x₀, y₀) and (x₁, y₁).

In general, if the space weight of the node is k, it means that there are k nodes which must pass this node to the space core to send packets, which represent the degree near the center. The greater the space weight is, the nearer the node to the cluster core is, and vice versa.

After figuring out the space weight vector W′_i,j, the system can easily get the data quantity in every node, namely generating the data weight, because the data quantity vector is determinant, as shown in Table 2.

According to the Table 2, because the node B(2, 1) is the maximal value 10, it is the data core.

The relatively Weighted Vectors Generation algorithm in the m-D Sensor grid is given in Algorithm 2.

In Algorithm 2, Steps 4–7 can be executed in time O(n). Steps 9–18 can be improved by using binary searching algorithm that yields an O(ln(n)) complexity. But for the brevity of discussion, it is necessary to keep the linear search algorithm here. The algorithm may find out multiple spatial central nodes, but just choose one of them at random and others can be back-ups. Figure 4 illustrates the spatial center nodes (2, 2) selection in one cluster of 2-D sensor grid. It is known that the spatial center node should be in the area [1, 1] × [5, 6]. Therefore, it can be checked that the spatial center node’s x coordinate must be 2 while y coordinate could be 2 or 3 for f((2, 2)) = f((2, 3)) = 26. Node (2, 2) is the member and is preferential (2, 3).

In the sensor grid, the data quantities of nodes are very different, for instance, about 20% nodes process 80% of the data quantity of the whole system. These nodes are very important in the multicast data transfer, therefore when researching the algorithm in multicast, the node’s data quantity weight W″ should be taken into account. On the other hand, the spatial nodes weight W′ in the Algorithm 3 (Least Weighted Path Tree Generation) should be noticed too. The relationship between the two vectors is a very challenging question, we tried in this paper to begin from the linear relative between them.

In general, the Relative Weighted Vectors Generation algorithm, finds the spatial core c_i,a in every cluster C_i, calculates the space weight of every node W′_i,j, searches the data quantity weight of the very node W″_i,j in cluster C_i, and finally finds the data core c_i,b.

Theorem 2 If, two linear no-relationship vector W′_i, W″_i, their linear combination W_i,j = α_iW′_i,j + β_iW″_i,j, α_i, β_i is linear relation modulus, α_i, β_i ∈ r, α_i, β_i ≥ 0, then following express is satisfied: α i + β i = 1 ,   0 < α i ,   β i < 1 ,   α i ,   β i ∈ r

Proof: The rationality of the equation is obvious:

Let W i , j * = α i *   W i , j ′ + β i *   W i ″ ,   α i * ,   β i * ∈ r ,   α i * ,   β i * ≥ 0.

W i , j * α i * + β i * = α i * α i * + β i *   W i , j ′ + β i * α i * + β i *   W i , j ″

Theorem 3 If two linear no-relationship vector, W′_i = (w′_i,0, . . ., w′_i,j, . . ., w′_i,m−1), W″_i = (w″_i,0, . . ., w″_i,j, . . ., w″_i,m−1) their linear combination W_i = (w_i,0, . . ., w_i,j, . . ., w_i,m−1), and W_i = α_iW′_i + β_iW″_i, α_i, β_i is linear relation modulus, α_i, β_i ∈ r, α_i, β_i ≥ 0. The game balance point of W′_i and W″_i is W i ⋅ W i ′ ‖ W i ′ ‖ = W i ⋅ W i ″ ‖ W i ″ ‖.

Proof: Because W′_i = (w′_i,0, . . ., w′_i,j, . . ., w′_i,m−1), W″_i = (w″_i,0, . . ., w″_i,j, . . ., w″_i,m−1) their linear combination W_i = (w_i,0, . . ., w_i,j, . . ., w_i,m−1), and W_i = α_iW′_i + β_iW″_i, as show in Figure 6.

Because cos θ 1 = W i ⋅ W i ′ ‖ W i ‖ ⋅ ‖ W i ′ ‖, cos θ 2 = W i ⋅ W i ′ ‖ W i ‖ ⋅ ‖ W i ′ ‖, we want to find the game and balance point to make θ₁=θ₂, then cosθ₁=cosθ₂.

On other words, the game balance point between W′_i and W″_i should be the point that has the equal correlation coefficient to each of the vectors. That is to say: W i ⋅ W i ′ ‖ W i ‖ ⋅ ‖ W i ′ ‖ = W i ⋅ W i ″ ‖ W i ‖ ⋅ ‖ W i ″ ‖, then we get: W i ⋅ W i ′ ‖ W i ′ ‖ = W i ⋅ W i ″ ‖ W i ″ ‖.

Combining (4) and (5), the paper builds the liner binary simple equations: { α i + β i = 1 W i ⋅ W i ′ ‖ W i ′ ‖ = W i ⋅ W i ″ ‖ W i ″ ‖

For W_i = α_iW′_i + β_iW″_i ( α i W i ′ + β i W i ″ ) · W i ′ ‖ W i ′ ‖ = ( α i W i ′ + β i W i ″ ) ⋅ W i ″ ‖ W i ″ ‖   then ( α i W i ′ + β i W i ″ )   ⋅   W i ′   ⋅   ‖ W i ″ ‖ = ( α i W i ′ + β i W i ″ )   ⋅   W i ″   ⋅   ‖ W i ′ ‖

For α_i + β_i = 1, then α i = ‖ W i ″ ‖ ‖ W i ′ ‖ + ‖ W i ″ ‖ = ∑ j = 0 m − 1   w i , j ″ 2 ∑ j = 0 m − 1   w i , j ′ 2 + ∑ j = 0 m − 1   w i , j ″ 2 β i = ‖ W i ′ ‖ ‖ W i ′ ‖ + ‖ W i ″ ‖ = ∑ j = 0 m − 1   w i , j ′ 2 ∑ j = 0 m − 1   w i , j ′ 2 + ∑ j = 0 m − 1   w i , j ″ 2

So: 0 < α_i, β_i < 1, α_i, β_i ∈ r

According to the above data table, the algorithm figures out α_i = 0.53, β_i = 0.47.

According to the above all, the algorithm gets the weight vector (as Table 3) and chooses the maximum value node as the cluster core C * = ( c 0 * , … , c i * , … , c m − 1 * ), in this cluster it chooses (2, 1) as cluster core c_i.

Given a shortest path, the path weight is the sum of all on-path node weights. For example, the weight of path < (2, 2), (2, 3), . . ., (2, 5), . . ., (5, 5) > is ∑_i α_iW′_i + ∑_i β_iW″_i.

Let the cluster with n′ members be: C = { C 0 = ( c 0 , 0 ,   c 0 , 1 , … , c 0 , m − 1 ) , … , C i = ( c i , 0 ,   c i , 1 , … , c i , m − 1 ) , … , C n ′ − 1 = ( c n ′ − 1 , 0 ,   c n ′ − 1 , 1 , … , c n ′ − 1 , m − 1 ) }

After obtaining the space weight vector W′, the data weight vector W″, the spatial core c_i,a and the data core c_i,b, the paper will calculate the weight vector W and the cluster core c j *.

The main idea of the least weighted path tree generation algorithm can be sketched as follows. After obtaining the space weight vector and the data quantity weight vector, we tried to find a new vector composed by linear combination of the two old ones. And it builds binary simple equations between them, seeks linear parameters. Then they generate a least weighted path tree, namely multicast tree. The least weighted path tree generation algorithm is shown in Algorithm 3.

For example: in the cluster, the spatial center node is A(2, 2), and the maximal data quantity node is B(2, 1). For the requirement of practice, in this cluster, let W i ⋅ W i ′ ‖ W i ′ ‖ = W i ⋅ W i ″ ‖ W i ″ ‖, and α_i + β_i = 1. To resolve the liner binary simple equations, the paper gets the liner relation modulus α_i, β_i = f(W′_i, W″_i), then gets α_i = 0.53, β_i = 0.47. At last, the algorithms attain new weight vector W.

Notice that since the dimension of the space weight vector and the data quantity weight vector may be different, the data quantity weight vector may stem from a function of the nature data quantity.

At present, the algorithm in this paper mostly focuses on optimal data transfer strategy in 2-D space. The reason is that recently mostly applications are still limited within 2-D. With the development of technology, Sensor grid architecture steps into outer space, into 3-D.

In the paper the space weight vector is complex, but the data quantity weight vector is definite, therefore the system can gain it directly. However, in some situation the data quantity could be changing, so that we should study special sub-algorithm for it. For example, in Pervasive Computing, the data quantity weight vector is not definite, therefore it is a function: W″ = f(x).

Moreover, we just discussed about the liner relationship of two vectors in this paper. On the other level, the above method can just resolve the linear relationship of two vectors. But, in practice, the system sometimes does not transform linearly but exponentially. The paper can resolve one exponential equation in the situation. For example, we can define the nodes weights as W i , j = ( α i W i , j ′ ) 2 + ( β i W i , j ″ ) 2. In the meantime the system can use some other methods to control, such as controlling theory method, differential method, etc.

Furthermore, after discussed two vectors correlation: the space weight vector and data weight vector, the paper can easily be extended to three weight vectors correlation, for instance the economy weight vector. In this situation, the costs of server in different place vary, therefore the clients prefer the node with cheaper cost. Therefore the system should take the economy factor into account, while constructing multicast tree. The three weight vectors are game and balance with each other. The relationship of the weight should be W_i,j = α_iW′_i,j + β_iW″_i,j + λ_iW″′_i,j. And it can easily extend the equation array as (7) { α i + β i + γ i = 1 W i ⋅ W i ′ ‖ W i ′ ‖ = W i ⋅ W i ″ ‖ W i ″ ‖ = W i ⋅ W i ‴ ‖ W i ‴ ‖

After discussing two and three vectors correlation, the algorithm can be extended to N-vectors correlation. Because in the realistic world, people should consider a number of factors while constructing the multicast tree, for example: space, data, economy, politics, military, etc. In this mode, every factor is a vector. As long as the physic mean of different factor is independent, the weight vector is linear (non-relative). Even if the two vectors are linear related, there are the ways to turn it to be linear (non-relative). All these factors can be denoted by a series of weight vectors. All of them game and balance with each other.

The relationship of the weight can be defined as W i , j = α i ( 1 ) W i , j ( 1 ) + … + α i ( k ) W i , j ( k ) + … + α i ( n ) W i , j ( n ). And the equation can be extended to { α i ( 1 ) + … + α i ( k ) + … + α i ( n ) = 1 W i ⋅ W i ( 1 ) ‖ W i ( 1 ) ‖ … = W i ⋅ W i ( k ) ‖ W i ( k ) ‖ … = W i ⋅ W i ( n ) ‖ W i ( n ) ‖

It can be solved by mathematical induction.

Moreover, these factors are based on linear non-relationship condition, so their cardinal number is accountable infinite. But in realty many factors are linear relationship, so that we can turn these to be linear non-relationship. If factors are of linear relationship, then their cardinal number is unaccountable infinite.