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