Ehrlich, G. (1973) Loopless algorithms for generating permutations, combinations, and other combinatorial objects. J. ACM, 20, 500--513.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 straightforward to ....

....[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 straightforward to convert to ones with O(1) time as in [6]. If we relax the requirement from O(1) time to a fixed number of changes from object to object, which we refer to as O(1) changes, there are numerous results: Nijenhuis and Wilf [2] and Eades and McKay [12] for combinations, Proskurowski and Ruskey [13] and Ruskey and Proskurowski [14] for ....

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Ehrlich, G., Loopless algorithms for generating permutations, combinations, and other combinatorial configurations, JACM, 20 (1973) 500-513.

....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 sequences. RRR 59 2001 PAGE 7 The proposed extension of Gray code preserves the property that it enumerates the elements of L in such a way that two consecutive ....

G. Ehrlich, Loopless algorithms for generating permutations, combinations and other configurations. J. ACM 20, 3 (

....2 (L) Minfnj( n b n 2 c ) kg d2 log ke Proof. Notice that all dlog ke elements subsets of an d2 log ke elements set are incomparable for the inclusion order. It is easy to verify that ( d2log ke blog kc ) k. To generate k dlog ke elements subsets, a nice algorithm can be found in [8] (Algorithm 7, page 510) 1234 34 14 24 4 3 1 13 12 23 Fig. 3. A coatomic lattice having a Sperner encoding Remark. The above example of a k diamond lattice shows the hardness for dim 2 computations, since dim 2 (k Gamma diamond) 2 O(log k) and C Delta(k Gamma diamond) k 1. ....

Gideon Ehrlich. Loopless algorithms for generating permutations, combinations, and other combinatorial configurations. J. of ACM, 20(3):500--513, july 1973.

....generating combinatorial objects so that successive objects differ in some pre specified, usually small, way. However, the origins of minimal change listings can be found in the early work of Gray [Gra53] Wells [Wel61] Trotter [Tro62] Johnson [Joh63] 2 Lehmer [Leh65] Chase [Cha70] Ehrlich [Ehr73], and Nijenhuis and Wilf [NW78] and in the work of campanologists [Whi83] In his article on the origins of the binary Gray code, Heath describes a telegraph invented by Emile Baudot in 1878 which used the binary reflected Gray code [Hea72] According to Heath, Baudot received a gold medal for ....

....1; t. If m i s for i = 1; t, C(s; m 1 ; m t ) is the set of s combinations of a t element set. If X is the multiset consisting of m i copies of element i for i = 1; t, then C(s; m 1 ; m t ) is the collection of s element submultisets, or s combinations of X . In [Ehr73], Ehrlich provides a loopless algorithm to generate multiset combinations so that successive elements differ in only two positions, but not necessarily by just Sigma1 in those positions. It is shown in [RS95] that a Gray code still exists when the two position can change by only Sigma1, thereby ....

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G. Ehrlich. Loopless algorithms for generating permutations, combinations, and other combinatorial configurations. Journal of the ACM, 20:500--513, 1973.

....arrangements of permutations were given by Johnson [32] and Trotter [53] in the early 1960 s. Other Hamiltonian circuits through all n permutations (satisfying different constraints) arise in bell ringing ( 49] 57] 60] Although several other generalizations of Gray codes have appeared ( 3] [22], 33] 37] 42] 48] 52] we believe our version is new. The theorem is proved in 2, and 3 gives some examples. In particular we give specific Gray codes for all the examples ( 1 , 2 , 3 , 2 , 3 , 2 , 3 , 2 (m) in dimensions n 3. It is worth remarking that not all Cayley ....

....1 , 1 , 0 , 0) with n 1 coordinates adding to zero. The corresponding reflections are the transpositions (1 , 2) 2 , 3) n , n 1) of the n 1 coordinates, and so n is isomorphic to the symmetric group S n 1 . Several algorithms for obtaining Gray codes are known ( 18] [22], 32] 53] The following algorithm (which arises from the proof of the theorem) seems as simple as any. It will also be used when we construct Gray codes for the other groups. We specify a Gray code C n for n 1 = S n in the third form, as in Eq. 5) taking w = 1 , 2 , n) ....

G. Ehrlich, Loopless algorithms for generating permutations, combinations, and other combinatorial configurations, J. ACM, 20 (1973), 500-513.

....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 using a loopless algorithm [5] remains open as does the resolution of Conjectures 1 and 2 of Section 2. 5 Acknowledgement We wish to thank Brendan McKay for suggesting the problem. ....

G. Ehrlich, Loopless Algorithms for Generating Permutations, Combinations, and Other Combinatorial Configurations, Journal of the ACM, 20 (1973) 500513.

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Ehrlich, G. (1973) Loopless algorithms for generating permutations, combinations, and other combinatorial objects. J. ACM, 20, 500--513.

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