行程編碼字串間最長公共子字串的次線性時間量子演算法

簡易檢索 / 詳目顯示

回結果列表

研究生：	李子敬 Lee, Tzu-Ching
論文名稱：	行程編碼字串間最長公共子字串的次線性時間量子演算法 A Sublinear Time Quantum Algorithm for Longest Common Substring Problem between Run-length Encoded Strings
指導教授：	林瀚仚 Lin, Han-Hsuan
口試委員:	韓永楷 Hon, Wing-Kai 賴青沂 Lai, Ching-Yi
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：	2023
畢業學年度：	112
語文別：	英文
論文頁數：	38
中文關鍵詞：	最長公共子字串、量子演算法、次線性時間演算法、行程編碼
外文關鍵詞：	longest common substring, quantum algorithm, sublinear time algorithm, run-length encoding
相關次數：	點閱：56 下載：0
分享至:	分享至facebook 分享至twitter

查詢本校圖書館目錄查詢臺灣博碩士論文知識加值系統勘誤回報

在本論文中，我們研究輸入為「行程編碼」字串的「最長公共子字串」問題。在行程的前綴和已知的前提下，我們給出了一個次線性（Õ(n^(5/6))O(polylog(ñ))）時間的量子演算法，其中 n 與 ñ 分別是編碼後與編碼前的字串長度。此外，我們證明若前綴和未知，輸入為行程編碼字串的最長公共子字串問題有近線性的查詢複雜度下界 Ω(n/log^2n) 。

In this thesis, we study the Longest Common Substring (LCS) problem with run-length encoded (RLE) inputs. Assuming the prefix-sum of the runs are given as extra oracles, we give a sublinear Õ(n^(5/6))O(polylog(ñ)) time quantum algorithm, where n and ñ are the encoded and decoded length of the input strings, respectively. Additionally, we demonstrate a near linear Ω(n/log^2n) query lower-bound on finding the LCS with RLE inputs without access to the prefix-sum oracle.

摘要 ...................................................  i
Abstract ............................................... ii
Introduction ........................................  1
1. Related work ......................................  3
2. Overview of the algorithm .........................  4
3. Paper organization ................................  5
Preliminaries .......................................  6
1. Conventions and Notations .........................  6
2. Computation Model .................................  7
3. Primitives ........................................  7
4. Deﬁnitions ........................................  8
LCS from two RLE strings with Preﬁx-sum Oracles ..... 11
1. Inverse Preﬁx-Sum ................................. 12
2. Algorithm for short answers ....................... 12
2.1. Quantum walk search ............................. 14
2.2. Proof of Theorem 16 (Data structure) ............ 17
3. Algorithm for long answers ........................ 23
3.1. Good pairs ...................................... 23
3.2. Search for a long answer from a good pair ....... 25
4. Putting things together ........................... 28
Lower Bounds ........................................ 30
1. Lower Bound on 𝖣𝖫-𝖫𝖢𝖲-𝖱𝖫𝖤 ......................... 30
2. Lower Bounds on 𝖤𝖫-𝖫𝖢𝖲-𝖱𝖫𝖤 and 𝖫𝖢𝖲-𝖱𝖫𝖤 ............ 31
Bibliography ........................................... 35

                                

[1] D. E. Knuth, J. H. Morris Jr., and V. R. Pratt, “Fast Pattern Matching in Strings”, SIAM J. Comput., no. 2, pp. 323–350, 1977, doi: 10.1137/0206024.
[2] R. S. Boyer and J. S. Moore, “A Fast String Searching Algorithm”, Commun. ACM, no. 10, pp. 762–772, 1977, doi: 10.1145/359842.359859.
[3] R. M. Karp and M. O. Rabin, “Eﬃcient Randomized Pattern-Matching Algorithms”, IBM J. Res. Dev., no. 2, pp. 249–260, 1987, doi: 10.1147/rd.312.0249.
[4] P. Weiner, “Linear Pattern Matching Algorithms”, IEEE Computer Society, 1973, pp. 1–11. doi: 10.1109/SWAT.1973.13.
[5] M. Farach, “Optimal Suﬃx Tree Construction with Large Alphabets”, IEEE Computer Society, 1997, pp. 137–143. doi: 10.1109/SFCS.1997.646102.
[6] E. Fredkin, “Trie memory”, Commun. ACM, no. 9, pp. 490–499, 1960, doi: 10.1145/367390.367400.
[7] S. B. Needleman and C. D. Wunsch, “A general method applicable to the search for similarities in the amino acid sequence of two proteins”, Journal of Molecular Biology, no. 3, pp. 443–453, 1970, doi: https://doi.org/10.1016/0022-2836(70)90057-4.
[8] S. C. Hinds, J. L. Fisher, and D. P. D'Amato, “A document skew detection method using run-length encoding and the Hough transform”, IEEE, 1990, pp. 464–468. doi: 10.1109/ICPR.1990.118147.
[9] J. Ziv and A. Lempel, “A universal algorithm for sequential data compression”, IEEE Trans. Inf. Theory, no. 3, pp. 337–343, 1977, doi: 10.1109/TIT.1977.1055714.
[10] ITU-T, “T.4 : Standardization of Group 3 facsimile terminals for document transmission”, Feb. 2004. Available: https://www.itu.int/rec/T-REC-T.4-200307-I/en
[11] ISO, ISO\slash IEC 10918-1:1994: Information technology — Digital compression and coding of continuous-tone still images: Requirements and guidelines. Geneva, Switzerland: International Organization for Standardization, 1994, p. 182. Available: http://www.iso.ch/cate/d18902.html
[12] ISO, ISO 12639:1998: Graphic technology — Prepress digital data exchange — Tag image ﬁle format for image technology (TIFF/IT). Geneva, Switzerland: International Organization for Standardization, 1998, p. 43. Available: http://www.iso.ch/cate/d2181.html
[13] A. Apostolico, G. M. Landau, and S. Skiena, “Matching for Run-Length Encoded Strings”, J. Complex., no. 1, pp. 4–16, 1999, doi: 10.1006/jcom.1998.0493.
[14] S. Hooshmand, N. Tavakoli, P. Abedin, and S. V. Thankachan, “On Computing Average Common Substring Over Run Length Encoded Sequences”, Fundam. Informaticae, no. 3, pp. 267–273, 2018, doi: 10.3233/FI-2018-1743.
[15] K.-Y. Chen and K.-M. Chao, “A Fully Compressed Algorithm for Computing the Edit Distance of Run-Length Encoded Strings”, Algorithmica, no. 2, pp. 354–370, 2013, doi: 10.1007/s00453-011-9592-4.
[16] R. Cliﬀord, P. Gawrychowski, T. Kociumaka, D. P. Martin, and P. Uznanski, “RLE Edit Distance in Near Optimal Time”, P. Rossmanith, P. Heggernes, and J.-P. Katoen, Eds., in LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019, pp. 1–13. doi: 10.4230/LIPIcs.MFCS.2019.66.
[17] R. Hariharan and V. Vinay, “String matching in {\tilde O}(\sqrt{n}+\sqrt{m}) quantum time”, J. Discrete Algorithms, no. 1, pp. 103–110, 2003, doi: 10.1016/ S1570-8667(03)00010-8.
[18] L. K. Grover, “A Fast Quantum Mechanical Algorithm for Database Search”, G. L. Miller, Ed., ACM, 1996, pp. 212–219. doi: 10.1145/237814.237866.
[19] U. Vishkin, “Deterministic Sampling-A New Technique for Fast Pattern Matching”, H. Ortiz, Ed., ACM, 1990, pp. 170–180. doi: 10.1145/100216.100235.
[20] F. LeGall and S. Seddighin, “Quantum Meets Fine-Grained Complexity: Sublinear Time Quantum Algorithms for String Problems”, M. Braverman, Ed., in LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022, pp. 1–23. doi: 10.4230/LIPIcs.ITCS. 2022.97.
[21] S. Akmal and C. Jin, “Near-Optimal Quantum Algorithms for String Problems”, J. (Seffi) Naor and N. Buchbinder, Eds., SIAM, 2022, pp. 2791–2832. doi: 10.1137/1.9781611977073.109.
[22] D. Kempa and T. Kociumaka, “String synchronizing sets: sublinear-time BWT construction and optimal LCE data structure”, M. Charikar and E. Cohen, Eds., ACM,
2019, pp. 756–767. doi: 10.1145/3313276.3316368.
[23] F. Magniez, A. Nayak, J. Roland, and M. Santha, “Search via Quantum Walk”, SIAM J. Comput., no. 1, pp. 142–164, 2011, doi: 10.1137/090745854.
[24] D. Gibney and S. V. Thankachan, “Compressibility-Aware Quantum Algorithms on Strings”, CoRR, 2023, doi: 10.48550/arXiv.2302.07235.
[25] A. M. Childs and J. M. Eisenberg, “Quantum algorithms for subset ﬁnding”, Quantum Inf. Comput., no. 7, pp. 593–604, 2005, doi: 10.26421/QIC5.7-7.
[26] G. Brassard and P. Høyer, “An Exact Quantum Polynomial-Time Algorithm for Simon's Problem”, IEEE Computer Society, 1997, pp. 12–23. doi: 10.1109/ISTCS.1997.595153.
[27] L. K. Grover, “Quantum Computers Can Search Rapidly by Using Almost Any Transformation”, Phys. Rev. Lett., no. 19, pp. 4329–4332, 1998, doi: 10.1103/PhysRevLett. 80.4329.
[28] C. Dürr and P. Høyer, “A Quantum Algorithm for Finding the Minimum”, CoRR, 1996, Available: http://arxiv.org/abs/quant-ph/9607014
[29] A. Ambainis, “Quantum Walk Algorithm for Element Distinctness”, SIAM J. Comput., no. 1, pp. 210–239, 2007, doi: 10.1137/S0097539705447311.
[30] P. Høyer, M. Mosca, and R. de Wolf, “Quantum Search on Bounded-Error Inputs”, J. C. M. Baeten, J. K. Lenstra, J. Parrow, and G. J. Woeginger, Eds., in Lecture Notes in Computer Science. Springer, 2003, pp. 291–299. doi: 10.1007/3-540-45061-0\_25.
[31] W. W. Pugh, “Skip Lists: A Probabilistic Alternative to Balanced Trees”, Commun. ACM, no. 6, pp. 668–676, 1990, doi: 10.1145/78973.78977.
[32] H. Buhrman, B. Loff, S. Patro, and F. Speelman, “Limits of Quantum Speed-Ups for Computational Geometry and Other Problems: Fine-Grained Complexity via Quantum Walks”, M. Braverman, Ed., in LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022, pp. 1–12. doi: 10.4230/LIPIcs.ITCS.2022.31.

簡易檢索 / 詳目顯示

相關論文