Savage, C. (1997), `A survey of combinatorial gray codes', SIAM Review 39(4), 605--629.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....circuit in Q n . Frank Gray [3] described an elementary family of re ected Gray codes (RGC) which has seen countless applications. Certain applications in engineering, statistics and computer science require specialized Gray codes with properties not possessed by the RGC. We refer to Savage [6] for more information on such variations. This paper is concerned with Gray codes for which any two edges which have equal transitions are well separated along the circuit. Research supported by the Natural Sciences and Engineering Research Council of Canada Formally, the minimum run length of ....

Carla Savage, A survey of combinatorial Gray codes, SIAM Rev. 39 (1997), 605-629.

....S 0018 9448(99)07312 5. hashing [10] puzzles, such as the Chinese Rings and Tower of Hanoi [13] ordering of documents on shelves [19] signal encoding [20] data compression [22] and circuit testing [23] Finally, for an excellent survey on Gray codes the interested reader is referred to [24]. The classic example of a Gray code is the reflected Gray code [14] 15] This code is a list of the binary tuples in the following way. For the list consists of the words and . Given the list of the binary tuples, we generate the list of the binary tuples by attaching a ZERO as a prefix to ....

C. Savage, "A survey of combinatorial Gray codes," SIAM Rev., vol. 39, pp. 605--629, 1997.

....of size r. 1. Introduction Generation of combinatorial objects such as combinations, permutations, and well formed parenthesis strings, or parenthesis strings for simplicity, is a well studied area, documented in Reingold, Nievergelt and Deo [1] Nijenhuis and Wilf [2] Wilf [3] and Savage[4]. We consider the generation of combinations and parenthesis strings in this paper. Since there are at least an exponential number of those kinds of combinatorial objects, it is not hard to see that the lexicographic order generation of those objects based on a recursive algorithm takes O(f(n) ....

....place at any level of the tree, called the difference point, and the corresponding changing position down the tree, called the solution point. The concept of this paper can be viewed as a refinement of combinatorial Gray code introduced in Joichi, White and Williamson [19] Wilf [3] and Savage [4], which is in turn a generalization of the generation of binary reflected Gray code [20] Based on this approach, we derive a new O(1) time algorithm for generating parenthesis strings in an order different from that in [11] and an in place algorithm for generating combinations of n elements out ....

Savage, C., A survey of combinatorial Gray codes, SIAM Review, 39 (1997) 605- 629.

.... Reflected Gray Mapping A particular mapping of bits onto symbols that has the appealing property that adjacent bit vectors are separated by one single bit was patented in 1953 by Frank Gray [13] A great wealth of work has been done to study 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, ....

Carla Savage, "A survey of combinatorial Gray codes," SIAM Rev., vol. 39, no. 4, pp. 605--629, 1997.

....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 ....

C. Savage, "A survey of combinatorial Gray codes," SIAM Rev., vol. 39, no. 4, pp. 605--629, 1997. TABLE I RECURSIVE CONSTRUCTION OF THE BRGCS OF ORDER m =2, 3, AND 4 FROM THE BRGC OF ORDER m =1.

....of the n cube is prevalent in the literature of coding theory with the work of Frank Gray and the Gray Codes . It is well known that paths in an n cube may be represented with Gray Codes. In fact other combinatorial objects such as permutations can be made to have Gray Code like orderings [13]. Also, the n cube may be viewed as the lattice of the power set of the set f0; 1; 2; n 1g ordered by set inclusion [2] Of course some edges are removed. Since shortest paths in the n cube are the primary focus of this paper it is instructive to de ne a shortest path predicate. s; t; ....

C. Savage. A Survey of Combinatorial Gray Codes. SIAM Review, 39(4):605-629, Dec. 1997.

....5; 5) with the Q(3; 5; 5) case remaining unsettled. In Section 5 we consider the 2 dimensional case, proving that a bent Hamilton cycle exists in Q (n; m) if and only if both n and m are even. Our results fall within the area of combinatorial Gray codes (for an excellent survey, see Savage [5]) In general, it is a simple matter to nd Hamilton paths and cycles in d dimensional grids. There are not many papers that have considered restricted types of Hamilton paths and cycles in grids; one that does is the paper of Trotter and Erd os [6] about the existence of Hamilton cycles in the ....

C.D. Savage, A survey of combinatorial Gray codes, SIAM Review 39 (1997) 605-629.

....of the n cube is prevalent in the literature of coding theory with the work of Frank Gray and the Gray Codes . It is well known that paths in an n cube may be represented with Gray codes. In fact other combinatorial objects such as permutations can be made to have Gray Code like orderings [14]. Also, the n cube may be viewed as the lattice of the power set of the set f0; 1; 2; n Gamma 1g ordered by set inclusion [3] Since shortest paths in the n cube are the primary focus of this paper it is instructive to define a shortest path predicate. Theta(s; t; u) is true if and only ....

C. Savage. A Survey of Combinatorial Gray Codes. SIAM Review, 39(4):605--629, Dec. 1997.

....which Research supported by the Natural Sciences and Engineering Research Council of Canada 1 have been used in countless applications. Certain applications in engineering, statistics and computer science require Gray codes with properties not possessed by the RGC. We refer the reader to [5, 7] for more information on specialized Gray codes and variations. This paper is concerned with binary Gray codes W for which any two edges xy, uv which are nearby along W satisfy xy 6= uv . Equivalently, no bit may change its value twice in quick succession as one traverses W . If W = w 0 w 1 ....

Savage, Carla, A survey of combinatorial Gray codes, SIAM Rev. 39 (1997), 605-629.

....independently arrived at this question, motivated in part by the search for Hamiltonian cycles on the cube connected cycle graph allowing simple traversal of the processors of certain parallel computers. Many types of restricted Gray codes, often motivated by applications, have been studied see [2, 4, 6] for surveys. The current work makes progress in both positive (constructing Gray codes that induce new graphs) and negative ( nding graphs G such that no G compatible code exists) directions. Section 2 introduces supercomposite Gray codes. Bultena and Ruskey [1] conjecture that all trees induced ....

C. Savage, A survey of combinatorial Gray codes, SIAM Rev. 39(1997) 605-629.

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Carla D. Savage. A survey of combinatorial gray codes. SIAM Rev., 39(4):605-629, 1997.

....bit [VS80, RC81] enforcing non locality conditions [Ram90] maximizing the gap, i.e. the shortest maximal consecutive sequence of 0 s (or 1 s) among all bit positions [GLN88] and requiring certain monotonicity properties [SW95] But this is just a small sample. For a survey of Gray codes see [Sav97] and for a beautiful treatment of generating bit strings in general see [Knu] In the Fall of 2000, Hunter Snevily posed a question about what he called antipodal Gray codes [Sne] An n bit antipodal Gray code must satisfy the requirements for an ordinary n bit Gray code, but in addition it has ....

Carla Savage. A survey of combinatorial Gray codes. SIAM Rev., 39(4):605-629, 1997.

....of a single generator or its inverse to its immediate predecessor This problem seems very difficult. For surveys of the general topic of Gray codes, for many more examples of such codes in a variety of combinatorial families, and for pointers to recent literature in the subject we suggest [1, 3, 6, 8]. 1.2. About this paper. In this paper, we study a numerical obstruction to being able to list in Gray code order a collection of subsets of f1; ng. We show that this obstruction grows exponentially in n for the collection of g blockfree subsets of f1; ng if and only if g 2. ....

C. D. Savage, A survey of combinatorial Gray codes, SIAM Review, 39, No. 4, to appear Dec. 1997.

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Savage, C. (1997), `A survey of combinatorial gray codes', SIAM Review 39(4), 605--629.

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C. D. Savage, A survey of combinatorial Gray codes, SIAM Review 39(4) (1997), pp. 605-629.

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C.D. Savage, A survey of combinatorial Gray codes, SIAM Review 39 (1997) 605--629.

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C.D. Savage, A survey of combinatorial Gray codes, SIAM Review 39 (1997) 605-629.

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Carla Savage, A survey of combinatorial Gray codes, SIAM Rev. 39 (1997), 605-629.

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C. Savage, A survey of combinatorial Gray codes, SIAM Rev. 39(1997) 605--629.

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C.D. Savage, A survey of combinatorial Gray codes, SIAM Review 39 (1997) 605-629.

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C.D. Savage, A survey of combinatorial Gray codes, SIAM Review 39 (1997) 605--629.

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Carla Savage, A survey of combinatorial Gray codes, SIAM Rev. 39 (1997), 605--629.

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Savage, C. D. (1997) A survey of combinatorial Gray codes. SIAM Rev., 39, 605--629.

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C.D. Savage, A survey of combinatorial Gray codes, SIAM Review 39 (1997) 605--629.

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Savage, Carla. A survey of combinatorial Gray codes. SIAM Review 39(1997), 605--629.

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