An Integer Representation Problem with Application to Postage Stamp Problem

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研究生：	鄭延琮 Cheng, Yen-Tsung
論文名稱：	An Integer Representation Problem with Application to Postage Stamp Problem 一個可應用於連續郵資問題的整數表示問題
指導教授：	李端興 Lee, Duan-Shin
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：	2009
畢業學年度：	97
語文別：	英文
論文頁數：	83
中文關鍵詞：	連續整數、演算法
外文關鍵詞：	integer representation
相關次數：	點閱：53 下載：0
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Postage stamp problem is a famous, classic, and old problem which appears on
many books related to algorithm and programming skill . Here is its' description:
There are N kinds of stamps whose value to be printed on them is still unknown;
the space on each envelope is limited, and there are only k stamps can be attached on
it. How to print the value on each kind of stamps to allow the widest choice of value
to be made up from $1?
Note that Postage stamp problem can be viewed as a maximum consecutive in-
teger representation problem. Except for brute force algorithm, there is no explicit
algorithm to solve Postage stamp problem so far. However, if we specify the or-
der of choosing stamps to be attached when we want to make up some value on
the envelope, this problem can be solve by a greedy algorithm whose complexity is
O((N + k)minfN;kg). We then proof it can actually nd the optimal solution. Besides
Postage stamp problem, this greedy algorithm also have some other applications in-
cluding building optical queues, node symmetric chordal ring

連續郵資問題是一個在許多介紹演算法與程式技巧中常出現有
名、經典、古老的問題。問題的敘述如下：
有N 種郵票，面額還尚待確定。又因為信封上的空位有限，每封
信封上只能貼最多k 張郵票，那麼如何去選定這N 張郵票的面額，使
得我們能夠在信封上從1 塊錢開始連續貼出來的金額能夠最大?
連續郵資問題可以被視為一個最大連續整數表現問題。目前除了
暴力演算法(brute force algorithm)並無明確的演算法可以解決該問題。
然而如果限定郵票被選取的順序，我們在這裡將會介紹一種時間複雜
度為O((M+k)min{M, k})的貪婪演算法(greedy algorithm)間以解決郵票問題。
我們也將會證明該演算法的正確性，並且說明一些其他的應用：如建
造某些光佇列(optical queue)、點對稱弦環(node symmetric chordal
ring)。

ABSTRACT : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : ii
LIST OF TABLES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : vi
LIST OF FIGURES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : vii
Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1
Problem description : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4
The greedy construction : : : : : : : : : : : : : : : : : : : : : : : : : : : 8
Proofs : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11
1 Properties of B(dM
; k) for non-decreasing sequence : : : : : : : : : 11
2 Optimality of the greedy construction : : : : : : : : : : : : : : : : : 31
3 Optimality in non-decreasing sequence : : : : : : : : : : : : : : : : 47
Building the sequence with the greedy construction in O(N + k) time : : 54
Related problems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 63
1 Optical queues : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 63
2 Node symmetric chordal ring : : : : : : : : : : : : : : : : : : : : : : 71
Numerical results : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 77
Conclusion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 79
LIST OF REFERENCES : : : : : : : : : : : : : : : : : : : : : : : : : : : : 80

                                

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