給定一條字串 S = {a1, a2, …, an}，最長遞增子序列問題是要找出該字串的子序列中遞增且長度為最長的，在之前的成果中，ㄧ條長度為n的字串中每個固定長度為w的滑動窗的最長遞增子序列可以在O(w log log n + OUTPUT) 時間內找出，其中O(w log log n + w2) 的時間是花在前處理，OUTPUT是所有輸出的東西的長度和，在這篇論文中我們解決了找尋字串S中每個子字串之最長遞增子序列的問題。 若是直接套用前人的演算法，前處理的時間為O(n3)，我們修改該演算法所用的資料結構，致使前處理的時間改進為O(n2)，輸出階段的時間和輸出的長度成正比，換句話說，我們的演算法可以在O(n2+OUTPUT)時間內找出每個子字串的最長遞增子序列。對於要輸出所有子字串的最長遞增子序列的情況，因為共有O(n2)條子序列，每條長度都是O(n)，我們的演算法已是最佳。

Given a string S = {a1, a2, a3, ..., an}, the longest increasing subsequence (LIS) problem is to find a subsequence of the given string such that the subsequence
is increasing and its length is maximal. In a previous result, to find the longest increasing subsequences of each sliding window with a fixed size w of a given string
with length n can be solved in O(w log log n+OUTPUT) time, where O(w log log n+ w^2) time is taken for preprocessing and OUTPUT is the sum of all output lengths. In this thesis, we solve the problem for finding the longest increasing subsequence of every substring of S. With the straightforward implementation of the previous result, the time required for the preprocessing would be O(n^3). We modify the data structure used in the algorithm, hence the required preprocessing time is improved to O(n^2). The time required for the report stage is linear to the size of the output. In other words, our algorithm can find the LIS of every substring in O(n^2+OUTPUT) time. If the LIS's of all substrings are desired to be reported, since there are O(n^2) substrings totally in a given string with length n, our algorithm is optimal.

ABSTRACT . . . 0
Chapter 1. Introduction . . . 1
Chapter 2. Previous Results . . . 3
Chapter 3. A Previous Result on LIS in Sliding windows . . . 6
Chapter 4. Our Algorithm for Finding LIS of Every Substring . . . 15
Chapter 5. Implementation Details . . . 19
Chapter 6. Conclusion and Future Work . . . 25
BIBLIOGRAPHY . . . 27
APPENDIXES . . . 30

[1] M. H. Albert, A. Golynski, A. M. Hamel, A. Lopez-Ortiz, S. S. Rao, and M. A. Safari, "Longest increasing subsequences in sliding windows," Theoretical Computer Science, pp. 413-432, 2004.
[2] D. Aldous and P. Diaconis, "Longest increasing subsequences: From patience sorting to the baik-deift-johansson theorem," BAMS: Bulletin of the American Mathematical Society, Vol. 36, pp. 413-432, 1999.
[3] S. Bespamyatnikh and M. Segal, "Enumerating longest increasing subsequences and patience sorting," Information Processing Letters, Vol. 76, No. 1-2, pp. 7-11, 2000.
[4] G. S. Brodal, K. Kaligosi, I. Katriel, and M. Kutz, "Faster algorithms for computing longest common increasing subsequences," Tech. Rep. BRICS-RS-05-37, BRICS, Department of Computer Science, University of Aarhus, Dec. 2005.
[5] W. T. Chan, Y. Zhang, S. P. Y. Fung, D. Ye, and H. Zhu, "Efficient algorithms for finding a longest common increasing subsequence," The 16th Annual International Symposium on Algorithms and Computation, Hainan, China, pp. 665-674, 2005.
[6] M. S. Chang and F. H. Wang, "Efficient algorithms for the maximum weight clique and maximum weight independent set problems on permutation graphs," Information Processing Letters, Vol. 43, No. 6, pp. 293-295, 1992.
[7] A. L. Delcher, S. Kasif, R. D. Fleischmann, J. Peterson, O. White, and S. L. Salzberg, "Alignment of whole genomes," Nucleic Acids Research, Vol. 27, No. 11, pp. 2369-2376, 1999.
[8] W. L. Hsu, "Maximum weight clique algorithms for circular-arc graphs and circle graphs," SIAM Journal on Computing, Vol. 14, No. 1, pp. 224-231, 1985.
[9] J. W. Hunt and T. G. Szymanski, "A fast algorithm for computing longest common subsequences," Communications of the ACM, Vol. 20, No. 5, pp. 350-353, 1977.
[10] I. Katriel and M. Kutz, "A faster algorithm for computing a longest common increasing subsequence," Research Report MPI-I-2005-1-007, Max-Planck-Institut fur Informatik, Stuhlsatzenhausweg 85, 66123 SaarbrÄucken, Germany, Mar. 2005.
[11] H. Kim, "Finding a maximum independent set in a permutation graph," Information Processing Letters, Vol. 36, No. 1, pp. 19-23, 1990.
[12] D. T. Lee and M. Sarrafzadeh, "Maximum independent set of a permutation graph in k tracks," International Journal of Computational Geometry and Applications, Vol. 3, No. 3, pp. 291-304, 1993.
[13] D. Liben-Nowell, E. Vee, and A. Zhu, "Finding longest increasing and common subsequences in streaming data," 11th International Computing and Combinatorics Conference, Kunming, China, pp. 263-272, 2005.
[14] E. MÄakinen, "On the longest upsequence problem for permutations," Tech. Rep. A-1999-7, Department of Computer Science, University of Tampere, 1999.
[15] F. Malucelli, T. Ottmann, and D. retolani, "Efficient labelling algorithms for the maximum noncrossing matching problem," Discrete Applied Mathematics, Vol. 47, No. 2, pp. 175-179, 1993.
[16] Y. Sakai, "A linear space algorithm for computing a longest common increasing subsequence," Information Processing Letters, Vol. 99, No. 5, pp. 203-207, 2006.
[17] C. Schensted, "Longest increasing and decreasing subsequences," Canadian Journal of Mathematics, Vol. 13, pp. 179-191, 1961.
[18] P. van Emde Boas, R. Kaas, and E. Zijlstra, "Design and implementation of an efficient priority queue, "Mathematical Systems Theory, Vol. 10, pp. 99-127, 1977.
[19] C. B. Yang and R. C. T. Lee, "Systolic algorithms for the longest common subsequence problem.," Journal of the Chinese Institute of Engineers, Vol. 10, No. 6, pp. 691-699, 1987.
[20] I. H. Yang, C. P. Huang, and K. M. Chao, "A fast algorithm for computing a longest common increasing subsequence.," Information Processing Letters, Vol. 93, No. 5, pp. 249-253, 2005.
[21] M. S. Yu, L. Y. Tseng, and S. J. Chang, "Sequential and parallel algorithms for the maximum-weight idependent set problem on permutation graphs," Information Processing Letters, Vol. 46, No. 1, pp. 7-11, 1993.