L. Goddyn, G. M. Lawrence, E. Nemeth, Gray Codes with Optimized Run Lengths, Utilitas Mathematica 34 (1988), 179-192.  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 ....

L. Goddyn, G.M. Lawrence and E. Nemeth, Gray codes with optimized run lengths, Utilitas Mathematica, 34 (1988) 179-192.

....and 4 bit antipodal Gray codes. are: restricting where bit ips can occur [BR96] requiring the same number of ips on any given 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 ....

Luis Goddyn, George M. Lawrence, and Evi Nemeth. Gray codes with optimized run lengths. Utilitas Math., 34:179-192, 1988.

....the books of Nijenhuis and Wilf [5] Reingold, Nievergelt, and Deo [6] and Wilf [8] For certain applications, however, other Gray codes are desired. Many other Gray codes have been proposed, both for specific values of n and general constructions. For example, Goddyn, Lawrence, and Nemeth [3], motivated by an issue in the design of photon detectors, study the problem of finding a Gray code that maximizes the minimum number of edges between the use of edges of the same dimension. In a recent paper, Savage and Winkler [7] find a Gray code in which all subsets of size k appear before any ....

L. Goddyn, G.M. Lawrence, and E. Nemeth, Gray Codes with optimized run lengths, Utilitas Mathematica, 34 (1988) 179-192.

....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 ....

L. Goddyn, G.M. Lawrence, and E. Nemeth, Gray codes with optimized run lengths, Util. Math. 34(1988) 179-192.

....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 ....

L. Goddyn, G.M. Lawrence and E. Nemeth, Gray codes with optimized run lengths, Utilitas Mathematica, 34 (1988) 179-192.

....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 ....

L. Goddyn, G.M. Lawrence and E. Nemeth, Gray codes with optimized run lengths, Utilitas Mathematica, 34 (1988) 179-192.

....first asked by Slater [10, 11] and investigated by Bultena and Ruskey [1] is of interest for several reasons. Various types of restricted Gray codes, often motivated by applications, have been studied. Savage [8, Section 2] Conway, Sloane, and Wilks [2] and Goddyn, Lawrence, and Nemeth [5] survey this literature. Bultena and Ruskey [1] point out that Gray codings of certain graphs may be useful: from any cyclic Cn code a particularly nice Hamiltonian cycle of the cube connected cycle graph can be constructed, allowing a simple traversal of the processors of certain parallel ....

Goddyn, Luis, George M. Lawrence, and Evi Nemeth. Gray codes with optimized run lengths. Utilitas Mathematica 34(1988), 179--192.

....of L n also differ in one bit position, the code is in fact a cycle. It can be implemented efficiently as a loop free algorithm [BER76] Note that a binary Gray code can be viewed as a Hamilton cycle in the n cube. In practice, Gray codes with certain additional properties may be desirable (see [GLN88] for a survey) For example, note that as the elements of L n are scanned, the lowest order (rightmost) bit changes 2 n Gamma1 times, whereas the highest order bit changes only twice, counting the return to the first element. In certain applications, it is necessary that the number of bit ....

....the following sense: Let a = b2 n =nc or b2 n =nc Gamma 1, so that a is even. For each n 1 there is a cyclic n bit Gray code in which each bit position changes either a or a 2 times [BS96] In other applications, the requirement is to maximize the gap in a Gray code, which is defined in [GLN88] to be the shortest maximal consecutive sequence of 0 s (or 1 s) among all bit positions. See Figure 1(c) for an example from [GLN88] in which the gap is 4, which is best possible for n = 5. Goddyn and Gvozdjak report a construction in which GAP(n) n goes to 1 as n goes to infinity [GG] Another ....

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L. Goddyn, G. M. Lawrence, and E. Nemeth. Gray codes with optimized run lengths. Utilitas Mathematica, 34:179--192, 1988.

....problem involving digitization of analogue data. Since # Supported in part by NSF grant DMS9302505 1 the electronic journal of combinatorics 3 (1996) #R25 2 then, binary Gray codes have been used in a wide variety of other applications including databases,experimentaldesign,andpuzzlesolving[4,5,6,7,8]. As discussed, for example, in [7] the BRGC scheme, though su#cient to solve the communications problem, is not adequate for certain other applications because of its lack of uniformity . The term uniformity refers to the manner in which the bits change in the Gray code. Several di#erent ....

....data. Since # Supported in part by NSF grant DMS9302505 1 the electronic journal of combinatorics 3 (1996) #R25 2 then, binary Gray codes have been used in a wide variety of other applications including databases,experimentaldesign,andpuzzlesolving[4,5,6,7,8] As discussed, for example, in [7] the BRGC scheme, though su#cient to solve the communications problem, is not adequate for certain other applications because of its lack of uniformity . The term uniformity refers to the manner in which the bits change in the Gray code. Several di#erent measures of uniformity and techniques to ....

[Article contains additional citation context not shown here]

L. Goddyn, G.M. Lawrence and E. Nemeth, Gray Codes with Optimized Run Lengths, Utilitas Mathematica, 34 (1988) 179-192.

....first and last elements of L n also differ in one bit position, the code is in fact a cycle. It can be implemented efficiently in the sense that successive elements can be generated in worst case constant time [2] In practice, Gray codes with certain additional properties may be desirable (see [8] for a survey) For example, note that as the elements of L n are scanned, the lowest order (rightmost) bit changes 2 n Gamma1 times, whereas the highest order bit changes only twice, counting the return to the first element. A balanced Gray code, in which the 2 n bit changes are distributed ....

.... has long been sought and heuristics have been proposed [15, 19] Only in the case where n is a power of two is a balanced Gray code known to exist [18] In other applications, the requirement is to maximize the shortest maximal consecutive sequence of zeroes (or ones) among all bit positions [8]. Here we consider a new constraint. Define the weight of a binary string to be the number of 1 s in the string. For some applications strings are treated differently according to their weights, in which case it may be desirable to generate first strings of weight 0, then weight 1, etc. Clearly ....

L. Goddyn, G. M. Lawrence, and E. Nemeth, Gray codes with optimized run lengths, Utilitas Mathematica 34 (1988), 179-192.

....problem involving digitization of analogue data. Since Supported in part by NSF grant DMS9302505 the electronic journal of combinatorics 3 (1996) #R25 2 then, binary Gray codes have been used in a wide variety of other applications including databases, experimental design, and puzzle solving [4, 5, 6, 7, 8]. As discussed, for example, in [7] the BRGC scheme, though sufficient to solve the communications problem, is not adequate for certain other applications because of its lack of uniformity . The term uniformity refers to the manner in which the bits change in the Gray code. Several different ....

.... Since Supported in part by NSF grant DMS9302505 the electronic journal of combinatorics 3 (1996) #R25 2 then, binary Gray codes have been used in a wide variety of other applications including databases, experimental design, and puzzle solving [4, 5, 6, 7, 8] As discussed, for example, in [7] the BRGC scheme, though sufficient to solve the communications problem, is not adequate for certain other applications because of its lack of uniformity . The term uniformity refers to the manner in which the bits change in the Gray code. Several different measures of uniformity and techniques ....

[Article contains additional citation context not shown here]

L. Goddyn, G.M. Lawrence and E. Nemeth, Gray Codes with Optimized Run Lengths, Utilitas Mathematica, 34 (1988) 179-192.

....the books of Nijenhuis and Wilf [5] Reingold, Nievergelt, and Deo [6] and Wilf [8] For certain applications, however, other Gray codes are desired. Many other Gray codes have been proposed, both for specific values of n and general constructions. For example, Goddyn, Lawrence, and Nemeth [3], motivated by an issue in the design of photon detectors, study the problem of finding a Gray code that maximizes the minimum number of edges between the use of edges of the same dimension. In a recent paper, Savage and Winkler [7] find a Gray code in which all subsets of size k appear before any ....

L. Goddyn, G.M. Lawrence, and E. Nemeth, Gray Codes with optimized run lengths, Utilitas Mathematica, 34 (1988) 179-192.

....the books of Nijenhuis and Wilf [5] Reingold, Nievergelt, and Deo [6] and Wilf [8] For certain applications, however, other Gray codes are desired. Many other Gray codes have been proposed, both for specific values of n and general constructions. For example, Goddyn, Lawrence, and Nemeth [3], motivated by an issue in the design of photon detectors, study the problem of finding a Gray code that maximizes the minimum number of edges between the use of edges of the same dimension. In a recent paper, Savage and Winkler [7] find a Gray code in which all subsets of size k appear before any ....

L. Goddyn, G.M. Lawrence, and E. Nemeth, Gray Codes with optimized run lengths, Utilitas Mathematica, 34 (1988) 179-192.

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L. Goddyn, G. M. Lawrence, E. Nemeth, Gray Codes with Optimized Run Lengths, Utilitas Mathematica 34 (1988), 179-192.

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L. Goddyn, G. M. Lawrence, E. Nemeth, Gray Codes with Optimized Run Lengths, Utilitas Mathematica 34 (1988), 179--192.

....mrl(n) be the maximum possible value of mrl(W ) among all Hamilton circuits W in Q n . We write lg n for log 2 n. In Section 4 we prove our main result. Theorem 1 For n 2 we have mrl(n) bn 2:001 lg nc. It is easy to see that mrl(n) n for n 2, so this settles a conjecture appearing in [2]: lim n 1 mrl(n) n = 1: Constructions in [2] show only that mrl(n) n 2=3 o(1) An earlier construction of Evdokimov [1] proves that mrl(n) n 1=2. Gray codes with large minimum run length are used in electronic devices such as the Codacon [4] spectrograph. The 10 bit code used in the ....

....) among all Hamilton circuits W in Q n . We write lg n for log 2 n. In Section 4 we prove our main result. Theorem 1 For n 2 we have mrl(n) bn 2:001 lg nc. It is easy to see that mrl(n) n for n 2, so this settles a conjecture appearing in [2] lim n 1 mrl(n) n = 1: Constructions in [2] show only that mrl(n) n 2=3 o(1) An earlier construction of Evdokimov [1] proves that mrl(n) n 1=2. Gray codes with large minimum run length are used in electronic devices such as the Codacon [4] spectrograph. The 10 bit code used in the Codacon is presented as a sporadic construction in ....

[Article contains additional citation context not shown here]

L. Goddyn, G. M. Lawrence, E. Nemeth, Gray Codes with Optimized Run Lengths, Utilitas Mathematica 34 (1988), 179-192.

....spread , or the gap of W . Let mrl(n) be the maximum possible value of mrl(W ) over all Hamilton circuits W in Q n . In Section 4 we prove the following. Theorem 1 mrl(n) bn 2:001 log 2 (n)c. It is easy to see that mrl(n) n for n 2, so we have lim n 1 mrl(n) n = 1 as was conjectured in [2]. Constructions provided in that paper show only that mrl(n) n 2=3 o(1) An earlier construction of Evdokimov [1] proves mrl(n) n 1=2. Gray codes with large minimum run length are used in electronic devices such as the Codacon [4] spectrograph. The 10 bit code used in the Codacon is described ....

....In the second case, 1 is separated by the other a 1 permutations which de ne T . Thus mrl(S) a 1. Putting this together with Lemma 3 yields our basic recurrence. Corollary 6 If (a 1) 2 a 2a 6) 2 b , then mrl(a b) 2 minfa 1; mrl(b)g: 5) Starting with lower bounds given in [2], it is easy to write a computer program that applies (5) in an optimal manner for small values of n. The result (Figure 1) suggests that mrl(n) n 2 log 2 n for all n. In the next section we prove a bound which is almost this good. 7 4 Proof of Theorem 1 We show that, for any c 2, if mrl(n) ....

[Article contains additional citation context not shown here]

L. Goddyn, G. M. Lawrence, E. Nemeth, Gray Codes with Optimized Run Lengths, Utilitas Mathematica 34 (1988), 179-192.

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L. Goddyn, G.M. Lawrence, and E. Nemeth, Gray codes with optimized run lengths, Util. Math. 34(1988) 179--192.

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