Abstract
A radio mean square labeling of a connected graph is motivated by the channel assignment problem for radio transmitters to avoid interference of signals sent by transmitters. It is an injective map from the set of vertices of the graph to the set of positive integers , such that for any two distinct vertices , the inequality holds. For a particular radio mean square labeling , the maximum number of taken over all vertices of is called its spam, denoted by , and the minimum value of taking over all radio mean square labeling of is called the radio mean square number of , denoted by . In this study, we investigate the radio mean square numbers and for path and cycle, respectively. Then, we present an approximate algorithm to determine for graph . Finally, a new mathematical model to find the upper bound of for graph is introduced. A comparison between the proposed approximate algorithm and the proposed mathematical model is given. We also show that the computational results and their analysis prove that the proposed approximate algorithm overcomes the integer linear programming model (ILPM) according to the radio mean square number. On the other hand, the proposed ILPM outperforms the proposed approximate algorithm according to the running time.
1. Introduction
In wireless networks, each radio station assigns a number called frequency. When different transmitters of district stations send signals, the receiver might get unnecessarily interference of the signals sent by transmitters in particular with close frequencies. This is the channel assignment problem introduced by Hale [1] in 1980 to minimize such interference. In 2001, Chartrand et al. [2] proposed converting this problem to graph theoretical problem using vertex labeling. Many researchers involved with this problem [3–16] and produced different methods to minimize the interference of signals [7]. Recently, Ramesh et al. [8] proposed a new method called radio mean square labeling, which is defined as follows. A radio mean square labeling of a connected graph is an injective function from its vertex set to the set of natural numbers , such that for any two distinct vertices and , holds, where denotes the distance between the two vertices, and represents the diameter of the graph [8]. For a radio mean square labeling , the maximum number of taken over all vertices of is called its spam, denoted by , and the minimum value of taking over all radio mean square labeling of is called the radio mean square number of , denoted as . The radio mean square number of , denoted by is the maximum number assigned to any vertex of . Ramesh et al. [8] determined the radio mean square number for some graphs such as in the centric subdivision of spoke wheel graph and biwheel graph.
Due to most of nontrivial coloring models, graph coloring is an NP-hard problem. Therefore, we take into consideration a graph coloring algorithm [9–11]. The judgment of the performance of the used algorithm does include its effectiveness and accuracy for a large number of vertices and the level of complexity regarding suboptimal solutions [12, 13]. Here, we introduce an approximate algorithm that leads to an upper bound of the radio mean square for a large number of vertices. Finally, we turn our attention to reformulate the radio square labeling as a linear programming model and then minimize the suggested linear function. We use of transforming the nonlinear constraint to become a linear constraint by using of some large integers dependably on the coefficient range of the given problem under path calculating conditions. Some illustrated examples and comparison between the techniques will be given.
It should be noted that all the considered graphs in this study are finite, simple, connected, and undirected.
The organization of the study goes as follows: in Section 2, the radio mean square number of cycles and paths are given. Section 3 is devoted to present an approximate algorithm that finds the upper bound of the radio mean square number of a given graph and an illustrative example is included. Section 4 deals with a new mathematical model for finding the upper bound of the radio mean square number of the given graph. Section 5 provides the experimental results. Analysis and statistical tests between the mathematical model and the proposed approximate algorithm are provided. The last section is considered for conclusion.
2. Results
The following theorem is about the for the path , and next, we will get the for cycle .
Theorem 1. For the path , and the radio mean square number is
Proof. Clearly, . Then, one can define as follows:
Case a. ForOne may label the vertices of as follows: Subcase a.1: is even: Subcase a.1: is odd:
Case 2. bFor and one may label the vertices of as the following subcases: Subcase b.1: is even: Subcase b.1: is odd:Therefore, for any pair , we have .
Hence, is a valid radio mean square labeling for , and therefore, . Since is injective, for all radio mean square labeling , and hence, . Therefore, the labeling defined above satisfies the radio mean square condition.
Example 1. The radio mean square numbers of , and are shown in Figure 1. It is clear that for and , but for , and .
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Theorem 2. The radio mean square numbers of the cycles , , are given by
Proof. It is clear that the dimension of is since its length is . Then, one can define as follows:
Case 3. aFor , we may label the vertices of by
Case 4. bFor , . We may label the vertices of by one of the following subcases: Subcase b.1: is even: Therefore, for any pair , we have . Subcase b.2: is odd:So, for any pair , the following inequality holds .
Hence, is a valid radio mean square labeling for , and therefore,Since is injective,For all radio mean square labeling ,Therefore, the labeling defined above satisfies the radio mean square condition.
Example 2. The radio mean square numbers of cycles , and are shown in Figure 2. It is clear that for and , but for and.
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3. A Novel Graph Radio Mean Square Algorithm
Here, we introduce an approximated algorithm. This algorithm finds an upper bound of the radio mean square for arbitrary graph . The main idea is to labeling some vertices (initial vertices) by . On the other hand, the algorithm chooses a different vertex as an initial vertex in each iteration.
The time complexity of an algorithm is defined as the number of instructions of this algorithm multiplied by the running time of each instruction. The time complexity is considered as a good metric to evaluate the given algorithm. Thus, Algorithm 1 has nine steps, and both Step 1 and Step 2 have the same instruction. Step 3 has a nested loop that has time complexity. On the other hand, Step 4, Step 5, and Step 6 have time complexity. Step 7 has one instruction, while Step 8 and Step 9 have and , respectively. Therefore, Algorithm 1 has the time complexity .
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In the coming example, we show and explain how to compute the radio mean square labeling problem for .
Example 3. Suppose that is the label of the vertex , . Therefore, 1 explores an upper bound of the radio mean square labeling problem as follows:
It is known that . We select a vertex and . Let , and for all , computeLet ; we choose a vertex , such that . Give and .Let ; we choose a vertex , where . Give and .Let ; we choose a vertex where . Give and .Let ; we select a vertex , where . Give and . Plainly, all vertices are labelled and
4. Formulation of the Radio Mean Square Labeling as a Mathematical Model
In this section, we present the integer linear programming model (ILPM) for the radio mean square labeling problem.
Let be a connected graph of order with and let be the distance matrix of , that is, for . Suppose that is the label of the vertex , . Now, we can propose the mathematical model for the radio mean square labeling problem as the ILPM. Let us suppose the function is .
4.1. Formulation 1
Minimize F subject to the constraints for , , and , and as mentioned before, the following steps will transform nonlinear constraints to become linear which is easy to deal with.
4.2. Formulation 2
Since , we have the following inequalities:
M is a large integer number which depends on the coefficients range of the problem.
The absolute value notation is used to get distinct values for , where .
Now, we can reformulate the radio mean square problem as follows:
Subject to ,
Here, the floor function is used because the values of , are integers.
Example 4. The details of the ILPM to compute the radio mean square labeling for . Assume that is the label of the vertex , such that . Thus, the mathematical model for the radio mean square labeling problem as the ILPM is prepared as follows:Subject toSince and for , the distance matrix of isTherefore, the above mathematical model can be written as follows:Subject toThe solution of the above model equals to 3.
5. Computational Study
In this article, we propose the analysis of the computational results that show the superiority of Algorithm 1 on the ILPM according to the radio mean square number. On the other hand, the proposed ILPM outperforms Algorithm 1 according to CPU time.
Paths and cycles are used to evaluate the proposed models. The computation environment is given in Table 1. MATLAB solver is used to solve the ILPM. In Tables 2 and 3, the following symbols standard RMS, , and CPU times are used to indicate the exact radio mean square number, the calculated mean square number, and the running time for path and cycles, respectively. The convergence between the calculated and exact upper bounds of the radio mean square number of paths is given in Table 2. Figures 3 and 4 show that the superiority of the proposed Algorithm 1 on the ILPM according to the radio mean square number. For example, the standard radio mean square number for is 54, but it is 56 and 344 by Algorithm 1 and the ILPM, respectively. Figures 5 and show the superiority of the proposed Algorithm 1 on the ILPM according to CPU time. Table 3 provides that the gap between the ILPM and the proposed Algorithm 1 is large according to the radio mean square number. It is clear that 1 is better than the ILPM according to the radio mean square number. According to the CPU time, Tables 2 and 3 explain the superiority of the proposed ILPM on Algorithm 1.
6. Conclusions
In this work, we determined the radio mean square numbers and for paths and cycles. Then, the proposed approximate algorithm is introduced to obtain for graph . In addition, a new mathematical model is proposed in order to find the upper bound of for graph , and a comparison between the proposed approximate algorithm and the proposed mathematical model is introduced. Finally, the computational results and their analysis have proved that the proposed approximate algorithm overcomes the ILPM according to the radio mean square number.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through General Research Project (R.G.P.1/208/41).