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

Authors: Dakun Zhang; Yonggang Lin; Guozhi Song
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

The necessary of classification research on common formula ofDn group (dihedral group) cycle decomposition expression is illustrated. It includes the reflection and rotation conversion, which derived six common formulae on cycle decomposition expressions of Dn group; it designed the generation algorithm on the cycle decomposition expressions of Dn 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 enumeration of the necklace combinational scheme, especially the structural problems of combinational scheme, by using group theory and computer.


---

## Body

## 1. Introduction

D
n 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 Dn group [6–8] and use Burnside lemma and Pólya theorem to compute the number of combination necklace schemes, which all depend on the cycle decomposition expressions of Dn group. There are lots of research on the enumeration problem of necklace problem based on Dn group [9–11], 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 Dn group cycle decomposition expression [12]. For the Dn group of low order, we can compute cycle decomposition expression of every group element manually, but for Dn 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 Dn group permutation expression and then convert the permutation expression into cycle decomposition expressions of Dn group; however, it is inefficient. If we can get the cycle decomposition expressions of Dn group, it is much easier to solve the problem. There are two types of Dn 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 Dn group respectively, and then can get each common formula to fulfill fast generation algorithm on the cycle decomposition expressions of Dn group.


## 2. Necklace Problem and Permutation Expression ofDn Group

### 2.1. Necklace Problem and Permutation Expression ofDn Group

The necklace problem is defined as follows. Let us suppose that a necklace can be made from beads ofm colors; then how many different necklaces with n beads can be made?Whenn and m are both small, we can work out all the different necklaces using exhaustive algorithm. But with the increasing size of n and m, 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 G be a group of permutations of the set of n objects; then the number P(G;m,m,…,m) of nonequivalent colorings is given by
(1)P(G;m,m,…,m)=1|G|(∑mλ1(g)+mλ2(g)+⋯+mλn(g)),
where λk(g) is the number of k-cycle in the permutation g [11].When we analyze the satellite status in the orbit by Pólya theorem,n satellites in the orbit are the coloring objects and m status is the m colors. The key for the problem is to solve the group of permutations of the set of n objects, that is, the Dn group.


### 2.2. The Permutation Expressions ofDn Group

Suppose thatX={0,1,2,…,n-1} (without loss of generality, we adapt 0~n-1 as sequence number) is the vertex set of the regular n(n≥3) quadrate and arranged counterclockwise, as shown in Figure 1.Figure 1
Regularn quadrate.As we rotate regularn quadrate according to 2π/n counterclockwise, vertex i has moved to the position originally occupied by vertex i+1 (mod n), so this rotation is the conversion on X, marked as R1:
(2)R1=(012⋯n-1123⋯0).
The conversion according to 2k(π/n) is marked as Rk:
(3)Rk=(012⋯n-1kk+1k+2⋯k+n-1),
where the addition and subtraction are the modulo n operation (as the same for the entire paper), R0 is the identity, and Rk can be shown as
(4)Rk(i)=k+i,i=0,1,…,n-1.
Another conversion is reflection in the symmetric axis according toπ, named as reflectivity conversion. Because there is n symmetric axis, we mark the axis through vertex 0 as L0, the axis through the vertex of the midpoint of edge [0,1] as L1,…, until Ln-1. The corresponding reflectivity conversion is marked as M0,M1,…,Mn-1. For instance,
(5)M0=(01⋯n-10n-1⋯1).
We can prove that
(6)Mk=k+n-i,k=0,1,…,n-1.Let(7)Dn={Rk,Mk∣k=0,1,…,n-1}.
Then Dn is closed under the composite operation of the conversion, the identity R0 exists, and each element has inverse, so Dn forms group, which is dihedral group.


## 2.1. Necklace Problem and Permutation Expression ofDn Group

The necklace problem is defined as follows. Let us suppose that a necklace can be made from beads ofm colors; then how many different necklaces with n beads can be made?Whenn and m are both small, we can work out all the different necklaces using exhaustive algorithm. But with the increasing size of n and m, 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 G be a group of permutations of the set of n objects; then the number P(G;m,m,…,m) of nonequivalent colorings is given by
(1)P(G;m,m,…,m)=1|G|(∑mλ1(g)+mλ2(g)+⋯+mλn(g)),
where λk(g) is the number of k-cycle in the permutation g [11].When we analyze the satellite status in the orbit by Pólya theorem,n satellites in the orbit are the coloring objects and m status is the m colors. The key for the problem is to solve the group of permutations of the set of n objects, that is, the Dn group.


## 2.2. The Permutation Expressions ofDn Group

Suppose thatX={0,1,2,…,n-1} (without loss of generality, we adapt 0~n-1 as sequence number) is the vertex set of the regular n(n≥3) quadrate and arranged counterclockwise, as shown in Figure 1.Figure 1
Regularn quadrate.As we rotate regularn quadrate according to 2π/n counterclockwise, vertex i has moved to the position originally occupied by vertex i+1 (mod n), so this rotation is the conversion on X, marked as R1:
(2)R1=(012⋯n-1123⋯0).
The conversion according to 2k(π/n) is marked as Rk:
(3)Rk=(012⋯n-1kk+1k+2⋯k+n-1),
where the addition and subtraction are the modulo n operation (as the same for the entire paper), R0 is the identity, and Rk can be shown as
(4)Rk(i)=k+i,i=0,1,…,n-1.
Another conversion is reflection in the symmetric axis according toπ, named as reflectivity conversion. Because there is n symmetric axis, we mark the axis through vertex 0 as L0, the axis through the vertex of the midpoint of edge [0,1] as L1,…, until Ln-1. The corresponding reflectivity conversion is marked as M0,M1,…,Mn-1. For instance,
(5)M0=(01⋯n-10n-1⋯1).
We can prove that
(6)Mk=k+n-i,k=0,1,…,n-1.Let(7)Dn={Rk,Mk∣k=0,1,…,n-1}.
Then Dn is closed under the composite operation of the conversion, the identity R0 exists, and each element has inverse, so Dn forms group, which is dihedral group.


## 3. The Generation Algorithm on the Cycle Decomposition Expressions ofDn Group Based on Permutation and Complexity Analysis

### 3.1. The Cycle Decomposition Expressions ofDn 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 Dn Group.


### 3.2. The Algorithm Design for Converting the Permutation Expression into the Cycle Decomposition Expression ofDn Group

Letpk[i,j](i=0,1;j=0,1,…,n-1) express the element (i column j row) in the permutation expression of each element of the Dn group and traverse all the elements of pk starting from pk[0,0]; the algorithm is shown as follows.Algorithm 1.
Consider the following.(1)
Start frompk[0,0], if pk[0,0]!=pk[0,1]; then go to (3).(2)
pk[0,0] is the fixed point which forms an independent cycle; it is denoted by (pk[0,0]); go to (7).(3)
Search for the element equal top[1,0] from pk[0,1] to pk[0,n-1]; suppose that the element is pk[0,j].(4)
Search forpk[1,j], if pk[1,j]!=pk[0,0]; then go to (6).(5)
pk[0,0] and pk[1,j] form a cycle; it is denoted by pk[1,j]; go to (7).(6)
Search for the element equal topk[1,j] at pk[0,i] which has not been written into the cycle; then judge the element whether it is equal to pk[0,0] or not; proceed the next searching until you get the element equal to pk[0,0] that form a cycle.(7)
During the next searching, delete all thepk[i,j] that have been written into the cycle from the original data structure; go to (1), until all the pk[i,j] have been written into the expression of the product of the cycles.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 Dn; a fast generation algorithm on the cycle decomposition expressions of Dn group based on permutation must be designed.


### 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 Mk, Rk(k=0,1,…,n-1); we estimate the time complexity. Formula (4) is corresponding to the second row of Mk; formula (6) is corresponding to the second row of Rk; for each Rk or Mk, 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 Mk and Rk. There are 2n elements in the Dn group, so the time complexity function T1(n) of the algorithm is
(8)T1(n)=n*(2n)=2n2.After obtaining the expression of permutation of all elements in theDn 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 columnpk[0,0], comparing pk[0,0] with pk[1,0], searching the element which is same to pk[1,0] in the first row if pk[0,0] and pk[1,0] are not equal. Comparing pk[1,j](j=1,…,n-1) with pk[1,0] one by one, at most (n-1) comparisons are made; then comparing pk[1,j] with pk[0,0], searching the element which is same to pk[1,j] in the first row if pk[1,j] and pk[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 T2(n) of the algorithm is
(9)T2(n)=n!.As there are 2n elements in the Dn group, the time complexity function of the generation algorithm based on permutation is
(10)T3(n)=n2+2n*n!=2n(n+n!).
We can observe from (10) that the complexity of the generation algorithm based on permutation is Q1(n*n!) with very low efficiency which is unable to fulfill the requirement in the solution of large size problems using Dn group. So a faster generation algorithm needs to be developed.


## 3.1. The Cycle Decomposition Expressions ofDn 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 Dn Group.


## 3.2. The Algorithm Design for Converting the Permutation Expression into the Cycle Decomposition Expression ofDn Group

Letpk[i,j](i=0,1;j=0,1,…,n-1) express the element (i column j row) in the permutation expression of each element of the Dn group and traverse all the elements of pk starting from pk[0,0]; the algorithm is shown as follows.Algorithm 1.
Consider the following.(1)
Start frompk[0,0], if pk[0,0]!=pk[0,1]; then go to (3).(2)
pk[0,0] is the fixed point which forms an independent cycle; it is denoted by (pk[0,0]); go to (7).(3)
Search for the element equal top[1,0] from pk[0,1] to pk[0,n-1]; suppose that the element is pk[0,j].(4)
Search forpk[1,j], if pk[1,j]!=pk[0,0]; then go to (6).(5)
pk[0,0] and pk[1,j] form a cycle; it is denoted by pk[1,j]; go to (7).(6)
Search for the element equal topk[1,j] at pk[0,i] which has not been written into the cycle; then judge the element whether it is equal to pk[0,0] or not; proceed the next searching until you get the element equal to pk[0,0] that form a cycle.(7)
During the next searching, delete all thepk[i,j] that have been written into the cycle from the original data structure; go to (1), until all the pk[i,j] have been written into the expression of the product of the cycles.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 Dn; a fast generation algorithm on the cycle decomposition expressions of Dn group based on permutation must be designed.


## 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 Mk, Rk(k=0,1,…,n-1); we estimate the time complexity. Formula (4) is corresponding to the second row of Mk; formula (6) is corresponding to the second row of Rk; for each Rk or Mk, 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 Mk and Rk. There are 2n elements in the Dn group, so the time complexity function T1(n) of the algorithm is
(8)T1(n)=n*(2n)=2n2.After obtaining the expression of permutation of all elements in theDn 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 columnpk[0,0], comparing pk[0,0] with pk[1,0], searching the element which is same to pk[1,0] in the first row if pk[0,0] and pk[1,0] are not equal. Comparing pk[1,j](j=1,…,n-1) with pk[1,0] one by one, at most (n-1) comparisons are made; then comparing pk[1,j] with pk[0,0], searching the element which is same to pk[1,j] in the first row if pk[1,j] and pk[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 T2(n) of the algorithm is
(9)T2(n)=n!.As there are 2n elements in the Dn group, the time complexity function of the generation algorithm based on permutation is
(10)T3(n)=n2+2n*n!=2n(n+n!).
We can observe from (10) that the complexity of the generation algorithm based on permutation is Q1(n*n!) with very low efficiency which is unable to fulfill the requirement in the solution of large size problems using Dn group. So a faster generation algorithm needs to be developed.


## 4. The Derivation of the Common Formula for the Cycle Decomposition Expressions ofDn Group

There are two types ofDn group’s elements: one is derived from reflectivity conversion Mk(k=0,1,…,n-1), and the other from rotation conversion Rk(k=0,1,…,n-1); the cycle decomposition expressions of these two kinds of elements adhere to different rules, so we can research cycle decomposition expressions of Dn group, respectively; then, we can get each common formula to fulfill fast generation algorithm on the cycle decomposition expressions of Dn group.TheDn group is corresponding to a regular n quadrate (as shown in Figure 1). For Dn 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 Dn group, we bring forward the common formula and then prove it by mathematical induction.(1)The Cycle Decomposition Expressions forDnGroup of Low Order. For instance, we enumerate all the cycle decomposition expressions of D6 group. D6 group is corresponding to the Regular 6 quadrate, as shown in Figure 2, where Rk(k=0,1,2,…,5) are the cycle decomposition expressions of the reflectivity conversion, and Mk(k=0,1,2,…,5) are the cycle decomposition expressions of the rotation conversion. The main text paragraph is as follows (see Figure 2):
(11)R0=(0)(1)(2)(3)(4)(5),M0=(0)(3)(15)(24),R1=(012345),M1=(01)(25)(34),R2=(024)(135),M2=(1)(4)(02)(35),R3=(03)(14)(25),M3=(12)(03)(45),R4=(042)(153),M4=(2)(5)(04)(13),R5=(054321),M5=(23)(14)(05).Figure 2
Regular 6 quadrate.(2)The Common Cycle Decomposition Expressions forDnGroup with Reflectivity Conversion. The cycle decomposition expressions forMkare not only related to the parity ofnin theDngroup but also to the parity of k in the elementMk, so the formula can be divided into four instances.(a)n is odd in the Dn group andk is also odd in Mk. Whilekequals 1, take reflectivity conversion thatL1 is the axis (as shown in Figure 1); 0 and 1 compose the transposition, that is,(1,0); 2 andn-1 compose the transposition, that is,(2,n-1), and so forth.There are(n-1)/2 transpositions and a fixed point((1-1)/2+(n-1)/2+1).Whilek is general,
(12)k-12+1andk-12compose the transposition,that is,(k-12+1,k-12);k-12+2andk-12+1compose the transposition,thatis,(k-12+2,k-12+1);….There are(n-1)/2transpositions and a fixed point((1-1)/2+(n-1)/2+1).Now we obtain the following common formula:(13)Mk=(k2)(k2+1,k2-1)(k2+2,k2-2)⋯(k2+n2-1,k2-n2+1)(n2+k2).We can prove that formula (8) is valid fork=1,k=2j+1andk=(2(j=1)+1)=2j+3by mathematical induction. So formula (8) is true for all positive integersk. In the same way we can obtain the following three formulae.(b)n is odd in theDngroup andkis even in Mk:
(14)Mk=(k2)(k2+1,k2-1)(k2+2,k2-2)⋯(k2+n-12,k2-n-12).There are(n-1)/2 transpositions.(c)n is even in the Dn group and k is odd in Mk:
(15)Mk=(k-12+1,k-12)(k-12+2,k-12-1)⋯(k-12+n2,k-12-n2-1).There aren/2 transpositions.(d)n is even in the Dn group and k is even in Mk:
(16)Mk=(k2)(k2+1,k2-1)(k2+2,k2-2)⋯(k2+n2-1,k2-n2+1)(k2+n2).There are(n-2)/2 transpositions and two fixed points.(3)The Common Cycle Decomposition Expressions for the ElementRkin the Group with Rotation Conversion. The type of Rk is (n/d)d,d=(k,n), and the type of Rk varies along with the value of k and n. There are two instances.(a)nis prime. While k equals 0, we deal with it as follows:
(17)R0=(0)(1)⋯(n-1).Whilek=1,2,…,n-1,d=(k,n)=1, the type of Rk is (n/d)d=n1; that is, every element Rk in the group makes up of a cycle, and there are n terms in the cycle, it is denoted by
(18)Rk=(0kk+k⋯k+k+⋯+k).(b)nis composite. When k equals 0, R0 is the same to formula (14).Whilek=1,2,…,n-1, the type of Rk is (n/d)d,d=(k,n), Rk makes up of d cycles, and there are (n/d) terms in each cycle; we can obtain
(19)Rk=(0,k,k+k,…)(1,1+k,1+k+k,…)⋯((d-1),(d-1)+k,(d-1)+k+k,…).


## 5. The Generation Algorithm on the Cycle Decomposition Expressions ofDn 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 ofDn group based on the classification Formulae is relatively simple. For Mk(k=0,1,…,n-1) with reflectivity conversion, first judge the parity of n and k; then substitute them into formulae (13)–(16), thus we can get the cycle decomposition expressions of n elements of the group. For Rk(k=0,1,…,n-1) with rotation conversion, substitute k=0 into formula (17) and get R0; while n is prime, substitute k into formula (18); while n is composite, substitute k into formula (19); thus, we can obtain all the cycle decomposition expressions of Rk(k=0,1,…,n-1).


### 5.2. Algorithm

Consider the following.(1)
Inputn,k=0.(2)
Ifn is even, then go to (6).(3)
Ifk is even, then go to (5).(4)
Substitutek into formula (8); we obtain Mk, k=k+1, k<n; go to (3); else go to (9).(5)
Substitutek into formula (9); we obtain Mk, k=k+1, k<n; go to (3); else go to (9).(6)
Ifk is even, go to (8).(7)
Substitutek into formula (10); we obtain Mk, k=k+1, k<n; go to (6); else go to (9).(8)
Substitutek into formula (13); we obtain Mk, k=k+1, k<n; go to (6); else go to (9).(9)
Substitutek=0 into formula (14); we obtain R0.(10)
Ifn is composite, go to (13).(11)
Substitutek into formula (15).(12)
k=k+1, k<n; go to (11); else go to (15).(13)
Substitutek into formula (16).(14)
k=k+1, k<n; go to (13).(15)
Stop.


### 5.3. The Time Complexity of the Generation Algorithm Based on Classification Formulae

The generation algorithm on the cycle decomposition expressions ofDn group based on the classification Formulae is relatively simple. Judge the parity of n and each k(k=1,2,…,n); substitute k into Formulae (8)~(13); thus get Mk the cycle expression. For Rk, the only thing is to judge the fraction of n. For every Mk or Rk, at most 2n additions are made. As there are 2n elements in the Dn group, so the time complexity function is
(20)T4=2n+n*(2n)=2n2+2n=2(n2+n).As a result, the time complexity of generation algorithm based on permutation isO1(n)=O(n*n!), while the time complexity of generation algorithm based on classification Formulae is O2(n)=O(n2), 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.


## 5.1. The Idea of the Algorithm Designing

The generation algorithm on the cycle decomposition expressions ofDn group based on the classification Formulae is relatively simple. For Mk(k=0,1,…,n-1) with reflectivity conversion, first judge the parity of n and k; then substitute them into formulae (13)–(16), thus we can get the cycle decomposition expressions of n elements of the group. For Rk(k=0,1,…,n-1) with rotation conversion, substitute k=0 into formula (17) and get R0; while n is prime, substitute k into formula (18); while n is composite, substitute k into formula (19); thus, we can obtain all the cycle decomposition expressions of Rk(k=0,1,…,n-1).


## 5.2. Algorithm

Consider the following.(1)
Inputn,k=0.(2)
Ifn is even, then go to (6).(3)
Ifk is even, then go to (5).(4)
Substitutek into formula (8); we obtain Mk, k=k+1, k<n; go to (3); else go to (9).(5)
Substitutek into formula (9); we obtain Mk, k=k+1, k<n; go to (3); else go to (9).(6)
Ifk is even, go to (8).(7)
Substitutek into formula (10); we obtain Mk, k=k+1, k<n; go to (6); else go to (9).(8)
Substitutek into formula (13); we obtain Mk, k=k+1, k<n; go to (6); else go to (9).(9)
Substitutek=0 into formula (14); we obtain R0.(10)
Ifn is composite, go to (13).(11)
Substitutek into formula (15).(12)
k=k+1, k<n; go to (11); else go to (15).(13)
Substitutek into formula (16).(14)
k=k+1, k<n; go to (13).(15)
Stop.


## 5.3. The Time Complexity of the Generation Algorithm Based on Classification Formulae

The generation algorithm on the cycle decomposition expressions ofDn group based on the classification Formulae is relatively simple. Judge the parity of n and each k(k=1,2,…,n); substitute k into Formulae (8)~(13); thus get Mk the cycle expression. For Rk, the only thing is to judge the fraction of n. For every Mk or Rk, at most 2n additions are made. As there are 2n elements in the Dn group, so the time complexity function is
(20)T4=2n+n*(2n)=2n2+2n=2(n2+n).As a result, the time complexity of generation algorithm based on permutation isO1(n)=O(n*n!), while the time complexity of generation algorithm based on classification Formulae is O2(n)=O(n2), 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.


## 6. Conclusions

This paper includes the reflectivity and rotation conversion, which derived six common Formulae on cycle decomposition expressions ofDn group; it designed the generation algorithm on the cycle decomposition expressions of Dn 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.



---

*Source: 176598-2013-02-26.xml*