R. Kaye, "A Gray code for set partitions," Information Processing Letters 5, No. 6 (1976) 171-173.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....family may be magnified by the bijection. For example, the partitions 1 = f1; 2; 5g; f3; 6g; f4g and 2 = f1; 3; 6g; f2; 5g; f4g differ only in that element 1 changes sets. However, the RG functions associated with 1 and 2 are 0 0 1 2 0 1 and 0 1 0 2 1 0, which differ in several positions. In [4], Kaye describes a Gray code L(n) for S(n) attributed to Knuth, where between successive partitions, only one element moves and that move is to an adjacent block. The listing is recursive with L(1) f1g; the list L(n) is obtained from L(n Gamma 1) by replacing every = B 1 ; B b( on ....

....be large [7] In his 1973 paper, Ehrlich [1] gave a loop free implementation of his Gray code algorithm for R(n) and a loop free algorithm for generating S b (n) results which have been overlooked by some later papers. Kaye s 1976 paper contains a CAT implementation of Knuth s Gray code for S(n) [4]. The solutions manual by Fill and Reingold [2] for the book by Reingold, Nievergelt, and Deo [5] also presents a Gray Code for S b (n) which they attibute to Brian Hansche. Ruskey gives a CAT implementation of his Gray code for S b (n) 7] We conjecture that our new Gray codes for RG tails, T ....

R. Kaye, "A Gray code for set partitions," Information Processing Letters 5, No. 6 (1976) 171-173.

.... and the complexities of the solutions to other problems have been improved [Gar, ChLeDu, ChChCh, Los, Ric] There are many examples of combinatorial families for which Gray codes are known, including permutations [Joh, Tro] combinations [BuWi, NiWi, Rus1] compositions [Kli] set partitions [Kay], integer partitions [Sav, RaSaWe] binary trees [RuPr, Luc, LuRoRu] and linear extensions [PrRu1, PrRu2, Rus2, Sta, Wes] When an application requires an exhaustive examination of all objects in a combinatorial family, Gray codes can be used to speed up the task. With a Gray code scheme, it is ....

R. Kaye, "A Gray code for set partitions," Information Processing Letters 5 (1976) 171-173.

.... 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 three non trivial actions are considered in isolation, the composition of more than two of the actions gives rise to equivalence classes for which no efficient generation algorithms were previously known. For example, composing (b) and (c) results in ....

....algorithm generating all canonical strings of length n. The algorithm, which we call gen, is given in Figure 2. A similar algorithm for generating all set partitions (i.e. when the maximum block size k is n) was given by Er [1] and other iterative algorithms are given by Hutchinson [7] Kaye [8], Semba [16] and Stanton and White [17] The call gen(l,m) generates all sequences s[l] s[n] such that 0 s[i] min(k 1; 1 maxfm; s[l] s[i 1]g) for i = l; l 1; n. Here, parameter m is procedure gen( l, m : integer ) var i : integer; begin if l n ( and ....

R.A. Kaye, "A Gray code for set partitions", Information Processing Letters, 5 (1976) 171-173.

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R. Kaye, A Gray code for set partitions, Info. Proc. Lett., 5 (1976), 171-173.

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