Chung, F., Diaconis, P., Graham, R.: Universal cycles for combinatorial structures. Discrete Math. 110 (1992) 43--59  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that for certain well known practical applications of de Bruijn sequences, including their use for position location (see, for example, 3] 5] the decoding problem is an important one. Over and above its practical significance, the decoding problem has been listed by Chung, Diaconis, and Graham [6] as one of the fundamental questions for the study of de Bruijn sequences. Previous work on the decoding problem can be summarized as follows. The obvious approach is the brute force method of storing a lookup table of the positions in the sequence of all possible v tuples. Alternatively, ....

F. Chung, P. Diaconis, and R. Graham, "Universal cycles for combinatorial structures," Discr. Math., vol. 110, pp. 43-59, 1992.

....mentioned in [3] and proved in [10] Theorem 1:1 0 . For all n; d and k (except n 2 = 2 if k is even) there is an ( R; n) k de Bruijn d torus with r 1 = k n1 and r j = Q j 1 i=1 r i ) n j 1 = k (n j 1) Q j 1 i=1 n i for j 1. This prompted Chung, Diaconis, and Graham [2] to ask whether it could be true that, in two dimensions, R = S and m = n (the so called square tori) The binary case was resolved with the following theorem of Fan, Fan, Ma, and Sin [5] Theorem 1.2. There is a (2 n 2 =2 ; 2 n 2 =2 ; n; n) 2 de Bruijn torus if and only if n is even. ....

F. R. K. Chung, P. Diaconis, and R.L. Graham, Universal cycles for combinatorial structures, Discrete Math. 110 (1992), 43-59.

....uniquely. This result went largely unnoticed until the sequences were rediscovered in 1946 by de Bruijn [B] and Good [G] and they have come to be known as de Bruijn Cycles. For interesting results, generalizations, applications, and history see [F] H] Recently, Chung, Diaconis, and Graham [C] have generalized such sequences so as to list combinatorial families other than k ary n tuples including permutations of [n] partitions of [n] k subsets of [n] and k dimensional subspaces of an n dimensional vector space over a nite eld. Such sequences they call Universal Cycles, or ....

....integer in the Ucycle. Notice that (NC) holds whenever n is relatively prime to k. Indeed, n k = n n 1 k 1 =k is an integer and n and k have no common factors, so n 1 k 1 =k must be an integer. Also notice that (NC) fails whenever n is a multiple of k. 2. Results. In [C] Chung, Diaconis, and Graham made the following conjecture, for which 100 is o ered. Conjecture 1. For all k there is an integer n 0 (k) such that, for n n 0 (k) Ucycles for k subsets of [n] exist if and only if (NC) holds. In [J1] we nd the following results. Theorem 2. Ucycles for ....

F.R.K. Chung, P. Diaconis, R.I. Graham, Universal Cycles for Combinatorial Structures, Discrete Math. 110 (1992), 43-59.

....[2] and [9] see Frederickson [8] for a survey of De Bruijn cycles) 2 dimensional De Bruijn tori are examined in [1,4 6,11 18] among others. A 2 dimensional De Bruijn torus (r 1 ; r 2 ; n 1 ; n 2 ; 1) 2 k is square if r 1 = r 2 = r and totally square if n 1 = n 2 = n as well. It was asked in [3] whether such totally square tori exist. Except for small values of n j , it has been shown (see [6] for k = 2 and [11] for general k) that the obvious necessary conditions r n and r 2 = k n 2 are also sucient for their existence. We have conjectured [11] as others have) that for general ....

F.R.K. Chung, P. Diaconis, and R.L. Graham, Universal cycles for combinatorial structures, Discrete Math. 110 (1992), 43-59.

....suggested over thirty years ago and again various applications have been suggested. In the 1980 s initial ground breaking work on these objects appeared in the work of Fan,Fan, Ma and Sui [29] Cock [20] Etzion [27] and others. In [5] we answered a question posed by Chung, Diaconis and Graham [19] by constructing 2 square two dimensional de Bruijn tori in nearly all (except n = 3; 5; 7; 9) cases in which such are possible and conjectured more general existence results. DeBruijn tori have also been called perfect maps and attempts to construct them produce the need for factored and ....

....of new classes of higher dimensional perfect multifactors. I expect that this general framework will set the stage for even more results regarding sufficiency in higher dimensions. Perfect factors fit into the general scheme of universal cycles, introduced by Chung, Diaconis and Graham [19] for cyclic representations of combinatorial objects. In [13] we construct a compact type of universal cycle for permutations, answering a question raised by Chung, Diaconis and Graham for such compact representations, although using a different tack than they suggested. Inconsistencies Under ....

F.R.K. Chung, P. Diaconis, and R.L. Graham, Universal Cycles for Combinatorial Structures, Discrete Math. 110 (1992), 43--59.

....Bruijn torus. If k r = m, say, then the all 0 s matrix is found m times. The suciency of these relations seems to be a rather tricky problem, and we conjecture that, except possibly for very small values of m and n, the conditions r s = mn, k r m and k s n are sucient for all k. In [2], Chung, Diaconis, and Graham ask whether it is possible that square tori exist for even n. That is, can it be that r = s and m = n This question was resolved for the binary case by Fan, Fan, Ma, and Siu [4] who proved Theorem 1.2. There exists a (2 r ; 2 r ; n; n) 2 de Bruijn torus if ....

....case by Fan, Fan, Ma, and Siu [4] who proved Theorem 1.2. There exists a (2 r ; 2 r ; n; n) 2 de Bruijn torus if and only if n is even. Again, one should notice that r = n 2 =2, so either n is even or k is a perfect square. Our purpose here is to prove a bit more than was conjectured in [2]. Theorem 1.3. a) For k odd there is a (k r ; k r ; n; n) k de Bruijn torus if and only if n is even or k is a perfect square, and b) For k even and n 10, there is a (k r ; k r ; n; n) k de Bruijn torus if and only if n is even or k is a perfect square. 2 For part b) we will ....

F.R.K. Chung, P. Diaconis and R.L. Graham, Universal Cycles for Combinatorial Structures, to appear in Discrete Math.

....100. It is well known that k ary de Bruijn Cycles of order n exist for all k and n, and there are many interesting algorithms which are used to construct them. It is also known exactly how many there are (see [1] 3] 4] 5] 6] A generalization introduced by Chung, Diaconis, and Graham [2] is the Universal Cycle, or Ucycle, for k subsets of [n] f1; 2; ng. It is a cyclic sequence of n k integers with the property that every k subset of [n] appears exactly once as a contiguous subsequence. The ambitious reader may verify that the following sequence is an example ....

....condition. Fact 1. If there is a t cover Ucycle for k subsets of [n] then n divides t n k . Proof of Fact 1. A t cover Ucycle has length t n k and each integer occurs equally often. For t = 1 this necessary condition is conjectured to be sucient for n large enough in [2]. We believe the same should be true for general t. Theorem 2. If k = 2; 3; 4, or 6, gcd(n; k) 1, and n n 0 (k) then U(n; k) 1. The proof is to be found in [7] see also [8] with values of n 0 (2) 5, n 0 (3) 8, n 0 (4) 9, and n 0 (6) 17. In this paper we show that U(n; k) is ....

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F.R.K. Chung, P. Diaconis, R.I. Graham, Universal Cycles for Combinatorial Structures, Discrete Math., to appear.

.... in computer architecture, where the de Bruijn graph is recognized as a bounded degree derivative of the shuffle exchange network [ABR90] Chung, Diaconis, and Graham generalized the notion of a de Bruijn sequence for binary numbers to universal cycles for other families of combinatorial objects [CDG92]. Universal cycles for combinations were studied by Hurlbert in [Hur90] and some interesting problems remain open. A universal cycle of order n for permutations is a circular sequence x 0 : xN Gamma1 of length N = n of symbols from f1; Mg in which every permutation 1 : n of 1 : ....

....Mg in which every permutation 1 : n of 1 : n is order isomorphic to some contiguous subsequence x t 1 : x t n . Order isomorphic means that for 1 i; j n, i j iff x t i x t j . As an example, the sequence 123415342154213541352435 is a universal cycle of order 4 with M = 5. In [CDG92], the goal is to choose M as small as possible to guarantee the existence of a universal cycle. It is clear that M must satisfy M n 1 for n 2. It is conjectured in [CDG92] that M = n 1 for all n 2, although the best upper bound they were able to obtain was M n 6. Even for n = 5 it is ....

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F. Chung, P. Diaconis, and R. Graham. Universal cycles for combinatorial structures. Discrete Mathematics, 110:43--60, 1992.

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Chung, F., Diaconis, P., Graham, R.: Universal cycles for combinatorial structures. Discrete Math. 110 (1992) 43--59

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F. Chung, P. Diaconis, R. Graham, Universal cycles for combinatorial structures, Discrete Math. 110 (1992), 43-59.

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F. Chung, P. Diaconis, and R. Graham, "Universal cycles for combinatorial structures," Discr. Math., vol. 110, pp. 43--60, 1992.

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F. Chung, P. Diaconis and R. Graham, Universal cycles for combinatorial structures, Discrete Math. 110 (1992) 43--55.

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