二元樹序列旋轉及多元樹之列舉｜國立臺灣科技大學博碩士論文系統

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研究生：	吳若禹 Ro-Yu Wu
論文名稱：	二元樹序列旋轉及多元樹之列舉 Binary Tree Sequence Rotations and t-ary Tree Enumerations
指導教授：	王有禮 Yue-Li Wang 盧希鵬 Hsi-Peng Lu
口試委員:	張瑞雄 Ruay-Shiung Chang 洪西進 Shi-Jinn Horng 徐力行 Lih-Hsing Hsu 徐俊傑 Hsu, Chiun-Chieh 趙坤茂 Kun-Mao Chao 張肇明 Jou–Ming Chang 杜迪榕 Dyi-Rong Duh
學位類別：	博士 Doctor
系所名稱：	管理學院 - 資訊管理系 Department of Information Management
論文出版年：	2007
畢業學年度：	95
語文別：	英文
論文頁數：	89
中文關鍵詞：	旋轉、遞迴樹、多元樹、次序演算法、反次序演算法、葛瑞碼、無迴圈演算法、哈密爾敦迴路.
外文關鍵詞：	rotation, recursion tree, t-ary tree, ranking algorithm, unranking algorithm, gray-code, loopless algorithm, Hamiltonian cycle.
相關次數：	點閱：225 下載：3
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有許多研究致力於兩棵ｎ個點二元樹的轉換距離，即是否能在多項式的時間內利用一般的旋轉計算出兩棵二元樹轉換的距離，這個問題仍是一個待解決的開放問題。我們提出一些新形式的旋轉，這些旋轉只在二元樹左臂和右臂上的點作旋轉。對於二元樹的轉換，在左臂上做左旋和右旋及右臂上做右旋和左旋能提供快速且有效的轉換，我們改進轉換任意兩棵ｎ點二元樹之演算法，且能在至多(n-1)的時間來轉換兩棵ｎ點二元樹的左權重序列和右權重序列。特別的是對一序列的旋轉以合計方法分析之，每一旋轉能在一常數攤還時間完成。除了ｎ點二元樹的轉換距離之外，利用左臂上左旋和右旋及右臂上右旋和左旋，可以建構出所有ｎ點二元樹之間的旋轉圖Gn，利用在右臂上左旋，從Gn我們可以建構出一棵特殊的旋轉樹Tn，當走訪此旋轉樹Tn時，我們可以辭彙編纂的方式列舉出所有ｎ點二元樹的左權重序列，也就是列舉出所有ｎ點二元樹。

我們對ｎ個點多元樹用一簡單的右距離序列表示之，其和 Zaks 所提出的Z序列存在著密切關係。運用多元遞迴樹及其伴隨關係表使我們在ｎ點多元樹作有系統的研究。因此，在辭彙編纂方式下，決定一ｎ點多元樹的排列次序及在轉換一正整數為其相對應的ｎ點多元樹，我們皆有效地改進次序演算法及反次序演算法。這兩演算法毋需建立任何關係表皆能在O(tn)時間內被執行完。

此外，利用右距離序列表示，我們也提出一無迴圈演算法可以列舉所有ｎ點多元樹，此一無迴圈演算法只需要3n+O(1)記憶空間並且比現存的演算法更有效率。由於它在右距離序列所表示的Gray-code圖上列舉所有ｎ點多元樹過程就是哈密爾敦路徑。最後，我們在右距離序列所表示的Gray-code圖上找到了哈密爾敦迴路。

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 right-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.
Then, we use 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-formated sequences suggested by Zaks [Theoret. Comput. Sci.
10 (1980) 63-82]. Using a t-ary recursion 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., the ranking algorithm), and for converting a positive integer to its corresponding RD-sequence (i.e., the unranking algorithm). Both the ranking and unranking algorithms can be run in O(tn) time and without really building any auxiliary table.
In addition, we also present a loopless algorithm to enumerate Gray-codes of t-ary trees using RD-sequences. The proposed algorithm requires 3n+O(1) memory space and is more efficient than all the existing algorithms. Because it fits a Hamiltonian path P in RDtn, we find that there exists a Hamiltonian
cycle in RDtn.

Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Rotation Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 Enumerating, Ranking and Unranking Problems . . . . . . . . . . . . . . . . . 5
4 Loopless Generating t-ary Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . .  7
5 Organization of this Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Tree Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . 9
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 LeftWeight Sequences and RightWeight Sequences . . . . . . . .. . . . . . . 11
3 Left-arm and Right-arm Rotations . . . . . . . . . . . . . . . . . .. . . . . . . . . 15
4 A Tree Transformation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5 An Improved Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23
Enumerations of Binary Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26
1 A-order, B-order and Related Representations . . . . . . . . . . . . . . . . . 26
2 Enumerated Sequences of Binary Trees . . . . . . . . . . . . . . . . . . . . . . . 38
3 Enumerate Binary Trees by Using the Rotation Tree . . . . . . . . . . . . . .39
Ranking and Unranking of t-ary Trees . . . . . . . . . . . . . . . . . . . . . . . . . 43
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2 Structural Representations of t-ary Trees . . . . . . . . . . . . . . . . . . . . . 44
3 A Ranking Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55
4 An Unranking Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5 Enumerate Sequences of t-ary Trees. . . . . . . . . . . . . . . . . . . . . . . . . 58
A Loopless Algorithm to Generate Gray-codes of t-ary Trees . . . . . .. . 61
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2 RD-representation Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63
3 Loopless Gray-code Generation for t-ary Trees. . . . . . . . . . . . . . . . . 67
4 Hamiltonian RD-representation Graphs . . . . . . . . . . . . . . . . . . . . . . .73
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .77
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .81

List of Tables
1 A-order and B-order definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2 left weight sequence and left distance sequence . . . . . . . . . . . . . . . . 37
1 The 3-ary recursion table for n = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2 The 3-ary accumulation table for n = 6 . . . . . . . . . . . . . . . . . . . . . . . 52
1 A summary of loopless algorithms for generating sequences of
binary trees and t-ary trees with n internal nodes . . . . . . . . . . . . . . . . . . .63
2 The RD-representation graph RD3n for all n-node 3-ary trees . . . . . .66
1 The diameter and the average rotation distance for various types
of rotation graph Gn with n = 3, 4, 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

List of Figures
1 The corresponding representations including triangulations of a 5-gon,
the ways to parenthesize the product x1x2x3x4, 3-node binary trees,
3-pair balanced parentheses strings, and all X-sequences of 3-node
binary trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 A left or right rotation at node x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4
1 The LW-sequence and the RW-sequence of a tree T. . . . . . . . . . . . . . . 12
2 The left-arm and right-arm rotations. . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 An example of tree transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1 Three binary trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27
2 A 9-node binary tree and its X-sequence. . . . . . . . . . . . . . . . . . . . . . . 28
3 The X-sequences of all 4-node binary trees. . . . . . . . . . . . . . . . . . . . . 29
4 The Z-sequences and Y -sequences of all 4-node binary trees. . . . . . . 29
5 The P-sequences and level sequences of all 4-node binary trees. . . . . .30
6 The permutations of all 4-node binary trees. . . . . . . . . . . . . . . . . . . . . 31
7 The inversion tables of all 4-node binary trees. . . . . . . . . . . . . . . . . . . 32
8 The left weight sequences and L-sequences of all 4-node binary trees. 33
9 The tree permutations and A-sequences of all 4-node binary trees. . . . 34
10 The node number sequences of all 4-node binary trees. . . . . . . . . . . . 34
11 The inorder left weight sequences and left distance sequences of
all 4-node binary trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35
12 The rotation tree T4 whose nodes in each level are labeled in
lexicographic order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1 The 3-ary trees with 3 internal nodes. . . . . . . . . . . . . . . . . . . . . . . .. . . 45
2 A 4-ary tree T such that every internal node i 2 T is labeled by di/zi . . .46
3 The recursion tree, RD-sequences, Z-sequences, and the rank of 3-ary trees .48
4 The label for each node corresponds to the number of leaves in the
subtree rooted at the node of 3-ary recursion tree. . . . . . . . . . . . . . . . . . . 49
5 The sum of labels corresponded by the path (d1, d2, d3) is
Rank(d1, d2, d3) in the 3-ary recursion tree (n = 3) . . . . . . . . . . . . . . . . . . 54
1 A trinary recursion tree and a Gray-code graph RD33 . . . . . . . . . . . . . .64
2 A trinary recursion tree and a Gray-code graph Z33 . . . . . . . . . . . . . . . 67
3 A Hamiltonian path in the graph RD34 . . . . . . . . . . . . . . . . . . . . . . . . . 68
4 A listing of RD-sequences for 3-ary trees with 4 internal nodes
in Gray-code order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69
5 The flowchart of procedure NextRD() . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6 A binary recursion tree, a corresponding Hamiltonian path and
a Hamiltonian cycle in RD25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75
7 A trinary recursion tree, two corresponding paths P1 and P2 and
a Hamiltonian cycle in RD34. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

                                

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