T. Wang and C. Savage, "A new algorithm for generating necklaces," Report TR-90-20, Department of Computer Science, North Carolina State University (1990). 20 Figure Captions:  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Generating Necklaces In this section, we describe a new algorithm for generating necklaces in which the number of strings examined is never more that twice the number of necklaces. For simplicity, we focus on the case for two colors, but the generalization to k colors is described in detail in [WaSa]. Our idea for generating necklaces of two color beads was inspired by a result in [LiHiCa] that a certain variation on the shuffle exchange graph is hamiltonian. The graph of [LiHiCa] is actually the deBruijn graph, which is known to be hamiltonian. When k = 2, the n tuples are bit strings ....

T. Wang and C. Savage, "A new algorithm for generating necklaces," Report TR-90-20, Department of Computer Science, North Carolina State University (1990). 20 Figure Captions:

....necklaces for (n; k) 5; 2) 7; 2) and (3; 3) Wilf asked if it is possible to generate necklaces efficiently, possibly in constant time per necklace. A proposed solution, the FKM algorithm of Fredricksen, Kessler, and Maiorana, had no proven upper bound better than O(nk n ) FK86, FM78] In [WS90] a new algorithm was presented with time complexity O(nN n k ) where N n k is the number of n bead necklaces in k colors. Subsequently, a tight analysis of the original FKM algorithm showed that it could, in fact, be implemented to run in time O(N n k ) giving an optimal solution [RSW92] ....

....one We know of no counterexamples. To construct a slightly different set of objects, call two k ary strings equivalent if one is a rotation or a reversal of the other. The equivalence classes under this relation are called bracelets. Lisonek [Lis93] shows how to modify the necklace algorithm of [WS90] to generate bracelets. We know of no Gray code for bracelets and it is open whether it is possible to generate bracelets in constant amortized time. When beads have distinct colors, bracelets are the rosary permutations of [Har71, Rea72] Define a new relation R on n bead binary necklaces by xRy ....

T. M. Y. Wang and C. D. Savage. A new algorithm for generating necklaces. In Proceedings, Twenty-Eighth Annual Allerton Conference on Communication, Control, and Computing, pages 72--81, 1990.

....smallest representatives of the n bit necklaces. That is, L(n) fx 2 Sigma n j x oe i (x) for 1 i ng: Let L(n; d) be the subset of L(n) of strings of density d. As the backbone of our Gray code construction, we will use a tree of elements of L(n) which was introduced in [WaSa]. For n 1, let : Sigma n Sigma n be the function (x 1 x 2 : x n ) x 1 x 2 : x n : Then the tree, TREE(n) is defined recursively by (i) 0 n is the root of TREE(n) and (ii) if x is a node of TREE(n) then for 1 i n, oe i (x) is a child of x if and only if oe i (x) 2 ....

....k ff10 i 1, where k; i 0. But then y = oe i 1 (0 k i 1 ff1) and x = 0 k i 1 ff1 2 L(n; d Gamma 1) By induction, x is in TREE(n) at level d Gamma 1 and therefore, by definition of TREE(n) y is a child of x at level d. The following result, crucial to our construction, was proved in [WaSa]. Theorem 1 For node x = 0 k 1ff in TREE(n) with k 0 and ff 2 Sigma , and for i satisfying 1 i k, if oe i (x) 62 L(n) then oe i 1 62 L(n) 2 As a consequence of Theorem 1, if a node x = 0 k 1ff in TREE(n) has exactly c 0 children, then those children are oe 1 (x) oe 2 (x) ....

T. M. Wang and C. D. Savage, "A new algorithm for generating necklaces," Proceedings, Twenty-eighth Annual Allerton Conference on Communication, Control, and Computing, Allerton, October 1990.

....Generating Necklaces In this section, we describe a new algorithm for generating necklaces in which the number of strings examined is never more that twice the number of necklaces. For simplicity, we focus on the case for two colors, but the generalization to k colors is described in detail in [WaSa]. Our idea for generating necklaces of two color beads was inspired by a result in [LiHiCa] that a certain variation on the shuffle exchange graph is hamiltonian. The graph of [LiHiCa] is actually the deBruijn graph, which is known to be hamiltonian. When k = 2, the n tuples are bit strings ....

T. Wang and C. Savage, "A new algorithm for generating necklaces," Report TR-90-20, Department of Computer Science, North Carolina State University (1990).

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