Ro-Yu Wu

A linear time algorithm for binary tree sequences transformation using left-arm and right-arm rotations

In this paper, we consider a transformation on binary trees using new types of rotations. Each of the newly proposed rotations is permitted only at nodes on the left-arm or the fight-arm of a tree. Consequently, we develop a linear time algorithm with at most n - 1 rotations for converting weight sequences between any two binary trees. In particular, from an analysis of aggregate method for a sequence of rotations, each rotation of the proposed algorithm can be performed in a constant amortized time. Next, we show that a specific directed rooted tree called rotation tree can be constructed using one of the new type rotations. As a by-product, a naive algorithm for enumerating weight sequences of binary trees in lexicographic order can be implemented by traversing the rotation tree.

Loopless Generation of Non-regular Trees with a Prescribed Branching Sequence1

An ordered tree is called a non-regular tree with a prescribed branching sequence (or non-regular tree for short) if its internal nodes have a prespecified degree sequence in preorder list. We define a concise representation, called right distance sequences to describe such trees. A coding tree helps us to systematically investigate the structural representation of non-regular trees. Consequently, we present a loopless algorithm to generate Gray-codes of non-regular trees using right distance sequences.

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).

A loopless algorithm for generating multiple binary tree sequences simultaneously

Pallo and Wu et al. respectively introduced the left-weight sequences (LW-sequences) and right-weight sequences (RW-sequences) for representing binary trees. In this paper, we introduce two new types of binary tree sequences called the left-child sequences (LC-sequences) and right-child sequences (RC-sequences). Next, we propose a loopless algorithm associated with rotations of binary trees for generating LW-, RW-, LC-, and RC-sequences simultaneously. Moreover, we show that LW- and RW-sequences are generated in Gray-code order, and LC- and RC-sequences are generated so that each sequence can be obtained from its predecessor by changing at most two digits. Our algorithm is shown to be more efficient in both space and time than the existing known algorithms.

Incidence coloring on hypercubes

Let ? i ( G ) and Δ ( G ) denote the incidence coloring number and the maximum degree of a graph G, respectively. An easy observation shows that ? i ( G ) ? Δ ( G ) + 1 . In this paper, we consider incidence coloring number on hypercubes Q n . Based on the technique of Hamming codes, we present algorithms to obtain upper bounds of ? i ( Q n ) for n in certain forms of integers. Let p , q be positive integers. We show that (1) ? i ( Q n ) = n + 1 if n = 2 p - 1 for p ? 1 ; (2) ? i ( Q n ) = n + 2 if the following conditions hold: (i) n = 2 p - 2 for p ? 2 ; (ii) n = 2 p + 2 q - 2 for p , q ? 1 ; (iii) n = 2 p + 2 q - 3 for p , q ? 2 ; and (3) ? i ( Q n ) ? n + 2 otherwise.

A Parallel Construction of Vertex-Disjoint Spanning Trees with Optimal Heights in Star Networks

Constructing vertex-disjoint spanning trees (VDSTs for short) of a given network is an important issue in the research of network fault-tolerance and security. The star network was proposed as an attractive interconnection network model for competing with n-cube. Accordingly, Rescigno in [Inform. Sci. 137 (2001) 259–276] proposed an algorithm to construct \(n-1\) VDSTs rooted at a common node in an n-dimensional star network \(S_n\). In this paper, we point out that there exists a flaw in Rescigno’s algorithm, and thus the spanning trees constructed by this algorithm may not be vertex-disjoint. As a result, a correct scheme of constructing \(n-1\) VDSTs on \(S_n\) is presented. Moreover, based on the reversing rule of building certain paths of VDSTs in the amendatory scheme, we propose a new algorithm to construct \(n-1\) VDSTs with optimal heights on \(S_n\). In particular, the proposed algorithm is more efficient and can easily be implemented in parallel.

Amortized Efficiency of 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. [19] for representing binary trees. In particular, 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 such a characterization, there is an algorithm to generate all LC-sequences of binary trees with n internal nodes in lexicographic order. In this paper, we extend our study to the ranking and unranking problems. By integrating a measure called “left distances” introduced by Makinen [8] to represent binary trees, we develop efficient ranking and unranking algorithms for LC-sequences in lexicographic order. With a help of aggregate analysis, we show that both ranking and unranking algorithms can be run in amortized cost of \(\mathcal {O}(n)\) time and space.

Ranking and unranking of well-formed parenthesis strings in diverse representations

Well-formed parenthesis (w.f.p.) strings can be represented by different types of integer sequences including P-sequences, X-sequences, and T-sequences. In this paper, we introduce a new class of sequences called L-sequences which comes from the so-called RD-sequences for representing k-ary trees. We then search out the relationships among these representations and deal with the problems of ranking and unranking of w.f.p. strings in lexicographic order under these diverse representations. A result shows that the ranking and unranking of w.f.p. strings can be done in O(n) time for all such representations, where n is the number of pairs of balanced parentheses.

Constructing Independent Spanning Trees on Bubble-Sort Networks

A set of spanning trees in a graph G is called independent spanning trees (ISTs for short) if they are rooted at the same vertex, say r, and for each vertex \(v(\ne r)\) in G, the two paths from v to r in any two trees share no common vertex except for v and r. Constructing ISTs has applications on fault-tolerant broadcasting and secure message distribution in reliable communication networks. Since Cayley graphs have been used extensively to design interconnection networks, the study of constructing ISTs on Cayley graphs is very significative. It is well-known that star networks \(S_n\) and bubble-sort network \(B_n\) are two of the most attractive subclasses of Cayley graphs. Although it has been dealt with about two decades for the construction of ISTs on \(S_n\) (which has been pointed out that there is a flaw and has been corrected recently), so far the problem of constructing ISTs on \(B_n\) has not been dealt with. In this paper, we present an efficient algorithm to construct \(n-1\) ISTs of \(B_n\). It seems that our work is the latest breakthrough on the problem of ISTs for all subclasses of Cayley graphs except star networks.

A Constant Amortized Time Algorithm for Generating Left-Child Sequences in Lexicographic Order

Wu et al. (Theoret. Comput. Sci. 556:25–33, 2014) recently introduced a new type of sequences, called left-child sequences (LC-sequences for short), 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, Pai et al. (International conference on combinatorial optimization and applications. Springer, Cham, pp. 505–518, 2016) showed that 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 this algorithm is not efficient at all. In this paper, we 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.