計算排列中值的改良式分支及限制演算法｜國立清華大學博碩士論文庫

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研究生：	張成鈺 Cherng-Yuh Chang
論文名稱：	計算排列中值的改良式分支及限制演算法 Improved Branch And Bound Algorithm For Computing Median Permutation By Block-Interchange
指導教授：	唐傳義 Chuan-Yi Tang
口試委員:
學位類別：	碩士 Master
系所名稱：	電機資訊學院 - 資訊工程學系 Computer Science
論文出版年：	2005
畢業學年度：	93
語文別：	英文
論文頁數：	38
中文關鍵詞：	區塊交換、中值、排列、分支及限制
外文關鍵詞：	block-interchange, median, permutation, branch and bound
相關次數：	點閱：3 下載：0
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摘要

生物演化的過程是奇妙的，長久以來人們為了解開生命的奧祕而發展了許多的理論和技術。演化樹訴說了物種演進的過程，我們以基因或蛋白質序列的操作距離為作為依據，建立物種之間可能的演化樹，進而分析物種之間演化的關係。發生在序列的操作(operation)有許多種，包括插入(insertion)，刪除(deletion)，取代(substitution)，乃至於較大片斷(large fragment)的操作有反轉(reversal)，置換(transposition)，易位(translocation)，區塊交換(block-interchange)等多種。建立演化樹其重要的一環是找出物種之間共同的祖先(ancestor)，對應於演算法即中值問題(median problem)。以反轉操作為基礎的中值問題已被證明屬於NP-Complete，面對較新的區塊交換操作，中值問題尚未被證明其問題的複雜度，乃至於此，我們改善C. Siepel等人在反轉操為基礎的中值問題所提出的分支及限制演算法(branch and bound)架構，在搜尋的過程中憑藉Cycle-Graph與Breakpoints的特性，試圖先找尋那些共同位在兩段路徑上的點，並使用在區塊操作的中值問題，發現其效率比僅使用單純的分支及限制演算法還要好。

Abstract
DNA, RNA and protein sequences have helped us to understand the evolutionary relationship between organisms. There are various operations in genome rearrangement problem, including insertions, deletions, substitutions, reversals, transpositions, translocations, transreversals, fissions, fusions, and block-interchange.
Dobzhansky and Sturtevant [10] first proposed that degree to gene orders differ between species could be useful to phylogenetic inference in 1938. One of the important problems is median problem. The inversion median was proved NP-Complete problem, as to block-interchange median is a NP-Complete is still open. We use the same branch-and-bound architecture proposed by Adam C. Siepel et al. and improve it to solve the block-interchange median with the two properties associated with cycle-graph and common breakpoints. We get a better result with improved branch-and-bound algorithm than pure branch-and-bound architecture algorithm in dealing with block-interchange median problem.

Contents
Abstraction..................................................................I
Contents....................................................................II
List of Figures............................................................III
List of Tables..............................................................IV
1       Introduction.........................................................1
        1.1     History......................................................1
        1.2     Block-Interchange............................................4
        1.3     Block-Interchange Median.....................................4

2    Median problem and Branch and Bound Algorithm........................5
        2.1     Median Problem Definitions...................................5
        2.2     General median bounds with triangular inequality.............7
        2.3     Architecture of Branch and Bound Algorithm...................8
                2.3.1   Lower bound and Upper bound..........................8
        2.4     Branch and Bound Algorithm..................................10
                2.4.1   Normalization.......................................11
                2.4.2    Concept.............................................11

3       Improved Branch and Bound Algorithm.................................12
        3.1     Common Breakpoints..........................................12
        3.2     Invariant Elements..........................................16
                3.2.1   Circular permutation................................16
                3.2.2   Common block-interchange path on shortest path by
                        invariant elements..................................17
        3.3     Pruned Section..............................................22
        3.4     Improved Branch and Bound Algorithm for finding median......24

4       Comparison and Experiment...........................................26
        4.1     Comparison..................................................26
                4.1.1   Len 12..............................................26
                4.1.2   Len 15..............................................31
        4.2     Conclusion..................................................34
        4.3     Future work.................................................34

        Reference...........................................................35

List of Figures

        Figure1.2.1     Block-Interchange with parameters....................4
        Figure2.1.1     Shortest Path Median.................................6
        Figure2.1.2     Cycle-Graph..........................................7
        Figure2.3.1     Lower and Upper bounds...............................9
        Figure3.1.1     alpha-type cycle graph..............................14
        Figure3.1.2     bata1-type cycle graph..............................14
        Figure3.1.3     bata2-type cycle graph..............................14
        Figure3.2.1     Example.............................................19
        Figure3.3.1     Pruned section......................................23
        Figure4.1.1     Len 12 visited of pure of all.......................26
        Figure4.1.2     Len 12 visited of pure less than 10,000.............27
        Figure4.1.3     Len 12 visited of improved of all...................27
        Figure4.1.4     Len 12 visited of improved less than 10,000.........28
        Figure4.1.5     Len 12 ratio of all.................................28
        Figure4.1.6     Len 12 ratio less than -1...........................29
        Figure4.1.7     Len 12 ratio ranges 1~200...........................29
        Figure4.1.8     Len 12 push of pure.................................30
        Figure4.1.9     Len 12 push of improved.............................30
        Figure4.1.10    Len 12 ratio of push................................30
        Figure4.1.11    Len 15 visited of pure of all.......................31
        Figure4.1.12    Len 15 visited of pure less than 10,000.............31
        Figure4.1.13    Len 15 visited of improved of all...................31
        Figure4.1.14    Len 15 visited of improved less than 10,000.........31
        Figure4.1.15    Len 15 ratio of all 1% equal........................32
        Figure4.1.16    Len 15 ratio 36% less than -1.......................32
        Figure4.1.17    Len 12 ratio 52% ranges 1~500.......................32
        Figure4.1.18    Len 15 push of pure.................................33
        Figure4.1.19    Len 15 push of improved.............................33
        Figure4.1.20    Len 15 ratio of push................................33

List of Tables

        Table 2.4       Branch And Bound Algorithm..........................10
        Table 3.1       Finding permutation on common shortest path by
                        common breakpoints..................................15
    Table 3.2       Finding common block-interchange operations by
                        Invariants..........................................20
    Table 3.4       Improved Branch and Bound Algorithm for finding
                        median..............................................25

                                

Reference
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