Ranking and unranking algorithms for loopless generation of t-ary trees

Ahmadi-Adl, Amin; Nowzari-Dalini, Abbas; Ahrabian, H.

doi:10.1093/jigpal/jzp097

Cited by 11 publications

(14 citation statements)

References 23 publications

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“…Recalling from the previous section that t 1 and t 2 indicate the number of leaves and internal nodes after node k, respectively, this step takes O( n−k N ), since we have t1+t2 ≤ n − k. Updating P T , which is illustrated in the Algorithm 3, involves updating parts s 1 , s 2 and s 3 (for more information, see Sect. 3).…”

Section: Endmentioning

confidence: 99%

“…Thus, numerous sequential algorithms have been developed to generate different types of trees [9,[15][16][17]20,23,25,[31][32][33]35,37]. In most of these algorithms, trees are encoded as integer sequences, and then, these sequences are generated with a certain order, and consequently, their corresponding trees are generated in a specific order [1,13,18,21,34,36]. The most well-known orderings on trees are A-order and B-order [38].…”

Section: Introductionmentioning

confidence: 99%

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A parallel algorithm for generation of RNA secondary structures with length n and k base-pairs

Mohammadi

Nowzari-Dalini

2017

Iran J Comput Sci

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RNA secondary structures are important in many biological processes since there is a close relationship between structure and function in biology. There is a known bijection between the possible shapes of a single-stranded RNA molecule of fixed length and trees with n nodes and m leaves, T (n, m). In this article, we present a sequential algorithm for generating such trees in A-order. The worst case time complexity of the presented algorithm is O(n). We also develop a parallel version of the algorithm which is both cost-optimal and adaptive. We use Exclusive Read Exclusive Write Shared Memory Single Instruction Multiple Data multi-processor architecture. Moreover, the generated trees are in A-order.

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Section: Endmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A parallel algorithm for generation of RNA secondary structures with length n and k base-pairs

Mohammadi

Nowzari-Dalini

2017

Iran J Comput Sci

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“…Baronaigien and Ruskey [3] showed how to generate k-ary trees using "A-order", achieving O(kn)-time and O(kn log n)-time algorithms for ranking and unranking, respectively. Several algorithms have since been proposed with the goal of achieving efficient running times for tree generation, ranking and unranking simultaneously [1,2,17,33]. Each of these algorithms uses a precomputed table for ranking and unranking that takes O(kn 2 )-time to construct.…”

Section: K-ary Catalan Structuresmentioning

confidence: 99%

Cool-lex order and k-ary Catalan structures

Durocher

Mondal

et al. 2012

Journal of Discrete Algorithms

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For any given k, the sequence of k-ary Catalan numbers, C t,k = 1 kt+1 kt t , enumerates a number of combinatorial objects, including k-ary Dyck words of length n = kt and k-ary trees with t internal nodes. We show that these objects can be efficiently ordered using the same variation of lexicographic order known as cool-lex order. In particular, we provide loopless algorithms that generate each successive object in O(1) time. The algorithms are also efficient in terms of memory, with the k-ary Dyck word algorithm using O(1) additional index variables, and the k-ary tree algorithm using O(t) additional pointers and index variables. We also show how to efficiently rank and unrank k-ary Dyck words in cool-lex order using O(kt) arithmetic operations, subject to an initial precomputation. Our results are based on the cool-lex successor rule for sets of binary strings that are bubble languages. However, we must complement and reverse 1/k-ary Dyck words to obtain the stated efficiency.An important problem in discrete mathematics and theoretical computer science is the creation of efficient orders for combinatorial objects. The research area that studies this problem is combinatorial generation, and it is overviewed by Knuth in the most recent volume of The Art of Computer Programming [12]. In this article we will show that cool-lex order is an efficient order for k-ary Catalan objects. To frame our results, this section provides background on combinatorial generation, k-ary Catalan objects, and cool-lex order. Combinatorial generationTo understand combinatorial generation, we explain each of the terms in "efficient orders for combinatorial objects" and then provide several examples. By 'combinatorial objects' we are referring to a set of objects of a particular size and type, such as binary strings of length n, permutations of [n] = {1, 2, . . . ,n} in one-line notation, trees on n vertices, and so on.The size of the object is usually denoted by n, and the number of objects by m. An 'order' is a list in which each object in the set appears exactly once. Finally, we associate three interrelated operations with each order, for which we evaluate 'efficiency' according to the worst-case time complexity:

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“…Accordingly, a loopless algorithm is implemented non-recursively by using, after the initialization of the first object, only assignment statements and "if-then-else" statements. The reader is referred to [2], [5]- [9], [15], [17] for loopless generation of combinatorial objects. In particular, see [12] for an excellent survey of generating combinatorial objects in Gray-code orders.…”

Section: Introductionmentioning

confidence: 99%

“…Many ranking algorithms [1]- [3], [10], [11], [13], [16], [19] and unranking algorithms [1]- [3], [11], [16] for diverse representations of regular trees have been proposed. Note that all proposed algorithms mentioned above dealt with the rank of objects in lexicographic order except for [2] in a Gray-code order. In addition, Wu et al [14] dealt with the ranking and unranking problems of non-regular trees encoded by RD-sequences in lexicographic order and showed that, given a prescribed branching sequence , both ranking and unranking problems can be solved in time, where .…”

Section: Introductionmentioning

confidence: 99%

A Ranking Algorithm of Non-Regular Trees in Gray-Code Order

Wu¹,

Chang²,

Chen³

2012

IJMLC

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Abstract-A non-regular tree T with a prescribed branching sequence is an ordered tree whose internal nodes are numbered from 1 to n in preorder such that every node in T has a prescribed number of children. Recently, Wu et al. (2010) introduced a concise representation called RD-sequences to represent all non-regular trees and proposed a loopless algorithm for generating all non-regular trees in a Gray-code order. In this paper, based on such a Gray-code order, we show that a ranking algorithm can be done in quadratic time provided a preprocessing in advance.

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Ranking and unranking algorithms for loopless generation of t-ary trees

Cited by 11 publications

References 23 publications

A parallel algorithm for generation of RNA secondary structures with length n and k base-pairs

A parallel algorithm for generation of RNA secondary structures with length n and k base-pairs

Cool-lex order and k-ary Catalan structures

A Ranking Algorithm of Non-Regular Trees in Gray-Code Order

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