J.-P. Duval, Generation d'une section des classes de conjugaison et arbre des mots de Z. Lyndon de longueur bornee, Theoret. Comput. Sci. 60 1988 , 255#283. Z. 4. M. C. Er, A fast algorithm for generating set partitions, Comput. J. 31 1988 , 283#284.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the discussion. We also take some time to discuss Lazard s elimination process (section 3) and the combinatorics intrinsic to the Hopf algebra structure of the free associative algebra (section 4) which forms the envelopping algebra of the free Lie algebra) We conclude by discussing J. P. Duval [17, 18] beautiful algorithms on Lyndon words (section 5) 2 Rewriting sequence of generators Let us first set some notations and terminology, taken from [19] and [13] We shall always use an underlying alphabet A, that is a set of letters (or generators) a 2 A. We shall call a sequence (a 1 , a ....

....system work, whether in the Lyndon case, only the order on is needed. The low complexity of Duval s algorithm comes as a consequence of the order upon which the Lyndon words are built. Indeed, any comparison between two letters is counted as an elementary operation. Another algorithm by Duval [18], solving a problem posed by Schutzenberger, generates Lyndon words of bounded length, again in linear time complexity. 10 ....

J. P. DUVAL. G'en'eration d'une section des classes de conjugaison et arbre de mots de Lyndon de longueur born'ee. Theoretical Computer Science, 60:255 -- 283, 1988.

....first developed by Fredricksen and Kessler [4] and Fredricksen and Maiorana [5] although they did not prove that they were e#cient. They were proven to be e#cient by Ruskey, Savage, and Wang [8] Closely related algorithms for generating Lyndon words (aperiodic necklaces) were developed by Duval [3] and shown to be e#cient by Berstel and Pocchiola [1] Subsequently, a recursive algorithm was developed that was more flexible and easier to analyze than the earlier algorithms, which were all iterative [2] In many applications not all necklaces are required, but rather only those of fixed ....

....another array b to indicate the values of the nonzero characters. The ith element of the array a represents the position of the ith nonzero character, and the ith element of the array b represents the value of the ith nonzero character. Thus if we generate a necklace with length 7 with a = [3, 4, 5, 7] and b = 1, 3, 2, 1] the corresponding necklace is 0013201. We can also maintain the original necklace structure by performing some extra constant time operations. Note that in the binary case, the second array b is not necessary since all nonzero characters must be 1. We use the parameter t ....

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J.-P. Duval, Generation d'une section des classes de conjugaison et arbre des mots de Lyndon de longueur bornee, Theoret. Comput. Sci., 60 (1988), pp. 255--283.

.... algorithms for generating necklaces were developed by Fredricksen and Kessler [5] and Fredricksen and Maiorana [6] these algorithms were proven to be efficient by Ruskey, Savage, and Wang [20] Closely related algorithms for generating Lyndon words (aperiodic necklaces) were developed by Duval [3] and shown to be efficient by Berstel and Pocchiola [2] In case (d) the representative strings are usually called restricted growth functions and efficient algorithms for generating them have been developed by Er [4] Kaye [12] and others. In contrast to the case where our three non trivial ....

J-P. Duval, G'en'eration d'une section des classes de conjugaison et arbre des mots de Lyndon de longueur born'ee, Theoretical Computer Science, 60 (1988) 255-283.

....developed by Fredricksen and Kessler [4] and Fredricksen and Maiorana [5] although they did not prove that they were efficient. They were proven to be efficient by Ruskey, Savage, and Wang [7] Closely related algorithms for generating Lyndon words (aperiodic necklaces) were developed by Duval [3] and shown to be efficient by Berstel and Pocchiola [1] Subsequently, a recursive algorithm was developed that was more flexible and easier to analyze than the earlier algorithms, which were all iterative [2] In many applications not all necklaces are required, but rather only those of fixed ....

....another array b to indicate the values of the non zero characters. The ith element of the array a represents the position of the ith non zero character, and the ith element of the array b represents the value of the ith non zero character. Thus if we generate a necklace with length 7 with a = [3,4,5,7] and b = 1,3,2,1] the corresponding necklace is 0013201. We can also maintain the original necklace structure by performing some extra constant time operations. Note that in the binary case, the second array b is not necessary since all non zero characters must be 1. We use the parameter t to ....

[Article contains additional citation context not shown here]

J-P. Duval, G'en'eration d'une section des classes de conjugaison et arbre des mots de Lyndon de longueur born'ee, Theoretical Computer Science, 60 (1988) 255-283.

....is proved to be asymptotically equal to (q 1) q Gamma 1) where q is the size of the underlying alphabet. In particular, the average cost is independent of the length of the words generated. A precise evaluation of the constants is also given. 1 Introduction Several years ago, J. P. Duval [4] has presented an amazingly simple algorithm for generating all Lyndon words up to a given length in lexicographic order. He observed that the worst case behavior of his algorithm, for computing the next Lyndon word is linear, and he left as an open problem to determine the average case running ....

....3.0417 3.0449 3 2.18 2.27 2.09 2.10 4 1.77 1.79 1.716 1.719 10 1.248 1.249 1.2354 1.2356 This shows that our bound is rather good. 5 Conclusion We have shown that the computation of the next Lyndon word in the set of Lyndon words up to some fixed length requires constant time. In the same paper [4], Duval has presented another algorithm that generates all Lyndon words of fixed length in lexicographic order. It is an easy consequence of our result that the average cost of this second algorithm is asymptotically bounded by (q 1) q. However, we were unable to give a sharp asymptotic ....

J.-P. Duval. G'en'eration d'une section des classes de conjugaison et arbre des mots de Lyndon de longueur born'ee. Theoret. Comput. Sci., 60:255-- 283, 1988.

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J.-P. Duval, Generation d'une section des classes de conjugaison et arbre des mots de Z. Lyndon de longueur bornee, Theoret. Comput. Sci. 60 1988 , 255#283. Z. 4. M. C. Er, A fast algorithm for generating set partitions, Comput. J. 31 1988 , 283#284.

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J.-P. Duval, Generation d'une section des classes de conjugaison et arbre des mots de Lyndon de longueur bornee, Theoret. Comput. Sci., 60 (1988), pp. 255--283.

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