Classification Formula and Generation Algorithm of Cycle Decomposition Expression for Dihedral Groups

and Applied Analysis 3 3.3. The Time Complexity of Generation Algorithm Based on Permutation. Computational complexity is divided into two kinds: one is time complexity, and the other is space complexity. The analysis of space complexity is similar to that of time complexity, and the analysis of space complexity is more simple [12]; in this paper, the two algorithms’ space complexities are the same on the whole, so we limit our study to the time complexity. First apply formulae (4) and (6) to solve M k , R k (k = 0, 1, . . . , n − 1); we estimate the time complexity. Formula (4) is corresponding to the second row of M k ; formula (6) is corresponding to the second row of R k ; for each R k or M k , we need n additions (modulo n); thus we obtain the second row of the permutation, then express it as the form of formula (3), so we get the expression of permutation of M k and R k . There are 2n elements in theD n group, so the time complexity function T 1 (n) of the algorithm is T 1 (n) = n ∗ (2n) = 2n 2 . (8) After obtaining the expression of permutation of all elements in theD n group, we apply the conversion algorithm in Section 3.2 to every element in the group to get their cycle decomposition expression. The main operation is comparison in this conversion algorithm. Begin with the first row and the first column p k [0, 0], comparing p k [0, 0] with p k [1, 0], searching the element which is same top k [1, 0] in the first row ifp k [0, 0] andp k [1, 0] are not equal. Comparing p k [1, j] (j = 1, . . . , n − 1) with p k [1, 0] one by one, at most (n − 1) comparisons are made; then comparing p k [1, j] with p k [0, 0], searching the element which is same top k [1, j] in the first row ifp k [1, j] andp k [0, 0] are not equal, at most (n − 2) comparisons are made, and so forth, the rest may be deduced by analogy and the time complexity function T 2 (n) of the algorithm is T 2 (n) = n!. (9) As there are 2n elements in the D n group, the time complexity function of the generation algorithm based on permutation is T 3 (n) = n 2 + 2n ∗ n! = 2n (n + n!) . (10) We can observe from (10) that the complexity of the generation algorithm based on permutation is Q 1 (n ∗ n!) with very low efficiency which is unable to fulfill the requirement in the solution of large size problems using D n group. So a faster generation algorithm needs to be developed. 4. The Derivation of the Common Formula for the Cycle Decomposition Expressions of D n Group There are two types of D n group’s elements: one is derived from reflectivity conversion M k (k = 0, 1, . . . , n − 1), and the other from rotation conversion R k (k = 0, 1, . . . , n − 1); the cycle decomposition expressions of these two kinds of elements adhere to different rules, so we can research cycle 1 2


Introduction
group (Dihedral group) is a kind of important group which plays an important role in the research of the properties of group [1,2].It can be used to solve a variety of factual problems, such as the classical necklace problem, the enumeration problem of molecule structure [3], modelling of communication networks [4], construction of visual cryptography scheme [5], and analysis of satellite status in the orbit of LEO/MEO satellite network, which all adopt the   group [6][7][8] and use Burnside lemma and Pólya theorem to compute the number of combination necklace schemes, which all depend on the cycle decomposition expressions of   group.There are lots of research on the enumeration problem of necklace problem based on   group [9][10][11], but very few concentrates on the structural problems of combinational scheme.In the study of researching satellite status in the orbit of LEO/MEO satellite network and network route simulation, we should get each satellite status and take these as input, that is to say, the structural problems of combinational scheme, which is closely related to the   group cycle decomposition expression [12].For the   group of low order, we can compute cycle decomposition expression of every group element manually, but for   group of high order, it is not only time consuming but also error by hand, so it is essential to adapt computer and the fast generation algorithm is the key to fulfill this problem.Fu and Wang [13] presented the common formula of   group permutation expression and then convert the permutation expression into cycle decomposition expressions of   group; however, it is inefficient.If we can get the cycle decomposition expressions of   group, it is much easier to solve the problem.There are two types of   group's elements: one is derived from reflection conversion, and the other from rotation conversion; the cycle decomposition expressions of these two kinds of elements adhere to different rules, so we can research cycle decomposition expressions of   group respectively, and then can get each common formula to fulfill fast generation algorithm on the cycle decomposition expressions of   group.

Necklace Problem and Permutation
Expression of   Group then how many different necklaces with  beads can be made?When  and  are both small, we can work out all the different necklaces using exhaustive algorithm.But with the increasing size of  and , it gets more and more difficult by exhaustive algorithm, so the group method must be used, and simpler and more efficient method has not been found up to the present [3].Pólya Theorem.Let  be a group of permutations of the set of  objects; then the number (; , , . . ., ) of nonequivalent colorings is given by where   () is the number of -cycle in the permutation  [11].
When we analyze the satellite status in the orbit by Pólya theorem,  satellites in the orbit are the coloring objects and  status is the  colors.The key for the problem is to solve the group of permutations of the set of  objects, that is, the   group.
As we rotate regular  quadrate according to 2/ counterclockwise, vertex  has moved to the position originally occupied by vertex  + 1 (mod ), so this rotation is the conversion on , marked as  1 : The conversion according to 2 (/) is marked as   : where the addition and subtraction are the modulo  operation (as the same for the entire paper),  0 is the identity, and   can be shown as Another conversion is reflection in the symmetric axis according to , named as reflectivity conversion.Because there is  symmetric axis, we mark the axis through vertex 0 as  0 , the axis through the vertex of the midpoint of edge [0, 1] as  1 , . .., until  −1 .The corresponding reflectivity conversion is marked as  0 ,  1 , . . .,  −1 .For instance, We can prove that Let Then   is closed under the composite operation of the conversion, the identity  0 exists, and each element has inverse, so   forms group, which is dihedral group.

The Generation Algorithm on the Cycle Decomposition Expressions of 𝐷 𝑛 Group Based on Permutation and Complexity Analysis
3.1.The Cycle Decomposition Expressions of   Group.The representation of each permutation as a product of disjoint cycles and the decomposition is unique [3]; the product of disjoint cycles is named as cycle decomposition expression of elements of the group.We can devise an algorithm for converting the permutation expression into the cycle decomposition expression of   Group.

The Algorithm Design for Converting the Permutation Expression into the Cycle Decomposition Expression of 𝐷 𝑛
Group.Let   [, ] ( = 0, 1;  = 0, 1, . . .,  − 1) express the element ( column  row) in the permutation expression of each element of the   group and traverse all the elements of   starting from   [0, 0]; the algorithm is shown as follows.
(3) Search for the element equal to ( In the analysis of the algorithm, we know that this conversion method is of low efficiency from formula (14), so it cannot be used for problem of great size by group   ; a fast generation algorithm on the cycle decomposition expressions of   group based on permutation must be designed.

The Time Complexity of Generation Algorithm Based on
Permutation.Computational complexity is divided into two kinds: one is time complexity, and the other is space complexity.The analysis of space complexity is similar to that of time complexity, and the analysis of space complexity is more simple [12]; in this paper, the two algorithms' space complexities are the same on the whole, so we limit our study to the time complexity.
First apply formulae ( 4) and ( 6) to solve   ,   ( = 0, 1, . . .,  − 1); we estimate the time complexity.Formula ( 4) is corresponding to the second row of   ; formula ( 6) is corresponding to the second row of   ; for each   or   , we need  additions (modulo ); thus we obtain the second row of the permutation, then express it as the form of formula (3), so we get the expression of permutation of   and   .There are 2 elements in the   group, so the time complexity function  1 () of the algorithm is After obtaining the expression of permutation of all elements in the   group, we apply the conversion algorithm in Section 3.2 to every element in the group to get their cycle decomposition expression.The main operation is comparison in this conversion algorithm.
Begin with the first row and the first column   [0, 0], comparing   [0, 0] with   [1, 0], searching the element which is same to   [1,0] in the first row if   [0, 0] and   [1,0] are not equal.Comparing   [1, ] ( = 1, . . .,  − 1) with   [1,0] one by one, at most ( − 1) comparisons are made; then comparing   [1, 𝑗] with   [0, 0], searching the element which is same to   [1, 𝑗] in the first row if   [1, 𝑗] and   [0, 0] are not equal, at most ( − 2) comparisons are made, and so forth, the rest may be deduced by analogy and the time complexity function  2 () of the algorithm is As there are 2 elements in the   group, the time complexity function of the generation algorithm based on permutation is We can observe from (10) that the complexity of the generation algorithm based on permutation is  1 ( * !) with very low efficiency which is unable to fulfill the requirement in the solution of large size problems using   group.So a faster generation algorithm needs to be developed.

The Derivation of the Common Formula for the Cycle Decomposition Expressions of 𝐷 𝑛 Group
There are two types of   group's elements: one is derived from reflectivity conversion   ( = 0, 1, . . .,  − 1), and the other from rotation conversion   ( = 0, 1, . . .,  − 1); the cycle decomposition expressions of these two kinds of elements adhere to different rules, so we can research cycle  decomposition expressions of   group, respectively; then, we can get each common formula to fulfill fast generation algorithm on the cycle decomposition expressions of   group.
The   group is corresponding to a regular  quadrate (as shown in Figure 1).For   group of low order, we can get the cycle decomposition expressions of each element in the group by eyes, observe the vertex's constituting rule in every cycle with induction.Based on an exhausted series of the cycle decomposition expressions of   group, we bring forward the common formula and then prove it by mathematical induction.
(1) The Cycle Decomposition Expressions for   Group of Low Order.For instance, we enumerate all the cycle decomposition expressions of  6 group. 6 group is corresponding to the Regular 6 quadrate, as shown in Figure 2, where   ( = 0, 1, 2, . . ., 5) are the cycle decomposition expressions of the reflectivity conversion, and   ( = 0, 1, 2, . . ., 5) are the cycle decomposition expressions of the rotation conversion.
The main text paragraph is as follows (see Figure 2): (2) The Common Cycle Decomposition Expressions for   Group with Reflectivity Conversion.The cycle decomposition expressions for   are not only related to the parity of  in the   group but also to the parity of  in the element   , so the formula can be divided into four instances.
(a)  is odd in the   group and  is also odd in   .While  equals 1, take reflectivity conversion that  1 is the axis (as shown in Figure 1); 0 and 1 compose the transposition, that is, (1, 0); 2 and  − 1 compose the transposition, that is, (2,  − 1), and so forth.
Now we obtain the following common formula: We can prove that formula ( 8) is valid for  = 1,  = 2 + 1 and  = (2( = 1) + 1) = 2 + 3 by mathematical induction.So formula (8) is true for all positive integers .In the same way we can obtain the following three formulae.
(b)  is odd in the   group and  is even in   : There are ( − 1)/2 transpositions.
(c)  is even in the   group and  is odd in   : There are /2 transpositions.
(d)  is even in the   group and  is even in   : There are ( − 2)/2 transpositions and two fixed points.
(3) The Common Cycle Decomposition Expressions for the Element   in the Group with Rotation Conversion.The type of   is (/)  ,  = (, ), and the type of   varies along with the value of  and .There are two instances.
(a)  is prime.While  equals 0, we deal with it as follows: While  = 1, 2, . . .,  − 1,  = (, ) = 1, the type of   is (/)  =  1 ; that is, every element   in the group makes up of a cycle, and there are  terms in the cycle, it is denoted by (b)  is composite.When  equals 0,  0 is the same to formula (14).

The Time Complexity of the Generation
As a result, the time complexity of generation algorithm based on permutation is  1 () = ( * !), while the time complexity of generation algorithm based on classification Formulae is  2 () = ( 2 ), so the time complexity of generation algorithm based on permutation is much more greater than that of generation algorithm based on classification Formulae.The results of the process show that the generation algorithm which is based on the classification formula is of superiority.

Conclusions
This paper includes the reflectivity and rotation conversion, which derived six common Formulae on cycle decomposition expressions of   group; it designed the generation algorithm on the cycle decomposition expressions of   group, which is based on the method of replacement conversion and the classification formula; algorithm analysis and the results of the process show that the generation algorithm which is based on the classification formula is outperformed by the general algorithm which is based on replacement conversion, it has great significance to solve the necklace problem and the combinational scheme of the satellite status in the orbit of LEO/MEO satellite network, especially the structural problems of combinational scheme, by using group theory and computer.

5. The Generation Algorithm on the Cycle Decomposition Expressions of 𝐷 𝑛 Group Based on the Classification Formulae and Complexity Analysis 5
.1.The Idea of the Algorithm Designing.The generation algorithm on the cycle decomposition expressions of   group based on the classification Formulae is relatively simple.For   ( = 0, 1, ..., −1) with reflectivity conversion, first judge the parity of  and ; then substitute them into formulae (13)-(16), thus we can get the cycle decomposition expressions of  elements of the group.For   ( = 0, 1, . . .,  − 1) with rotation conversion, substitute  = 0 into formula (17) and get  0 ; while  is prime, substitute  into formula (18); while  is composite, substitute  into formula (19); thus, we can obtain all the cycle decomposition expressions of   ( = 0, 1, . . .,  − 1).
Algorithm Based on Classification Formulae.The generation algorithm on the cycle decomposition expressions of   group based on the classification Formulae is relatively simple.Judge the parity of  and each  ( = 1, 2, . . ., ); substitute  into Formulae (8)∼(13); thus get   the cycle expression.For   , the only thing is to judge the fraction of .For every   or   , at most 2 additions are made.As there are 2 elements in the   group, so the time complexity function is  4 = 2 +  * (2) = 2 2 + 2 = 2 ( 2 + ) .