On Rotations and the Generation of Binary Trees

Lucas, Joan M.; Vanbaronaigien, D.R.

doi:10.1006/jagm.1993.1045

Cited by 91 publications

(51 citation statements)

References 18 publications

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“…Based on these concepts, many algorithms can be developed to perform query translation automatically even if schema conflicts exist. A second difficult problem is the generation of balanced join trees from a join sequence since, in essence, this problem is NP-hard [25]. To get a better approximately optimal solution to this problem, we have extended the basic transformation step used in [14] and developed a recursive algorithm to obtain more balanced join trees.…”

Section: Discussionmentioning

confidence: 99%

Query evaluation for distributed heterogeneous relational databases

Chen

Benn²

1998

Proceedings 3rd IFCIS International Conference on Cooperative Information Systems (Cat No 98EX122) COOPIS-98

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In this article, we consider the query evaluation problem in relational multidatabases and develop a method for generating optimal plans for queries submitted to such a system. Three aspects will be discussed: query transformation, join tree balance and node allocation. For query translation, the concept of relation structure terms (RST) is introduced. Based on RSTs, we can transform a query into another form automatically by constructing derivation rules for them. Further, we extend the approach for balancing a join tree proposed by Du et al. so that more balanced join trees can be obtained. Lastly, we present the concept of dynamic time tables for performing node allocation in a dynamic programming manner.

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Section: Discussionmentioning

confidence: 99%

Query evaluation for distributed heterogeneous relational databases

Chen

Benn²

1998

Proceedings 3rd IFCIS International Conference on Cooperative Information Systems (Cat No 98EX122) COOPIS-98

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“…. , n} by adding or removing a single element [Gra,Ehr73,BER76], (3) generating all k-element subsets of an [n] by exchanging a single element [Ehr73, BER76, EHR84, EM84, Rus88], (4) generating all binary trees with n vertices by single rotation operations [Luc87,LvBR93].…”

Section: Introductionmentioning

confidence: 99%

A constant-time algorithm for middle levels Gray codes

Mütze

Nummenpalo

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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For any integer n ≥ 1 a middle levels Gray code is a cyclic listing of all n-element and (n + 1)-element subsets of {1, 2, . . . , 2n + 1} such that any two consecutive subsets differ in adding or removing a single element. The question whether such a Gray code exists for any n ≥ 1 has been the subject of intensive research during the last 30 years, and has been answered affirmatively only recently [T. Mütze. Proof of the middle levels conjecture.

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“…In most of these algorithms, t-ary trees are encoded to integer sequences and all different sequences are generated in a particular order, especially in lexicographic order [1], [4], [12], [13], [15], [18], [21], [22] or Gray-code order [2], [6], [11], [19]. Besides, a few algorithms employ the usual pointer in computer representation to generate t-ary trees [5], [7], [8], [20]. Accordingly, we customarily say t-ary trees to mean their representations.…”

Section: Introductionmentioning

confidence: 99%

“…A challenge arisen from this research is to ask whether the pre-computation for building coefficient table is necessary in order to compute the ranking and/or unranking function. Lucas et al claimed that, in fact, they have been unable to reduce the running time of the ranking algorithm or unranking algorithm to less than O(n 2 ) time even if they consider for binary trees (see p. 354 in [8]). …”

Section: Introductionmentioning

confidence: 99%

Ranking and Unranking of t-Ary Trees Using RD-Sequences

Chang

Wang

2011

IEICE Trans. Inf. & Syst.

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SUMMARYIn this paper, we introduce a concise representation, called right-distance sequences (or RD-sequences for short), to describe all t-ary trees with n internal nodes. A result reveals that there exists a close relationship between the representation and the well-formed sequences suggested by Zaks [Lexicographic generation of ordered trees, Theoretical Computer Science 10 (1980) 63-82]. Using a coding tree and its concomitant tables, a systematical way can help us to investigate the structural representation of t-ary trees. Consequently, we develop efficient algorithms for determining the rank of a given t-ary tree in lexicographic order (i.e., a ranking algorithm), and for converting a positive integer to its corresponding RDsequence (i.e., an unranking algorithm). Both the ranking and unranking algorithms can be run in O(tn) time and without computing all the entries of the coefficient table.

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On Rotations and the Generation of Binary Trees

Cited by 91 publications

References 18 publications

Query evaluation for distributed heterogeneous relational databases

Query evaluation for distributed heterogeneous relational databases

A constant-time algorithm for middle levels Gray codes

Ranking and Unranking of t-Ary Trees Using RD-Sequences

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