Eades, P and McKay, B, Al algorithm for generating subsets of fixed size with a strong minimal change property, Info. Proc. Lett., 19 (1984) 131-133.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... 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 parenthesis strings, etc. Ruskey and Proskurowski generate parenthesis strings by adjacent transpositions only when n is even where the length of a string is 2n. Recently Walsh [15] successfully converted their ....

Eades, P and McKay, B, Al algorithm for generating subsets of fixed size with a strong minimal change property, Info. Proc. Lett., 19 (1984) 131-133.

....many algorithms in the literature for generating the k sets of an n set. The revolving door algorithm in [18] generates all k sets so that successive sets differ in only one element. If we used this algorithm to generate 2 sets, neither property (1) nor (2) would be satisfied. Other algorithms ([6], 3] 10] can in some cases generate k sets with the minimal change property, that is, so that successive sets differ in exactly one element and this element has either increased or decreased by 1. For k = 2, this would amount to using only horizontal and vertical edges in G n . However, it is ....

....only horizontal and vertical edges in G n . However, it is known that there is no listing of k sets with the minimal change property if and only if either n is even and k is odd or if k is one of 0, 1, n, n Gamma 1. More closely related to our listing is the strong minimal change (SMC) property [6]. That is, k sets are generated so that if a k set is represented as a sorted sequence, successive k sets differ in only one position. This is what property (1) requires: a listing of the n Gamma 2 sets of [n] with the SMC property, starting with 34 : n and ending at 1256 : n. We cannot ....

[Article contains additional citation context not shown here]

P. Eades and B. McKay, "An algorithm for generating subsets of fixed size with a strong minimal change property," Inform. Process. Lett., 19 (1984), pp. 131-133.

....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 k sets ....

P. Eades and B. McKay, "An algorithm for generating subsets of fixed size with a strong minimal change property," Information Processing Letters 19 (1984) 131-133.

.... 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 ....

....156 124 234 126 146 145 235 136 145 245 135 135 245 345 125 134 345 135 145 234 346 235 245 235 356 125 345 236 456 Figure 6: Examples of Gray codes for combinations. simple recursive expression. A more stringent requirement is to list all k sets with the strong minimal change property [EM84]. That is, if a k set is represented as a sorted k tuple of its elements, successive k sets differ in only one position (see Figure 6(b) Eades and McKay have shown that such a listing is always possible. An earlier solution was reported by Chase in [Cha70] Perhaps the most restrictive Gray ....

P. Eades and B. McKay. An algorithm for generating subsets of fixed size with a strong minimal change property. Information Processing Letters, 19:131--133, 1984.

....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. Thus our ....

P. Eades and B. McKay, An Algorithm for Generating Subsets of Fixed Size with a Strong Minimal Change Property, Information Processing Letters, 19 (1984) 131-133.

....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 are adjacent under this ....

P. Eades and B. McKay, An Algorithm for Generating Subsets of Fixed Size with a Strong Minimal Change Property, Information Processing Letters, 19 (1984) 131-133.

....defining the combinatorial Gray code should mirror some simple recursion for counting the combinatorial objects, in this case an = 2an Gamma1 with a 0 = 1. Very simple constructions have been obtained in the cases of combinations (e.g. Reingold, Nievergelt, and Deo [ReNiDe] or Eades and McKay [EaMc]) compositions (e.g. Wilf [Wi] and well formed parentheses (e.g. Ruskey and Proskurowski [RuPr] Less simple recursive constructions, that are nevertheless based on reversing sublists, have been obtained for numerical partitions (Savage [Sa] and set partitions (Fill and Reingold [FiRe] ....

....In each case they are straightforward inductive arguments. The real difficulties lie in coming up with the recursive definitions and the statements of the lemmata. 2 The equivalence of two well known lists The following recursively defined list of k combinations of n is due to Eades and McKay [EaMc]. C(n; k) 8 : 0 n if k = 0 10 n Gamma1 ffi 010 n Gamma2 ffi Delta Delta Delta ffi 0 n Gamma1 1 if k = 1 C(n Gamma 1; k) Delta 0 ffiC(n Gamma 2; k Gamma 1) Delta 01 ffiC(n Gamma 2; k Gamma 2) Delta 11 if 1 k n 1 n if k = n (1) This list ....

P. Eades and B. McKay, An Algorithm for Generating Subsets of Fixed Size with a Strong Minimal Change Property, Information Processing Letters, 19 (1984) 131-133.

....2n. We show that the elements of T(n) can be listed so that successive strings differ by the transposition of a left and a right parenthesis. Furthermore, between the two parentheses that are transposed, only left parentheses occur. Our listing is a modification of the well known Eades McKay [4] algorithm for generating combinations. Like that algorithm, ours generates strings from the lexicographically greatest string to the lexicographically least and can be implemented so that each string is generated in constant time, in an amortized sense. 1 Introduction Among the classes of ....

....case of a homogeneous transposition, but not in the case of a non homogeneous transposition. Define B(n; k) to be the set of all bitstrings of length n that contain exactly k 1 s. Algorithms for generating the elements of B(n; k) by homogeneous transpositions have been developed by Eades and McKay [4], Chase [2] and Ruskey [7] The objects being listed are strings over the binary alphabet f0; 1g. If L is a list of strings and x is a symbol, then L Delta x denotes the list of strings obtained by appending an x to each string of L. For example if L = h01; 10i, then L Delta 1 = h011; 101i. ....

[Article contains additional citation context not shown here]

P. Eades and B. McKay, An Algorithm for Generating Subsets of Fixed Size with a Strong Minimal Change Property," Information Processing Letters, 19 (1984) 131-133.

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