H. Fredricksen and J. Maiorana, "Necklaces of beads in k colors and k-ary de Bruijn sequences," Discr. Math., vol. 23, pp. 207--210, 1978.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....binary complement of . Two length words are said to be equivalent if there exists an integer such that , where is consecutive applications of . The equivalence classes under the shift operator are called necklaces. Efficient algorithms for producing necklaces of a given length are given in [11] [12], and [21] A length word is called self dual if for each , Finally, for any two positive integers and , denotes the greatest common divisor of and . Definition 1: Let be a length word. We define the cyclic order of as and the complementary cyclic order of as If we say that has full cyclic ....

H. Fredricksen and J. Maiorana, "Necklaces of beads in k colors and k-ary de Bruijn sequences," Discr. Math., vol. 23, pp. 207--210, 1978.

....every rotation of a given k ary n tuple represents the same phenomenon, it would be more efficient to work with only one representative of each necklace, rather than all k n of the n tuples. A necklace is thus identified with its representative. A simple and elegant algorithm was proposed in [FrMa] and [FrKe] to generate for each necklace the lexicographically smallest element. We will refer to this as the FKM algorithm. A disadvantage of the FKM algorithm is that there can be gaps in which as many as b(n Gamma 1) 2c non necklaces are examined between any two necklaces generated. For the ....

....successor of ff, succ(ff) is obtained from ff as follows. Definition 2 For ff (k Gamma 1) n , succ(ff) ff 1 : ff i Gamma1 (ff i 1) t ff 1 : ff j , where i is the largest integer 1 i n such that ff i k Gamma 1 and t; j are such that ti j = n and j i. It is shown in [FrMa] that no necklace can lie strictly between two elements of F(k; n) so that all necklaces appear on F(k; n) Thus, discarding non necklaces of F(k; n) would result in a list of all necklaces in increasing order. Figure 1 shows examples of lists F(k; n) where non necklaces are indicated by ....

H. Fredricksen and J. Maiorana, "Necklaces of beads in k colors and k-ary de Bruijn sequences," Discrete Mathematics 23, No. 3 (1978) 207-210.

....bit positions. We will show that such a Gray code is always possible and that it gives rise to the most efficient algorithm known for generating necklaces of fixed density. A simple and elegant algorithm for listing the lexicographically smallest representatives of all n bit necklaces was given in [FrMa, FrKe] and we refer to this as the FKM algorithm. It was shown in [RuSaWa] that the time required by the FKM algorithm is O(N(n) that is, constant average time per necklace, which is best possible. The efficiency here is achieved by amortization, rather than a Gray code, since successive ....

H. Fredricksen and J. Maiorana, "Necklaces of beads in k colors and k-ary de Bruijn sequences," Discrete Mathematics 23, 3 (1978) 207-210.

....base k which is known to be efficient for k 2. For (b) efficient algorithms were developed by Ruskey [15] In case (c) the equivalence classes are usually called necklaces. Efficient algorithms for generating necklaces were developed by Fredericksen and Kessler [2] and Fredericksen and Maiorana [3]; these algorithms were proven to be efficient by Ruskey, Savage, and Wang [14] In case (d) the representative strings are usually called restricted growth functions and efficient algorithms for generating them have been developed by Er [1] Kaye [8] and others. In contrast to the case where our ....

H. Fredricksen and I.J. Maiorana, "Necklaces of beads in k colors and k-ary de Bruijn sequences", Discrete Mathematics, 23 (1978) 207-210.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC