Bitner, J. R., Ehrlich, G. and Reingold, E. M. (1976) Efficient generation of the binary reflected Gray code and its applications. Commun. ACM, 19, 517--521.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....n. That is, it is easy to generate those objects in O(1) time per object on average. A less trivial problem is whether we can generate those objects in O(1) time per object in the worst case. There are such algorithms with O(1) worst case time. To name just a few, Bitner, Ehrlich and Reingold [5], Ehrlich [6] and Lehmer [7] for combinations, Johnson [8] and Heap [9] for permutations, Korsh and Lipschutz [10] for multiset permutations, and Mikawa and Takaoka [11] for parenthesis strings. Johnson s and Heap s algorithms for permutations take O(n) time from object to object, but it is ....

....in Nijenhuis and Wilf [11] with the revolving door function which requires O(n) time in the worst case. Eades and McKay also considered an in place algorithm, which generate combinations with O(1) changes but requires O(n) time. O(1) worst case time is known for combination generation in [5], 6] 7] and so forth, at the cost of O(r) space of a binary vector of size r to represent a set of n elements out of r elements. 2. Constant Change Generation Let S = s 0 , s r 1 be an alphabet for combinatorial objects. A combinatorial object is a string a 1 . a n of length n ....

Bitner, J.R., G. Ehrlich and E.M. Reingold, "Efficient Generation of Binary Reflected Gray Code and its Applications," CACM 19 (1976) pp. 517-521.

.... in Ko [6] 10 4 Concluding Remarks When all n i equal 1, Versions A and B produce the same list of #n permutations, and this list appears to be different than any of those produced by the permutation generation algorithms surveyed in Sedgewick [12] or Lipski [7] The proof technique used in [1], 9] to show the interchange property is different than that used here. Their proof is inductive and is based on the starting permutation being 0 and the ending permutation being 10 n1 Gamma1 . For t 1 the ending permutations are not so easy to specify. For example, 303131122233 and ....

J.R. Bitner, G. Ehrlich, and E.M. Reingold. Efficient generation of the binary reflected gray code and its applications. Comm. ACM, 19:517--521, 1976.

....1000 1010 1011 1001 0001 0011 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Figure 5: Adjacency in 4 bit Hamming space for Gray and standard Binary encodings. The Binary representation destroys half of the connectivity of the original function. code [6]; but there are exponentially many possible Gray codes. A Gray code is a bit encoding where adjacent integers are also Hamming distance 1 neighbors in Hamming space. Over all possible discrete functions that can be mapped onto bit strings, the space of all Gray codes and the space of all Binary ....

James R. Bitner, Gideon Ehrlich, and Edward M. Reingold. Efficient Generation of the Binary Reflected Gray Code and Its Applications. Communications of the ACM, 19(9):517--521, 1976.

....6.1, we only need to check whether a comparator network sorts all 2 n zeroone inputs. Suppose we check them in binary reflected Gray code order. This ordering of the inputs has the useful property that each input differs from the previous one in exactly one bit. Bitner, Ehrlich and Reingold [4] provide algorithms for generating the n bit binary reflected Gray code in time O(2 n ) that is, linear in the number of inputs generated) We will assume the existence of three sub algorithms; procedure firstinput which initializes the appropriate global variables to represent the all zero ....

.... of n=2 ternary digits (the ternary digit 0 represents the binary string 00, the ternary digit 1 represents the binary string 01 and the ternary digit 2 represents the binary string 11) in ternary reflected Gray code order in linear time by modifying the algorithms of Bitner, Ehrlich and Reingold [4]. Thus the time to check each first normal form comparator network is O(3 n=2 d) 6.2 The Last Level The last level of a sorting network can be constructed on the fly during the testing process if we use the algorithms of Section 6.1. We test each comparator network of depth d Gamma 1 to see ....

J. R. Bitner, G. Ehrlich, and E. M. Reingold. Efficient generation of the binary reflected gray code and its applications. Commun. ACM, 19(9):517--521, September 1976.

....or decreasing the number of local optima 3 Gray and Binary Representations Another much debated issue in the evolutionary algorithms community is the relative merit of Gray code versus Standard Binary code bit representations. Generally, Gray code refers to Standard Binary Reflected Gray code [2]. In general, a Gray code is any bit encoding where adjacent integers are also Hamming distance 1 neighbors in the bit space. There are exponentially many Gray codes with respect to bit length L. Over all possible discrete functions that can be mapped onto bit strings, the space of all Gray codes ....

James R. Bitner, Gideon Ehrlich, and Edward M. Reingold. Efficient Generation of the Binary Reflected Gray Code and Its Applications. Communications of the ACM, 19(9):517--521, 1976.

....in the system of Sato and Inokuchi [Sato 87] the projector pixels are turned on and off over time, so that when a camera pixel records a particular on and off intensity pattern, the corresponding projector pixel can be identified. A popular choice for this on off pattern is a set of Gray codes [Bitner 76] though other approaches using black and white [Gartner 96] gray level [Horn 99] or color [Caspi 96] stripes, or swept laser stripes or dots [Rioux 94] have been examined. The common underlying assumption is temporal coherence of the scene, namely that neighborhoods of pixels can be ....

Bitner, J. R., Erlich, G, and Reingold, E. M. "Efficient Generation of the Binary Reflected Gray Code and its Applications," CACM, Vol. 19, No. 9, 1976.

....given in Mbytes; running times are given in seconds. 7 To verify that the solutions obtained from our algorithm are not too far from optimal, we also implemented a deterministic algorithm that guarantees the best solution. This algorithm is exponential, but by using binary reflected Gray codes [BER76] it was possible to run experiments for up to 30 vertices, which is not too far from real data. In all the tests we ran the solution found by the random algorithm was indeed the optimal. We also compared the best 127 edges obtained from the random algorithm on the whole graph (usually about 40 60 ....

Bitner J. R., G. Erlich, and E. M. Reingold, "Efficient generation of the binary reflected Gray code and its applications," Communications of the ACM, 19 (September 1976), pp. 517-521.

....code words have a Hamming distance of one. Gray codes can be used to reduce errors when an analog signal is converted to a digital signal, they can be used in distributed 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 ....

....other methods for generating it, including a direct method. They did further work [44] on this Gray code, as well. For another ff ary Gray code, the reader may wish to see Barasch, et al. 1] A good discussion of the binary reflected Gray code and some of its uses is given by Bitner, et al. [2]. 5.2 Iterating the Gray Code G(ff; In this section we prove a theorem on the number of unique codes produced when N (ff; is iteratively mapped using K Gamma1 . That is, we start with N (ff; and repeatedly apply the permutation given by K Gamma1 . Let N i j (ff; K Gamma1 ....

Bitner, James R., Gideon Ehrlich, and Edward M. Reingold. (1976). "Efficient Generation of the Binary Reflected Gray Code and its Applications." Comm. ACM, vol. 19, no. 9, pp. 517-521.

....before one with characteristics , because the former precedes the latter in value. Enumeration is accomplishing using a function that counts within the Gray code, denoted as that provides the Gray code for the value after Gray code representation , with similarly representing the predecessor of (Bitner, Ehrlich, Reingold, 1976; Er, 1985; Ludman Sampson, 1981) The ability to count forward and backward from any point is necessary if we are to analytically model browsing performance (Losee, 1995) In a traditional library, it is only necessary that the user (and book 6 shelvers) know whether one book goes before or ....

Bitner, J. R., Ehrlich, G., & Reingold, E. M. (1976). Efficient generation of the binary reflected Gray code and its applications. Communications of the ACM, 19(9).

....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 minimal change algorithms or combinatorial Gray codes, in honor ....

J. R. Bitner, G. Ehrlich, and E. M. Reingold, "Efficient generation of the binary reflected Gray code and its applications," Communications of the ACM 19 No. 9 (1976), 517-521.

....f1,2,4g f1,4g f1,4,5g f4,5g f3,4,5g f3,5 g f2,3,5g f2,5g f1,2,5g f1,5g f1,2,5g Figure 1: Revolving door algorithm listing of the 2 subsets of f1,2,3,4,5g does not give Hamilton cycle in the middle two levels of B 5 . in which successive elements differ in one element (see, e.g. NW] [BER], EM] Then, by taking unions of successive pairs of elements, create a list of k 1 sets. Alternating between the list of k subsets and the corresponding list of k 1 subsets would satisfy properties (ii) and (iii) but, unfortunately, not property (i) at least not for any known listing of ....

J. R. Bitner, G. Ehrlich, and E. M. Reingold, "Efficient generation of the binary reflected Gray code and its applications," Communications of the ACM 19 No. 9 (1976) 517-521.

.... Gray codes include (1) listing all permutations of 1 : n so that consecutive permutations differ only by the swap of one pair of adjacent elements [Joh63, Tro62] 2) listing all k element subsets of an n element set in such a way that consecutive sets differ by exactly one element [BER76, BW84, EHR84, EM84, NW78, Rus88a], 3) listing all binary trees so that consecutive trees differ only by a rotation at a single node [Luc87, LRR93] 4) listing all spanning trees of a graph so that successive trees differ only by a single edge [HH72, Cum66] 5) listing all partitions of an integer n so that in successive ....

....to every number. So, for example, 6 L 2 = 00; 01; 11; 10, L 3 = 000; 001; 011; 010; 110; 111; 101; 100 and L 5 is shown in Figure 1(a) Since the 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 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 ....

[Article contains additional citation context not shown here]

J. R. Bitner, G. Ehrlich, and E. M. Reingold. Efficient generation of the binary reflected Gray code. Communications of the ACM, 19(9):517--521, 1976.

....independent and different ways by Eades, Hickey, and Read [3] Buck and Wiedemann [2] and Ruskey [16] We will imitate the approach of [16] In order to do this we need to add an additional parameter k to our problem. The addition of this parameter helps immensely; we can now follow the lead of [1] for the Gray code question, and of [16] for the strong Gray code question. Let T(n; k) denote the set of all bitstrings in T(n) with prefix 1 k 0. These bitstrings correspond to binary trees with leftmost leaf at level k. If 1 k 0s is an element of T(n; k) then the 1 s in s will be call the ....

....here and those of [11] Some of these algorithms run in constant average time but none of them run in constant worst case time. It now seems possible to produce such an algorithm by a more in depth analysis of the transposition algorithm presented in this paper. This is the same route followed in [1] (see also [13] for bitstrings representing combinations: A careful analysis of the recursive generation algorithm produced a constant average time algorithm. It is remarkable that Theorem 2 is almost exactly the same as theorems in [2] 3] and [16] except that the set T(n; k) is replaced by ....

J.R. Bitner, G. Ehrlich, and E.M. Reingold. Efficient generation of the binary reflected gray code and its applications. Comm. ACM, 19:517-- 521, 1976.

....example, L 2 = 00; 01; 11; 10; L 3 = 000, 001, 011, 010, 110, 111, 101, 100. Since the 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 ....

J. R. Bitner, G. Ehrlich, and E. M. Reingold, Efficient generation of the binary reflected Gray code and its applications, Communications of the ACM 19 No. 9 (1976), 517-521.

....The running times are in seconds on a DEC 5000 240. To verify that the solutions obtained from our algorithm are not too far from optimal, we also implemented a deterministic algorithm that guarantees the best solution. This algorithm is exponential, but by using binary reflected Gray codes [BER76] it was possible to run experiments for up to 30 vertices, which is not too far from real data. In all our tests the solution found by the random algorithm was indeed the optimal. We also compared the best 127 edges obtained from the random algorithm on the whole graph (usually about 40 60 ....

Bitner J. R., G. Erlich, and E. M. Reingold, "Efficient generation of the binary reflected Gray code and its applications," Communications of the ACM, 19 (September 1976), pp. 517-521.

....for producing the Gray Code listing; this algorithm is based on a preorder traversal of the forest. If this algorithm is applied to the antichain A n the algorithm reduces to an algorithm for generating the Binary Reflected Gray Code of size n (e.g. Ehrlich [5] Bitner, Ehrlich, and Reingold [1], and Wilf [22] In Section 4 we eliminate certain redundancy from the algorithm and in Section 5 we show how to attain a loopless algorithm. The loopless algorithm can be implemented on a restricted version of a pointer machine. For a formal definition of pointer machine, see Tarjan [21] From ....

J. Bitner, G. Ehrlich, and E. Reingold, "Efficient Generation of the Binary Reflected Gray Code and its Applications," Communications of the ACM, 19 (1976) 517-521.

....the 2 n vertices of the n cube, or equivalently an ordering of the 2 n binary vectors of length n such that each pair of adjacent vectors (including the first and last) differ in a single position. There is extensive literature, going back at least to 1872 see for example [1] 2] 4] [6], 9] 12] 16] 17] 19] 21] 23] 29] 38] 40] 47] 50] 51] 54] 55] 61] 62] As we will show, the classical version is the special case G = 1 n of the following. Theorem. Let G be a finite group generated by reflections R 1 , R n . Then there is a Hamiltonian ....

J. R. Bitner, G. Ehrlich and E. M. Reingold, Efficient generation of the binary reflected Gray code and its applications, Comm. ACM, 19 (1976), 517-521.

....where J(F ; k) fails to have a Hamilton path. In fact, J(F ; k) can be a path. If F is the poset antimatroid of a poset consisting of two disjoint chains, both containing n Gamma 1 elements, then J(F ; n) is a path of length n. A classic result of Ehrlich [5] and Bitner, Ehrlich and Reingold [2] is that J(P; k) is Hamiltonian if P is the poset antimatroid of an antichain. We extend this result to series parallel posets. The basic word graph G(L) of the antimatroid L is the graph whose vertex set is Basic(L) and such that two basic words are adjacent if they differ by the transposition ....

J. Bitner, G. Ehrlich, and E. Reingold, "Efficient Generation of the Binary Reflected Gray Code and its Applications," Communications of the ACM, 19 (1976) 517-521.

....generated. 1 Introduction A subject dear to the heart of every computational combinatorist is that of generating combinatorial objects. Subsets of an n set and k subsets of an n set, or combinations, are of fundamental importance and much has been written about generating them (see, for example, [2], 3] 4] 8] 11] A natural restriction of the problem of generating all subsets is the problem of generating all subsets of f1; 2; ng whose sum is, say, p. The study of algorithms for generating combinatorial objects often leads to a deeper understanding of the objects themselves. ....

....in constant amortized time. 4 Summary and Further Research We have given a provably constant amortized time algorithm for generating all subsets of an n set with a given sum. This algorithm uses a data structure similar to the stack simulation data structure used by Reingold, Neivergelt and Deo [2] to implement a Gray Code generation algorithm. We also presented a simpler algorithm that experimentally appears to run in constant amortized time. An algorithm for listing all subsets of an n set of a specific size that have a given sum is presented as well. The problem of listing these objects ....

J. Bitner, G. Ehrlich, and E. Reingold, Efficient Generation of the Binary Reflected Gray Code and its Applications, Communications of the ACM, Vol. 19, No. 9, Sept. 1976, pp 517-521.

....x q = y p Gamma 1. This concept of adjacency seems to be the most natural one for solutions in integers to an equation of the form x 0 x 1 Delta Delta Delta x t = k, possibly subject to some other side constraints. It has been applied to combinations (e.g. Bitner, Ehrlich and Reingold [1] or Eades and McKay [4] compositions (e.g. as attributed to Knuth in Wilf [14] and to integer partitions (e.g. Savage [13] Rasmussen, Savage and West [11] For each of these classes, it was shown that there is an exhaustive listing of the elements in which successive elements on the list ....

....observe that OE is a graph isomorphism between G(k; n) and G(n Gamma k; n) It is also clear that a left extreme vertex in G(k; n) is right extreme in G(k; n) and vice versa. 2 The following technical lemma will be used in Theorem 1 below to show the existence of a Gray code in C(k; n) Let [1] denote the 4 by 4 Boolean matrix with all entries 1. Lemma 5 Let A; B; C; and D be the four Boolean matrices in Figure 3. Let M be the Boolean product, M = M 1 M 2 Delta Delta Delta M k , where k 2 and M i 2 fA; B; C; D; 1]g for 1 i k. Then M = 1] unless M 1 = M 2 = Delta Delta Delta ....

[Article contains additional citation context not shown here]

J. Bitner, G. Ehrlich, and E. Reingold, Efficient Generation of the Binary Reflected Gray Code and its Applications, Communications of the ACM, 19 (1976) 517-521.

No context found.

Bitner, J. R., Ehrlich, G. and Reingold, E. M. (1976) Efficient generation of the binary reflected Gray code and its applications. Commun. ACM, 19, 517--521.

No context found.

J. R. Bither, G. Ehrlich, and E. M. Reingold, "Efficient generation of the binary reflected Gray code and its applications," Commun. ACM, vol. 19, pp. 517-521, 1976.

No context found.

J. Bitner, G. Ehrlich, and E. Reingold, "Efficient Generation of the Binary Reflected Gray Code and its Applications," Communications of the ACM, 19 (1976) 517-521.

No context found.

J. Bitner, G. Ehrlich, and E. Reingold, "Efficient Generation of the Binary Reflected Gray Code and its Applications," Communications of the ACM, 19 (1976) 517-521.

No context found.

J. Bitner, G. Ehrlich, and E. Reingold, "Efficient Generation of the Binary Reflected Gray Code and its Applications," Communications of the ACM, 19 (1976) 517-521.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC