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

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