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Figure 13. Hypothetical design of a cellular grid labeled with the L(3,1) coloring algorithm. Figure 13 shows a hypothetical design procedure for channel allocation sets to cellular base stations, where there are 12 channels in a cell cluster (i.e., N = 12).

Figure 13 Hypothetical design of a cellular grid labeled with the L(3,1) coloring algorithm. Figure 13 shows a hypothetical design procedure for channel allocation sets to cellular base stations, where there are 12 channels in a cell cluster (i.e., N = 12).

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Figure 2. A5 x 5 cellular graph tilled using the L(2,1) coloring. Figure 3. A 7 x 7 cellular graph network matrix colored with the L(2,1) coloring. 2.2. The L(2,1,1) Algorithm
Figure 2. A5 x 5 cellular graph tilled using the L(2,1) coloring. Figure 3. A 7 x 7 cellular graph network matrix colored with the L(2,1) coloring. 2.2. The L(2,1,1) Algorithm
Figure 5. Clique for the L(2,1,1) coloring.
Figure 5. Clique for the L(2,1,1) coloring.
Figure 6. A 5 x 5 matrix of a network graph colored using the L(2,1,1) coloring.
Figure 6. A 5 x 5 matrix of a network graph colored using the L(2,1,1) coloring.
Figure 7. A 7 X 7 matrix of a network graph colored using the L(2,1,1) coloring.
Figure 7. A 7 X 7 matrix of a network graph colored using the L(2,1,1) coloring.
Figure 8. Flow chart for the formulation of the L(3,1) coloring algorithm. la,b € T, |f(a) — f(b)| = 3 if d(a, b) = 1 and |f(a) — f(b)| = 1 if d(a, b) = 3. For this labeling, a pair o artices with a pair of consecutive colors should be separated by a minimum of one unit, while vertice: ith the same color must be separated by at least three units. Figure 8 shows the flowchart for the mulation of the L(3,1) algorithm.
Figure 8. Flow chart for the formulation of the L(3,1) coloring algorithm. la,b € T, |f(a) — f(b)| = 3 if d(a, b) = 1 and |f(a) — f(b)| = 1 if d(a, b) = 3. For this labeling, a pair o artices with a pair of consecutive colors should be separated by a minimum of one unit, while vertice: ith the same color must be separated by at least three units. Figure 8 shows the flowchart for the mulation of the L(3,1) algorithm.
Table 1. Vertex numbering of the L(3,1) algorithm (for a 10 x 10 matrix) with n row and m column.
Table 1. Vertex numbering of the L(3,1) algorithm (for a 10 x 10 matrix) with n row and m column.
Figure 9. Clique for the L(3,1) coloring.
Figure 9. Clique for the L(3,1) coloring.
Figure 11. Cellular grid of a5 x 5 matrix colored using the L@G,1) coloring Figure 12. A 10 x 10 matrix cellular grid colored using the L(3,1) coloring.
Figure 11. Cellular grid of a5 x 5 matrix colored using the L@G,1) coloring Figure 12. A 10 x 10 matrix cellular grid colored using the L(3,1) coloring.
Figure 10. Color spectrum of the L(3,1) algorithm.
Figure 10. Color spectrum of the L(3,1) algorithm.
Table 2. Relationships between successive vertices for the L(3,1) coloring algorithm. As shown in Table 2, each pair of consecutive colors [(0,1), (1,2), (2,3), (3,4), (4,5), (5,6), (6,7), (7,8), (8,9), (9,10) and (10,11)] has a Manhattan distance of at least 3. According to Table 1, the lowest spacing between colors should be at least 2. Other sets of successive color pairs, such as (0,1), (3,4), (4,5), (7,8), and (9,10), have a Manhattan distance of at least 2 and have a similar parity for the n and m coordinates of both of their component colors. Note that in Figure 14, any random vertex (n, m) and another vertex (n’, m’) have a Manhattan distance of 2. The separation between them will not be less than 1, except when n and n’, or m and m’, have unique parities. Hence, these two vertices are separated by a minimum of 2 coloring units. It can thus be deduced that the distance possessing the same color is >3. From the function o f a vertex (a) of the L(3,1) a between two vertices gorithm, it is possible to infer that vertices with similar colors also possess similar parities for coordinates i or j. As shown in Table 2, none of the parity pairs (i and j coordinates) for the consecutive co Therefore, from these observations, it can be seen that a distance of more than three units. This implies tha the vertices with the same or pairs are the same. color are separated by the second definition hold coloring of L(3,1) holds true for a given cellular gra s true. Hence, the f(a) ph matrix. Additionally, it can be observed from Figure 14 that the distance between two vertices of the same color is at least four units. Thus, the reuse distance is four (i.e., o = 4), which satisfies the third condition for correctness. This evaluation of the Manhattan distance and the minimum distance between pairs of consecutive colors therefore proves the correctness of the L(3,1) algorithm.
Table 2. Relationships between successive vertices for the L(3,1) coloring algorithm. As shown in Table 2, each pair of consecutive colors [(0,1), (1,2), (2,3), (3,4), (4,5), (5,6), (6,7), (7,8), (8,9), (9,10) and (10,11)] has a Manhattan distance of at least 3. According to Table 1, the lowest spacing between colors should be at least 2. Other sets of successive color pairs, such as (0,1), (3,4), (4,5), (7,8), and (9,10), have a Manhattan distance of at least 2 and have a similar parity for the n and m coordinates of both of their component colors. Note that in Figure 14, any random vertex (n, m) and another vertex (n’, m’) have a Manhattan distance of 2. The separation between them will not be less than 1, except when n and n’, or m and m’, have unique parities. Hence, these two vertices are separated by a minimum of 2 coloring units. It can thus be deduced that the distance possessing the same color is >3. From the function o f a vertex (a) of the L(3,1) a between two vertices gorithm, it is possible to infer that vertices with similar colors also possess similar parities for coordinates i or j. As shown in Table 2, none of the parity pairs (i and j coordinates) for the consecutive co Therefore, from these observations, it can be seen that a distance of more than three units. This implies tha the vertices with the same or pairs are the same. color are separated by the second definition hold coloring of L(3,1) holds true for a given cellular gra s true. Hence, the f(a) ph matrix. Additionally, it can be observed from Figure 14 that the distance between two vertices of the same color is at least four units. Thus, the reuse distance is four (i.e., o = 4), which satisfies the third condition for correctness. This evaluation of the Manhattan distance and the minimum distance between pairs of consecutive colors therefore proves the correctness of the L(3,1) algorithm.
Figure 14. Description of cell cluster. Co-channel reuse means that within a specific area of coverage there are many cells utilizing a similar group of channels [37]. Such cells are called co-channel cells. Co-channel cells also exhibit interference signals, which are referred to as co-channel interference. The only way to eliminate co-channel interference is by ensuring that the cells are separated by a sufficient distance [38]. In [39], the authors showed that in situations where cell sizes and base station power transmission are equal, the ratio of co-channel interference is often not a function of the power transmitted, but rather, that it depends on the cell radius (R) and the center spread of the nearest co-channel cells (D).
Figure 14. Description of cell cluster. Co-channel reuse means that within a specific area of coverage there are many cells utilizing a similar group of channels [37]. Such cells are called co-channel cells. Co-channel cells also exhibit interference signals, which are referred to as co-channel interference. The only way to eliminate co-channel interference is by ensuring that the cells are separated by a sufficient distance [38]. In [39], the authors showed that in situations where cell sizes and base station power transmission are equal, the ratio of co-channel interference is often not a function of the power transmitted, but rather, that it depends on the cell radius (R) and the center spread of the nearest co-channel cells (D).
Table 3. Comparison of running times (in milliseconds) for each of the coloring algorithms for different input sizes (number of subscribers).
Table 3. Comparison of running times (in milliseconds) for each of the coloring algorithms for different input sizes (number of subscribers).
Figure 15. Comparison of the running times of the three algorithms for different input sizes. Figure 15 shows a comparison of the running times of the three L(h,k) algorithms for different input sizes. The figure shows that, given a large amount of subscribers, the L(3,1) algorithm outperforms the L(2,1) algorithm (i.e., the L(3,1) algorithm has a smaller running time). This indicates that when there is a large number of users (e.g., in a densely populated area), the L(3,1) algorithm will perform optimally for channel allocation.
Figure 15. Comparison of the running times of the three algorithms for different input sizes. Figure 15 shows a comparison of the running times of the three L(h,k) algorithms for different input sizes. The figure shows that, given a large amount of subscribers, the L(3,1) algorithm outperforms the L(2,1) algorithm (i.e., the L(3,1) algorithm has a smaller running time). This indicates that when there is a large number of users (e.g., in a densely populated area), the L(3,1) algorithm will perform optimally for channel allocation.
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Computer ScienceFuture InternetFrequency allocation
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