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研究生： 顏志瑜
Chih-yu Yen
論文名稱： 穩定婚姻及穩定配偶的模擬
The Simulation of Stable Marriage and Stable Pairs
指導教授： 于振華
Jenn-hwa Yu
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
畢業學年度： 98
語文別： 中文
論文頁數： 60
中文關鍵詞： 穩定婚姻 、穩定配偶
外文關鍵詞： stable marriage, stable pairs
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    • 本文主要在於探討穩定配對中的穩定婚姻問題，提及穩定男伴、拆散配對以及P循環的概念，重新推導 Knuth, Motwani 與 Pittel 的結論:
    對任何女生的穩定男伴為至少 max(0,(1/2-epsilon)lnn ，最多為 (1+epsilon)lnn 個的機率趨近 1，當n → ∞ 時，其中 0< varepsilon <1 。並介紹兩種方法找到所有的穩定配對，透過 C++ 模擬結果比較各方法的優缺點，並與理論值作比對。

    In this paper we study the stable marriage and stable husbands problems of stable matching, using the concept of breakmarriage and p-cycle, and revisit the result of Knuth ,Motwani and, Pittel :
    any particular girl has at least max(0, (1/2−epsilon) ln n) and at most (1+epsilon) ln n different husbands, with probability approaching 1 as n → ∞, if 0 < epsilin < 1. We introduce two methods to find all stable matchings and simulate in C++ programming language to compare these two methods and theoretical results.

    1. 簡介................................................1 2. 穩定配對 ...........................................2 2.1 穩定男伴........................................2 2.2 隨機模型........................................6 2.3 機率性定理......................................7 3. 拆散配對...........................................23 3.1 基本觀念.......................................23 3.2 所有穩定配對...................................25 4. P循環..............................................30 4.1 所有穩定配偶...................................30 4.2 演算法.........................................31 4.3 利用 P 循環架構 G..............................34 4.4 樹狀圖 T.......................................37 5. 結論...............................................39 參考文獻..............................................40 附錄一................................................41 附錄二................................................44 附錄三................................................46 附錄四................................................50

    [1] D. Gale and L. S. Shapley (1962), College admissions and the stability of marriage, Am. Math. Monthly, 69, 9-15.
    [2] R. L. Graham, D. E. Knuth, and O.Patashnik (1989), Concrete Mathematics, AddisonWesley, Reading, MA.
    [3] D. Gusfield (1987), The fast algorithms for four problems in stable marriage, SIAM J. Comput. 16, 111-128.
    [4] D.E.Knuth (1976), Marriages Stable et leurs relations avec d''autres problemes combinatoires, Les Presses de l''Universite de Montreal, Montreal.
    [5] D.E. Knuth (1988), Personal Communication .
    [6] D.E. Knuth and R. Motwani and B. Pittel (1990), Stable husbands, Random Structures and Algorithms, Vol.1, No.1.
    [7] D. G. McVitie and L. B. Wilson(1971), The stable marriage problem, Commun. ACM, 14 , 486-492.
    [8] B. Pittel (1989), The average number of stable matchings, SIAM J. Discr. Math., to appear.

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