E.N. Gilbert, Gray codes and paths on the n-cube, Bell System Tech. J. 37(1958) 815--826.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....we mention a more familiar partial order on the same set: the inclusion relation. This de nes a poset on the subsets of [n] called the Boolean lattice B(n) Its cover graph is the n cube. The n cube is Hamiltonian and has Hamiltonian paths satisfying a wide variety of constraints (e.g. [2, 3, 4, 5, 7]) It is easy to obtain a Hamiltonian path in B(n) by induction. The cover graph of M(n) has a somewhat more complicated structure than that of B(n) and fewer edges: n 1)2 instead of n2 . 2 Necessary Conditions Let A n be the cover graph of M(n) For compactness, we will represent the ....

E.N. Gilbert, Gray Codes and paths on the n-cube, Bell System Technical Journal, Vol. 37 (1958) 815-826.

....there is still the possibility that another labeling has a sorted profile that is better in some places and worse in others, such that neither edge profile dominates the other. Section III E explores when the SU edge profile dominates all others. A. SU Labeling Gilbert s taxonomy of Gray codes [9] enumerates a number of different labeling structures that meet the GC condition and classifies them according to structure. In particular, Gilbert identifies composite and ultracomposite Gray codes. Below, we extend Gilbert s definitions of composite and ultracomposite GC difference lists to ....

E. N. Gilbert, "Gray codes and paths on the n-cube," Bell Syst. Tech. J., vol. 37, pp. 815--826, May 1958.

....and a merging method for cyclic Gray codes based on necklaces representatives. Index Terms Cyclic Gray codes, feedback shift register, linear complexity, necklaces, self dual sequences, single track codes. I. INTRODUCTION G RAY codes were found by Gray [15] and introduced by Gilbert [14] as a listing of all the binary tuples in a list such that any two successive tuples in the list differ in exactly one position. Generalization of Gray codes were given during the years. Such generalizations include the arrangements of other combinatorial objects in a such way that any two ....

....in the list differ in exactly one position. Generalization of Gray codes were given during the years. Such generalizations include the arrangements of other combinatorial objects in a such way that any two consecutive elements in the list differ in some prespecified, usually small way [14], 15] Other generalizations include listing subsets of the binary tuples in a Gray code manner, in such a way that the list has some more prespecified properties. These properties were usually forced by a specific application for the Gray code. As an example we have the uniformly balanced Gray ....

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E. N. Gilbert, "Gray codes and paths on the n-cube," Bell Syst. Tech. J., vol. 37, pp. 815--826, 1958.

....that any two adjacent codewords Wi and Wi differ in exactly one component. If this property holds for Wp x and W 0 as well, we say the Gray code is cyclic with period the number of different codewords P. Otherwise, we say the Gray code is acyclic. Constructions for Gray codes can be found in [1] [3] while Gray codes have found application in diverse areas including coding theory [4] 5] and in the design of combinatorial algorithms [6] 7] Another common use of Gray codes is in reducing quantization errors in various types of analog todigital conversion systems [1] 8] As a typical ....

.... can be found in [1] 3] while Gray codes have found application in diverse areas including coding theory [4] 5] and in the design of combinatorial algorithms [6] 7] Another common use of Gray codes is in reducing quantization errors in various types of analog todigital conversion systems [1], 8] As a typical example, a length n, period P Gray code can be used to record the absolute angular positions of a rotating wheel by encoding (e.g. optically) the codewords on n concentrically arranged tracks. n reading heads, mounted in parallel across the tracks suffice to recover the ....

E. Gilbert, "Gray codes and paths on the n-cube," Bell Syst. Tech. J. vol. 37, pp. 815-826, 1958.

....the general class of Gray mappings (or Gray codes) 14] However, the original scheme proposed by Gray is almost ubiquitous in communications. This particular mapping is referred to as the binary reflected Gray code (BRGC) the name stemming from the construction method of the code [15, pp. 226] [16]. The BRGC is what is called a cyclic Gray code, meaning that in the list containing the codewords all adjacent codewords differ in exactly one bit position, including the first and the last codeword in the list. This property is desirable for M PSK, where we want to label the quantization of a ....

E. N. Gilbert, "Gray codes and paths on the n-cube," Bell System Technical Journal, , no. 37, pp. 815--826, May 1958.

....property is desirable for M PSK, where we want to label the quantization of a circle. The BRGC scheme, originally proposed and patented by Frank Gray in 1953 [7] is actually only one special code in a large class of codes having the property that adjacent codewords differ in only one position [8, 9]. For m =1(M =2) the BRGC is simply 0, 1 . The BRGC of order m can be constructed recursively from the BRGC of order m 1 according to the following procedure: i) list the M 2=2 codewords of the BRGC of order m 1 two times rowwise over each other, first in the original order and then in ....

E. N. Gilbert, "Gray codes and paths on the n-cube," Bell System Technical Journal, vol. 37, pp. 815--826, May 1958.

....Gray code, Hamiltonian cycle, n cube 1 Introduction A binary Gray code is a listing of all the n bit binary strings, for a given n, such that only one bit changes. between successive items in the list, including the rst and last. The classical approach, known as the binary re ected Gray code [Gil58, Gra58], starts with the 1 bit Gray code 0; 1 ; for n 1, construct the n bit Gray code by rst appending 0 to each element of the (n 1) bit Gray code, then list the (n 1) bit Gray code in reverse, appending 1 to each element. It is easy to see how this produces a complete listing of all n bit strings, ....

E. N. Gilbert. Gray codes and paths on the n-cube. Bell System Tech. J, 37:815{ 826, 1958.

....the applications into a logical n dimensional hypercube. Each node is identified by a label (e.g. 010 ) which indicates the position of the node in the logical hypercube. In an overlay network with n nodes, the lowest n positions of a hypercube are occupied (according to a Gray ordering [18]) as shown in Figure 1.6. 000 001 100 101 110 111 011 010 Figure 1.6: Hypercube in the HyperCast design. One advantage of using a hypercube is that each node has only n# neighbors, where n is the total number of nodes. Also, the longest route in the hypercube is n#. A disadvantage of ....

E.N. Gilbert. Gray codes and paths on the n-cube. Bell Systems Technical Journal, 815-826(37), 1958.

....2; 1; 4; 1; 2; 1; 3; 1; 2; 1. The cyclic transition sequence for the BRGC is T n ; n. For the BRGC the graph of transitions is K 1;n 1 . The graph K 1;n 1 for n 3 is called a star, denoted S n . It is a tree with one central vertex and n 1 leaves. The following useful theorem is due to Gilbert [2]. Theorem 1.1 The following statements characterize non cyclic and cyclic transition sequences: 1 integers from [n] The sequence T is a transition sequence if and only if every non empty consecutive subsequence of T contains some integer an odd number of times. integers from [n] The ....

E.N. Gilbert, Gray Codes and paths on the n-cube, Bell Systems Technical Journal, 37 (1958) 815-826.

....for i from 1 to n do for all ( n i i x x x x x X , 0 , 1 1 2 1 = do for j from 1 i k downto 0 do ( n n n x x x x j x x x j x x h h h , 1 , 2 1 2 1 2 1 : 0 H V = P 3. 2 A Gray code basis transformation Gray codes (see [7]) i.e. sequences of binary vectors q V V V , 2 1 such that V and 1 i V differ in a single component, have been extensively studied ( 1] 4] 6] 9] 12] We shall discuss in this section an extension of the Koda and Ruskey ( 11] Gray code enumeration method for binary ....

E.N. Gilbert, Gray codes and paths on the n-cube. Bell Syst. Tech. J. 37, 9 (1958), 815-826.

....b i 1 di er. When B is cyclic, its closing transition N is the position where bN and b 1 di er. We say generates B when = B) As transition sequences can be characterized simply and determine codes up to isomorphism, we treat codes and sequences interchangeably. Proposition 1. 1 (Gilbert [3]) Let = 1 ; 2 ; N 1 ) where N = 2 n . 1) generates an n bit Gray code if and only if every contiguous subsequence k ; k 1 ; k l contains some element of [n] an odd number of times. 2) generates a cyclic Gray code if and only if generates a Gray code ....

....all 7 vertex graphs G for which G compatible codes exist, and pose some new questions. 2. Supercomposite Gray Codes 2.1. De nitions. Supercomposite Gray codes are all those that can be built by two simple operations: shifting and re ecting. The term supercomposite is inspired by Gilbert [3], who gave a similar de nition of ultracomposite Gray codes. GRAPHS INDUCED BY GRAY CODES 3 Breaking a cyclic Gray code between any two consecutive codewords yields another Gray code. De ne the k th shift of a cyclic code B to be S k (B) b k 1 ; b k 2 ; bN ; b 1 ; b 2 ; b ....

[Article contains additional citation context not shown here]

E.N. Gilbert, Gray codes and paths on the n-cube, Bell System Tech. J. 37(1958) 815-826.

....we mention a more familiar partial order on the same set: the inclusion relation. This defines a poset on the subsets of [n] called the Boolean lattice B(n) Its cover graph is the n cube. The n cube is Hamiltonian and has Hamiltonian paths satisfying a wide variety of constraints (e.g. [2, 3, 4, 5, 7]) It is easy to obtain a Hamiltonian path in B(n) by induction. The cover graph of M(n) has a somewhat more complicated structure than that of B(n) and fewer edges: n 1)2 n Gamma2 instead of n2 n Gamma1 . 2 2 Necessary Conditions Let A n be the cover graph of M(n) For compactness, we ....

E.N. Gilbert, Gray Codes and paths on the n-cube, Bell System Technical Journal, Vol. 37 (1958) 815-826.

....we mention another partial order on the same set. The inclusion relation defines a more familiar poset on the subsets of [n] called the Boolean lattice B(n) Its cover graph is the n cube. The n cube is Hamiltonian and has Hamiltonian paths satisfying a wide variety of constraints (e.g. [2, 3, 4, 5, 7]) It is easy to obtain a Hamiltonian path in B(n) by induction. The cover graph of M(n) has a somewhat more complicated structure than that of B(n) and fewer edges: n 1)2 n Gamma2 instead of n2 n Gamma1 . 2 2 Necessary Conditions Let A n be the cover graph of M(n) For compactness, we ....

E.N. Gilbert, Gray Codes and paths on the n-cube, Bell System Technical Journal, Vol. 37 (1958) 815-826.

....nodes only when the data array axes have lengths equal to powers of 2. For other axes lengths, adjacency cannot be preserved for a node efficient mapping [3, 4, 10] On the Connection Machine system CM 200, the default mapping of data arrays is based upon a binary reflected Gray code encoding [9, 11, 18] of the index along each axis separately. Only the part of the index corresponding to the node address is encoded in a binary reflected Gray code. Binary encoding is always used for local addresses. A d bit binary reflected Gray code, G d is a sequence of 2 d nonnegative numbers in the ....

E.N. Gilbert. Gray codes and paths on the n-cube. Bell Systems Technical Journal, 37:815--826, 1958.

....of elements are assigned to nodes. On the CM 200, the default assignment is such that a pair of successive indices along any axis are either mapped into the same memory unit, or into the memory units of adjacent nodes. This mapping, known as NEWS order, uses a binary reflected Gray code [6, 15] for the encoding of node addresses. The standard binary encoding is referred to as SEND order. In the default NEWS order, allocation blocks i and i 1 are assigned to adjacent nodes, while in the SEND order allocation block i is assigned to node i. In a SEND ordered assignment, blocks N 2 ....

E.N. Gilbert. Gray codes and paths on the n-cube. Bell Systems Technical Journal, 37:815--826, 1958.

....n) when k = 2, the FKM algorithm approaches this limit from above, whereas the new algorithm approaches it from below. We ask whether it is possible, by any strategy, to generate necklaces in worst case constant time As a variation, is it possible to list necklaces in some Gray code like order ([Gi], Jo] Lu] Ru] Sa] Acknowledgement Thanks to Herb Wilf (for suggesting this problem and for help with the proof of Lemma 5) and Pete Winkler (for helpful discussions on the FKM algorithm. ....

E. N. Gilbert, "Gray codes and paths on the n-cube," Bell Systems Technical Journal (1958) 815-826.

.... memory architectures that are based on the ff ary hypercubes [1] and can be used in various combinatorial applications [2] Gray codes also prove the existence of Hamiltonian paths on the hypercube graphs, and demonstrate the existence of Hamiltonian circuits if the Gray code is cyclic [20]. For any Gray code on an ff ary hypercube of length , there are (ff ) equivalent Gray codes that can be obtained by permuting the characters 0; 1; Delta Delta Delta ; ff in each column, and permuting each column. The Gray code discussed in this section will be represented by G(ff; ....

Gilbert, E.N. (1958). "Gray Codes and Paths on the n-Cube." Bell Sys. Tech. Journal, vol. 37. 72

....When = B) we say that generates the code B. Transition sequences determine Gray codes up to bit complementations. Because transition sequences of codes can be characterized simply, we can (and will) switch between codes and transition sequences when convenient. Proposition 1. 1 (Gilbert [4]) Let = 1 ; 2 ; N Gamma1 ) where N = 2 n . 1) generates an n bit Gray code if and only if every contiguous subsequence k ; k 1 ; k l contains some element of [n] an odd number of times. 1 GRAPHS INDUCED BY GRAY CODES 2 P 4 = 0 B B 0 0 0 0 ; 0 1 0 0 ; 1 1 0 ....

....reveals when supercomposite Gray codes induce trees. We find supercomposite Gray codes of certain trees of arbitrarily large diameter. We also show that many trees, including paths on at least 7 vertices, do not have supercomposite Gray codings. The term supercomposite is inspired by Gilbert [4], who defined ultracomposite Gray codes by a similar inductive procedure. 3.1. Definitions and properties. When a Gray code B = b 1 ; b 2 ; bN ) is cyclic, breaking the cycle between any two consecutive codewords yields a Gray code. More precisely, define S k (B) the k th shift of ....

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Gilbert, E.N. Gray codes and paths on the n-cube, Bell System Technical Journal 37(1958), 815--826.

....smaller than our algorithm. It would be nice to be able to prove this. More important, is there a way to generate all necklaces in total time O(N n k ) ignoring the time for output There are several combinatorial classes for which such a result is possible using Gray code like algorithms ([Gi], Jo] Lu] Ru] Sa] Alternatively, some clever encoding scheme, taking advantage of the structure of the tree, may allow one to reduce or eliminate the O(n) cost of necklace testing in our algorithm. ....

E. N. Gilbert, "Gray codes and paths on the n-cube," Bell Systems Technical Journal (1958) 815-826.

....ffi = 1. 3 1 Introduction Recent work in combinatorial enumeration has considered listing special sets so successive elements differ by a small, pre specified change. Examples include (1) generating permutations by adjacent transpositions [5, 16] 2) generating bit strings by changing one bit [4, 3], 3) generating subsets by changing one element [1, 8, 12] 4) generating binary trees by rotations [7] 5) generating Coxeter group elements by reflection [2] and (6) generating linear extensions of certain posets by transpositions [9, 10, 13, 15, 17] Such enumeration schemes are called ....

E. N. Gilbert, "Gray codes and paths on the n-cube," Bell Systems Technical Journal (1958), 815-826.

....to be possible, even for objects of moderate size, combinatorial generation methods must be extremely efficient. A common approach has been to try to generate the objects as a list in which successive elements differ only in a small way. The classic example is the binary reflected Gray code [Gil58, Gra53] which is a scheme for listing all n bit binary numbers so that successive numbers differ in exactly one bit. The advantage anticipated by such an approach is two fold. First, generation of successive objects might be faster. Although for many combinatorial families, a straightforward ....

....set partitions, and Catalan families. 2 Binary Numbers and Variations A Gray code for binary numbers is a listing of all n bit numbers so that successive numbers (including the first and last) differ in exactly one bit position. The best known example is the binary reflected Gray code [Gil58, Gra53] which can be described as follows. If L n denotes the listing for n bit numbers, then L 1 is the list 0, 1; for n 1, L n is formed by taking the list for L n Gamma1 and pre pending a bit of 0 to every number, then following that list by the reverse of L n Gamma1 with a bit of 1 prepended to ....

E. N. Gilbert. Gray codes and paths on the n-cube. Bell Systems Technical Journal, 37:815--826, 1958.

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E.N. Gilbert, Gray codes and paths on the n-cube, Bell System Tech. J. 37(1958) 815--826.

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E. Gilbert, "Gray codes and paths on the n-Cube." Bell Syst. Tech. J. 37 (1958), 815--826.

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E. N. Gilbert, "Gray codes and paths on the n-cube," Bell System Technical Journal, vol. 37, pp. 815--826, May 1958.

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E.N. Gilbert, "Gray Codes and Paths on the n-Cube", Bell Sys. Tech. J., Vol. 37, May 1958, pp. 815-826.

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