组合算法的研究与实现
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摘要
伴随着21世纪信息社会的到来,计算机的应用已渗透到了社会生活的各个领域,其基本实现从根本上说是通过软件来完成的,而软件开发与设计的核心则是算法的设计与实现,最终,组合算法又构成了计算机算法的中心内容。纵观全世界的经济与软件产业的发展情况,美国均处于霸主地位。从中,我们可以间接地看到组合算法的重要性,不仅在当今,将来更是如此,实践已经证明了这一点。
本文对组合算法进行了初步的研究,首先分析了该课题的历史背景及研究意义;其次提出了棋盘多项式的非递归算法,并与其它算法进行了比较,在速度上提高了3～5倍;再次是对组合数学中的常用算法:错位排列、正整数的分拆、排列组合三个算法进行了研究,并做了相应的改进和比较;随后对上述所研究的算法加以实现,同时进行了集成,使之成为一个小型的组合算法软件;最后对所研究的算法进行了总结,并针对难点提出了今后的努力方向。
综上所述,对组合算法的研究具有理论意义和实用价值,尤其是本文提出的棋盘多项式的非递归算法,较之递归算法不仅在速度上有了明显的提高,而且在给出总方案数的同时,也给出了具体的排列方案,操作方便、快捷。
With the twenty-first century's coming of the information society, the computers have already apply to many kinds of areas in the society , on the whole, which is basically completed by the software, at the same time, the core of the software development and designment is the designment and realization of the arithmetics, finally, combination arithmetics constitute the core of the computer arithmetics. From the development state of the economy and the software, America is on the hegemony state. By it, we can indirectly see the importance of the combination arithmetics, not only the nowadays, but also the future, which have testified by the practice.
The thesis researchs several basic arithmetics of the combination arithmetics. Firstly, it analyzes the historical background and the meaning of research; Secondly, it puts forward the non- recursive algorithm on the chessBoard polynomial, and compares with the other arithmetics, as a result, the non- recursive algorithm improves three to five times on the speed; Thirdly, it researchs three other basic arithmetics of the combinatorics: derangement, partition of the positive integer, arrangement and combination, makes some betterments and compare simultaneously; Then, it realizes and integrates the above arithmetics, which are formed a software of the combination arithmetics; Finally, it summarizes all the researchs, and provides the next goals aiming at the current problems.
To sum up, the research on the combination arithmetics has the academic significance and the applicability, especially to the the non- recursive algorithm on the chessBoard polynomial, which not only has a obvious improvement on the speed, but also gives the total and the material scheme at the same time, all of them make the operation more conveniently and fast.
本文对组合算法进行了初步的研究,首先分析了该课题的历史背景及研究意义;其次提出了棋盘多项式的非递归算法,并与其它算法进行了比较,在速度上提高了3～5倍;再次是对组合数学中的常用算法:错位排列、正整数的分拆、排列组合三个算法进行了研究,并做了相应的改进和比较;随后对上述所研究的算法加以实现,同时进行了集成,使之成为一个小型的组合算法软件;最后对所研究的算法进行了总结,并针对难点提出了今后的努力方向。
综上所述,对组合算法的研究具有理论意义和实用价值,尤其是本文提出的棋盘多项式的非递归算法,较之递归算法不仅在速度上有了明显的提高,而且在给出总方案数的同时,也给出了具体的排列方案,操作方便、快捷。
With the twenty-first century's coming of the information society, the computers have already apply to many kinds of areas in the society , on the whole, which is basically completed by the software, at the same time, the core of the software development and designment is the designment and realization of the arithmetics, finally, combination arithmetics constitute the core of the computer arithmetics. From the development state of the economy and the software, America is on the hegemony state. By it, we can indirectly see the importance of the combination arithmetics, not only the nowadays, but also the future, which have testified by the practice.
The thesis researchs several basic arithmetics of the combination arithmetics. Firstly, it analyzes the historical background and the meaning of research; Secondly, it puts forward the non- recursive algorithm on the chessBoard polynomial, and compares with the other arithmetics, as a result, the non- recursive algorithm improves three to five times on the speed; Thirdly, it researchs three other basic arithmetics of the combinatorics: derangement, partition of the positive integer, arrangement and combination, makes some betterments and compare simultaneously; Then, it realizes and integrates the above arithmetics, which are formed a software of the combination arithmetics; Finally, it summarizes all the researchs, and provides the next goals aiming at the current problems.
To sum up, the research on the combination arithmetics has the academic significance and the applicability, especially to the the non- recursive algorithm on the chessBoard polynomial, which not only has a obvious improvement on the speed, but also gives the total and the material scheme at the same time, all of them make the operation more conveniently and fast.
引文
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[2]徐利治,组合数学的发展趋势及关于发展研究的建议,曲阜师范大学学报,1994,20(3):1-7
[3]徐利治,对新世纪数学发展趋势的一些展望,高等数学研究,2001,4(3):4-6
[4]陈永川,组合数学及其应用的研究进展,中国科协首届学术年会,1999.10,杭州
[5]党亚茹,刘青,组合数学文献计量分析,世界图书,1992,12:10-15
[6]周闻,在奋进中崛起--记南开大学组合数学研究中心,中国基础科学·科研基地,2004,3:30-33
[7]William Y.C.Chen,Yu P.Deng and Laura L.M.Yang.Motzkin paths and reduced decompositions for permutations with forbidden patterns.Elect.J.Combin,2003:1-13
[8]William Y.C.Chen and James J.Y.Zhao.The gaussian coefficients and overpartitions.Discrete Mathematics,2005,305:1-5
[9]William Y.C.Chen,Guo-Guang Yah and Arthur L.B.Yang.The skew Schubert polynomials.Europ.J.Combinatorics,2004,25:1181-1196
[10]William Y.C.Chen,Wenchang Chu and Nancy S.S.Gu.Finite form of the quintuple product identity.J.Combin.,Theory Set.A,2006,113:185-187
[11]William Y.C.Chen and David C.Torney.Equivalence Classes of Matchings and Lattice-Square Designs.Discrete Applied Math.,2005,145:349-357
[12]胡冠章,组合计数的群论与计算机方法,数学进展,1997,26(1):2-13
[13]徐利治,对新世纪数学发展趋势的一些展望(续),高等数学研究,2001,4(4):2-4
[14]王若鹏,无约束优化的一个组合算法,兰州理工大学学报,2004,30(5):143-145
[15]韩毅,一种新的NBP网络增强组合算法,浙江大学学报(理学版),2004,4(11):44-47
[16]杨益民,一类具约束选址模型的组合算法,应用数学,2003,3(12):72-76
[17]陈光柱,组合查询的组合算法,计算机工程与应用,2003,33(64):197-198
[18]柴山等,离散变量结构优化设计的组合算法,应用数学和力学,1997,9(4):789-797
[19]钟珞,最佳并行排列组合算法,微机发展,1991,1(7):29-31
[20]Karp,Richard M.COMBINATORICS,COMPLEXITY,AND RANDOMNESS.Communications of the ACM,1986,29(2):98-109
[21]Fiore,Marcelo P.Mathematical models of computational and combinatorial structures.Lecture Notes in Computer Science,2005,3441:25-46
[22]Assmus,E.F.Jr.,Mattson,H.F.Jr.CODING AND COMBINATORICS.SIAM Review,1974,16(3):349-388
[23]de Werra,D.Combinatorics of timetabling.European Journal of Operational Research,1997,96(3):504-513
[24]Chazelle,Bernard;Edelsbrunner,Herbert;Guibas,Leonidas J.;Sharir,Micha.Lines in space - combinatorics,algorithms and applications.Proc Twenty First Annu ACM Symp Theory Comput,1989:382-393
[25]Bretto,A.;Azema,J.;Cherifi,H.;Laget,B.Combinatorics and image processing.Graphical Models and Image Processing,1997,59(5):265-277
[26]Koerner,J.;Marton,K.COUNTING IN COMBINATORICS AND GPAPH ENTROPY.1986 IEEE International Symposium on Information Theory,1986,51
[27]Pap,Gyula.A combinatorial algorithm to find a maximum even factor.Lecture Notes in Computer Science,v 3509,Integer Programming and Combinatorial Optimization:11th International IPCO Conference.Proceedings.2005,66-80
[28]高飞,王光兴,桥网络及无线通信网络的可靠性研究,博士学位论文,2003
[29]孔繁甲,王光兴,基于容斥原理与不交和公式的一个计算网络可靠性方法,电子学报,1998,11(28):117-119
[30]戴琼,罗铸锴,布尔代数上的自动机理论,硕士学位论文,2000
[31]吴国柱,郝端绪,容斥原理在组合数学中的若干应用,唐山师范学院学报,1994,6(8):23-25
[32]陈碧琴,容斥原理在数论中的应用实例,绵阳师范学院学报,2004,2(7):26-29
[33]徐学文,容斥原理,数学通讯,1997,4(17):40-42
[34]Kaniadakis,G.;Quarati,P.;Scarfone,A.M.Nonlinear canonical quantum system of collectively interacting particles via an exclusion-inclusion principle.Physical Review E.Statistical Physics,Plasmas,Fluids,and Related Interdisciplinary Topics,1998,58(5-A):5574-5585
[35]William Y.C.Chen:Gian-Carlo Rota.q-Analogs of the principle of inclusion--exclusion and permutations of restricted position.Discrete Mathematics,1992,104:7-22
[36]谢云鹏,谢孔彬,一类限位数排列的计数问题,山东理工大学学报(自然科学版),2003,3(16):60-62
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