Unranking Algorithms Research Articles

Combinatorial Generation Algorithms for Some Lattice Paths Using the Method Based on AND/OR Trees

Methods of combinatorial generation make it possible to develop algorithms for generating objects from a set of discrete structures with given parameters and properties. In this article, we demonstrate the possibilities of using the method based on AND/OR trees to obtain combinatorial generation algorithms for combinatorial sets of several well-known lattice paths (North-East lattice paths, Dyck paths, Delannoy paths, Schroder paths, and Motzkin paths). For each considered combinatorial set of lattice paths, we construct the corresponding AND/OR tree structure where the number of its variants is equal to the number of objects in the combinatorial set. Applying the constructed AND/OR tree structures, we have developed new algorithms for ranking and unranking their variants. The performed computational experiments have confirmed the obtained theoretical estimation of asymptotic computational complexity for the developed ranking and unranking algorithms.

Efficient linear-time ranking and unranking of derangements

In this paper, we propose linear-time algorithms for ranking and unranking derangements. Myrvold and Ruskey proposed the linear-time ranking and unranking of permutations that use a novel approach to convert a random generation algorithm into an unranking algorithm. Our algorithms achieve the exact linear-time ranking and unranking of derangements by applying a method specialized for derangements. We use a computational model that enables arithmetic operations and random access for O(log⁡n!)=O(nlog⁡n) bits to be executed in constant time.

Модификация метода построения алгоритмов комбинаторной генерации на основе применения производящих функций многих переменных и приближенных вычислений

This article proposes a modified method for developing combinatorial generation algorithms based on AND/OR trees. The proposed method differs from the original method and its modifications by using the method of obtaining explicit expressions for the coefficients of multivariate generating functions to find the expression for the cardinality function of a given combinatorial set (including combinatorial sets defined by several parameters). Also, the proposed method is distinguished by the use of approximation methods and the binary search to find the selected son of the OR node. This reduces the computational complexity of the unranking algorithms. In order to test the proposed method, a new ranking and unranking algorithms were developed for a combinatorial set of selfavoiding lattice paths. In this case, the use of binary search reduced on average the number of required computational operations and a better computational complexity compared to the original version of the algorithm were obtained.

Syntactic view of sigma-tau generation of permutations

We give a syntactic view of the Sawada-Williams (σ,τ)-generation of permutations. The corresponding sequence Seqn of στ-operations of length n!−1 is shown to have a compact description of size Θ(n2) in terms of straight-line programs. Using the compact description, we design almost linear time ranking and unranking algorithms.

Algorithms for ranking and unranking the combinatorial set of closed questionnaire answers

In this paper, we study the combinatorial set of closed questionnaire answers with a specified minimum number of correct answers. For this combinatorial set, we obtain an explicit formula for its cardinality function. Using the obtained cardinality function, we construct the corresponding AND/OR tree structure and determine the bijecion rules for this AND/OR tree and the considered combinatorial set. In addition, we develop ranking and unranking algorithms for the combinatorial set.

A loopless algorithm for generating (k, m)-ary trees in Gray code order

A family of (k, m)-ary trees was firstly introduced by Du and Liu when they studied hook length polynomial for plane trees. Recently, Amani and Nowzari-Dalini presented a generation algorithm to produce (k, m)-ary trees of order n encoding by Z-sequences in reverse lexicographic order. In this paper, we propose a loopless algorithm to generate all such Z-sequences in Gray code order. Hence, the worst-case time complexity of generating one Z-sequence is $${\mathcal {O}}(1)$$ , and the space requirement of our algorithm is $$2n+{\mathcal {O}}(1)$$ . Moreover, based on this ordering, we also provide ranking and unranking algorithms. The ranking algorithm can be run in $${\mathcal {O}}(\max \{kmn,n^2\})$$ time and $${\mathcal {O}}(kmn)$$ space, whereas the unranking algorithm requires $${\mathcal {O}}(kmn^2)$$ time and space.

Method for Developing Combinatorial Generation Algorithms Based on AND/OR Trees and Its Application

In this paper, we study the problem of developing new combinatorial generation algorithms. The main purpose of our research is to derive and improve general methods for developing combinatorial generation algorithms. We present basic general methods for solving this task and consider one of these methods, which is based on AND/OR trees. This method is extended by using the mathematical apparatus of the theory of generating functions since it is one of the basic approaches in combinatorics (we propose to use the method of compositae for obtaining explicit expression of the coefficients of generating functions). As a result, we also apply this method and develop new ranking and unranking algorithms for the following combinatorial sets: permutations, permutations with ascents, combinations, Dyck paths with return steps, labeled Dyck paths with ascents on return steps. For each of them, we construct an AND/OR tree structure, find a bijection between the elements of the combinatorial set and the set of variants of the AND/OR tree, and develop algorithms for ranking and unranking the variants of the AND/OR tree.

Improved algorithms for ranking and unranking (k, m)-ary trees in B-order

Du and Liu (Eur J Comb 28:1312–1321, 2007) introduced (k, m)-ary trees as a generalization of k-ary trees. In a (k, m)-ary tree, every node on even level has degree k (i.e., has k children), and every node on odd level has degree m (which is called a crucial node) or is a leaf. In particular, a (k, m)-ary tree of order n has exactly n crucial nodes. Recently, Amani and Nowzari-Dalini (Bull Iranian Math Soc 45(4):1145–1158, 2019) presented a generation algorithm to produce all (k, m)-ary trees of order n in B-order using Zaks’ encoding, and showed that the generated ordering of this encoding results in a reverse-lexicographical ordering. They also proposed the corresponding ranking and unranking algorithms for (k, m)-ary trees according to such a generated ordering. These algorithms take $$\mathcal {O}(kmn^2)$$ time and space for building a precomputed table in which (k, m)-Catalan numbers (i.e., a kind of generalized Catalan numbers) are stored in advance. Then, each ranking and unranking algorithm can be performed subsequently in $$\mathcal {O}(n)$$ and $$\mathcal {O}(n\log n)$$ time, respectively. In this paper, we revisit the ranking and unranking problems. With the help of an encoding scheme called “right-distance” introduced by Wu et al. (Math Comput Model 53:1331–1335, 2011a; IEICE Trans Inf Syst E94–D:226–232, 2011b), we propose new ranking and unranking algorithms for (k, m)-ary trees of order n in B-order using Zaks’ encoding. We show that both algorithms can be improved in $$\mathcal {O}(kmn)$$ time and $$\mathcal {O}(n)$$ space without building the precomputed table.

Ranking and unranking fixed-density necklaces and Lyndon words

We present the first polynomial-time ranking and unranking algorithms for fixed-density necklaces and Lyndon words. Using the unit-cost RAM model, the ranking algorithm runs in O(n3) time and the unranking algorithm runs in O(n4) time. By applying the ranking algorithms, the number of fixed-density necklaces and Lyndon words with a given prefix can also be computed in O(n3) time.

Efficient Generation, Ranking, and Unranking of (k, m)-Ary Trees in B-Order

In this paper, we present a new generation algorithm with corresponding ranking and unranking algorithms for (k, m)-ary trees in B-order. (k, m)-ary tree is introduced by Du and Liu. A (k, m)-ary tree is a generalization of k-ary tree, whose every node of even level of the tree has degree k and odd level of the tree has degree 0 or m. Up to our knowledge no generation, ranking or unranking algorithms are given in the literature for this family of trees. We use Zaks’ encoding for representing (k, m)-ary trees and to generate them in B-order. We also prove that, to generate (k, m)-ary trees in B-order using this encoding, the corresponding codewords should be generated in reverse-lexicographical ordering. The presented generation algorithm has a constant average time and O(n) time complexity in the worst case. Due to the given encoding, both ranking and unranking algorithms are also presented taking O(n) and $$O(n\log n)$$ time complexity, respectively.

Amortized efficiency of generation, ranking and unranking left-child sequences in lexicographic order

A new type of sequences called left-child sequences (LC-sequences for short) was recently introduced by Wu et al. (2014) for representing binary trees. They pointed out that such sequences have a natural interpretation from the view point of data structure and gave a characterization of them. Based on this characterization, there is an easily implementing algorithm that uses generate-and-test approach to filter all LC-sequences of binary trees with n internal nodes in lexicographic order, while in general it is not efficient at all. In this paper, we first design two novel rotations that allow us to drastically alter the shape of binary trees (and thus their corresponding LC-sequences). As an application, these operations can be employed to generate all LC-sequences in lexicographic order. Accordingly, we present a more efficient algorithm associated with the new types of rotations for generating all LC-sequences and show that it takes only constant amortized running cost. Moreover, we extend our study to the ranking and unranking problems. By integrating a measure called “left distances” introduced by Mäkinen (1987) to represent binary trees, we develop efficient ranking and unranking algorithms for LC-sequences in lexicographic order. With the help of aggregate analysis, we show that both ranking and unranking algorithms can be run in amortized cost of O(n) time and space.

Differential Spatial Modulation With Gray Coded Antenna Activation Order

Differential spatial modulation (DSM) was recently proposed to overcome the challenge of channel estimation in spatial modulation (SM). In this letter, we propose a gray code order of antenna index permutations for DSM. To facilitate the implementation, the well-known Trotter-Johnson ranking and unranking algorithms are adopted, which results in similar computational complexity to the existing DSM that uses the lexicographic order. The coding gain achieved by the proposed gray code order over the existing lexicographic order is also analyzed and verified via simulations, which reveals a maximum of about 1.2dB for the case of four transmit antennas. Based on the gray coding framework, we further propose a diversity-enhancing scheme named intersected gray (I-gray) code order for DSM, where the permutations of active antenna indices are selected directly from the odd (even) positions of the full permutations in the gray code order. From analysis and simulations, it is shown that the I-gray code order can harvests an additional diversity order at the expense of only one information bit loss for each transmission with respect to the gray code order.

Generation, Ranking and Unranking of Ordered Trees with Degree Bounds

We study the problem of generating, ranking and unranking of unlabeled ordered trees whose nodes have maximum degree of $\Delta$. This class of trees represents a generalization of chemical trees. A chemical tree is an unlabeled tree in which no node has degree greater than 4. By allowing up to $\Delta$ children for each node of chemical tree instead of 4, we will have a generalization of chemical trees. Here, we introduce a new encoding over an alphabet of size 4 for representing unlabeled ordered trees with maximum degree of $\Delta$. We use this encoding for generating these trees in A-order with constant average time and O(n) worst case time. Due to the given encoding, with a precomputation of size and time O(n^2) (assuming $\Delta$ is constant), both ranking and unranking algorithms are also designed taking O(n) and O(nlogn) time complexities.

Unranking of small combinations from large sets

An unranking algorithm for a finite set S is an algorithm that, given a number in {0,1,…,|S|−1}, returns an element of S that is associated with the number. We suppose that a number can be associated with any element in S so long as distinct elements are associated with different numbers. A ranking algorithm is the reverse of an unranking algorithm. In this paper, we present an unranking algorithm for the set of all m-element subsets of an n-element set. Our algorithm runs with O(m3m+3) arithmetic operations, which is independent of n and hence is effective when n is large. We also show that it admits a ranking algorithm with the same running time.

Lexicographic ranking and unranking of derangements in cycle notation

We present lexicographic ranking and unranking algorithms for derangements expressed in cycle notation. These algorithms run in O(nlogn) time, require O(n) space, and use O(n) arithmetic operations. Similar algorithms that require less than or equal to O(nlogn) time with O(n) space complexity have not previously been proposed.

Ranking and Unranking of Non-regular Trees in Gray-Code Order

A non-regular tree T with a prescribed branching sequence (s 1 , s 2 ,..., s n ) is a rooted and ordered tree such that its internal nodes are numbered from 1 to n in preorder and every internal node i in T has s i 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 present efficient ranking and unranking algorithms of non-regular trees with n internal nodes. Moreover, we show that the ranking algorithm and the unranking algorithm can be run in O(n 2 ) time and O(n 2 + nS n―1 ) time, respectively, provided a preprocessing takes O(n 2 S n―1 ) time and space in advance, where S n―1 = Σ n―1 i=1 (s i ―1).

Ranking and Unranking Algorithms for k-ary Trees in Gray Code Order

In this paper, we present two new ranking and unranking algorithms for k-ary trees represented by x-sequences in Gray code order. These algorithms are based on a gray code generation algorithm developed by Ahrabian et al.. In mentioned paper, a recursive backtracking generation algorithm for x-sequences corresponding to k-ary trees in Gray code was presented. This generation algorithm is based on Vajnovszki’s algorithm for generating binary trees in Gray code ordering. Up to our knowledge no ranking and unranking algorithms were given for x-sequences in this ordering. we present ranking and unranking algorithms with O(kn) time complexity for x-sequences in this Gray code ordering . Keywords—k-ary Tree Generation, Ranking, Unranking, Gray Code.

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

In 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 RD-sequence (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.

Ranking and unranking of non-regular trees with a prescribed branching sequence

Ordered trees are called non-regular trees with a prescribed branching sequence (or non-regular trees for short) if their internal nodes have a pre-specified degree sequence in preorder list. This article presents two main results. First, we develop a simple algorithm to generate all non-regular trees in lexicographic order using RD-sequences defined by [R.-Y. Wu, J.-M. Chang, Y.-L. Wang, Loopless generation of non-regular trees with a prescribed branching sequence, The Computer Journal 53 (2010) 661–666]. Then, by analyzing the structure of a coding tree, this algorithm is proved to run in constant amortized time. Next, we propose efficient ranking algorithm (i.e., determining the rank of a given non-regular tree in such an order) and unranking algorithm (i.e., converting a positive integer to its corresponding RD-sequence).

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

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