Uniquely Represented Data Structures for Computational Geometry

Blelloch, Guy E.; Golovin, Daniel; Vassilevska, Virginia

doi:10.1007/978-3-540-69903-3_4

“…1.2.1, one challenge is the lack of a comparison-based data structure for 2D range query that also satisfies the two requirements above. We remark that there exist comparison-based data structures with history-independence but only with expected time complexity (e.g., [92]). There also exist folklore data structures for integer coordinates that have history-independence and worst-case time complexity (e.g., Lemma 3.15).…”

Section: Dynamic Arrays Under Insertions and Deletionsmentioning

confidence: 99%

Near-Optimal Quantum Algorithms for String Problems

Akmal

¹

,

Jin

²

2023

Algorithmica

2

0

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We study quantum algorithms for several fundamental string problems, including Longest Common Substring, Lexicographically Minimal String Rotation, and Longest Square Substring. These problems have been widely studied in the stringology literature since the 1970s, and are known to be solvable by near-linear time classical algorithms. In this work, we give quantum algorithms for these problems with near-optimal query complexities and time complexities. Specifically, we show that: Longest Common Substring can be solved by a quantum algorithm in $$\tilde{O}(n^{2/3})$$ O ~ ( n 2 / 3 ) time, improving upon the recent $$\tilde{O}(n^{5/6})$$ O ~ ( n 5 / 6 ) -time algorithm by Le Gall and Seddighin (in: Proceedings of the 13th innovations in theoretical computer science conference (ITCS 2022), pp 97:1–97:23, 2022. https://doi.org/10.4230/LIPIcs.ITCS.2022.97). Our algorithm uses the MNRS quantum walk framework, together with a careful combination of string synchronizing sets (Kempa and Kociumaka, in: Proceedings of the 51st annual ACM SIGACT symposium on theory of computing (STOC 2019), ACM, pp 756–767, 2019. https://doi.org/10.1145/3313276.3316368) and generalized difference covers. Lexicographically Minimal String Rotation can be solved by a quantum algorithm in $$n^{1/2 + o(1)}$$ n 1 / 2 + o ( 1 ) time, improving upon the recent $$\tilde{O}(n^{3/4})$$ O ~ ( n 3 / 4 ) -time algorithm by Wang and Ying (in: Quantum algorithm for lexicographically minimal string rotation. CoRR, 2020. arXiv:2012.09376). We design our algorithm by first giving a new classical divide-and-conquer algorithm in near-linear time based on exclusion rules, and then speeding it up quadratically using nested Grover search and quantum minimum finding. Longest Square Substring can be solved by a quantum algorithm in $$\tilde{O}(\sqrt{n})$$ O ~ ( n ) time. Our algorithm is an adaptation of the algorithm by Le Gall and Seddighin (2022) for the Longest Palindromic Substring problem, but uses additional techniques to overcome the difficulty that binary search no longer applies. Our techniques naturally extend to other related string problems, such as Longest Repeated Substring, Longest Lyndon Substring, and Minimal Suffix.

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“…As mentioned in Section 1.2.1, one challenge is the lack of a comparison-based data structure for 2D range query that also satisfies the two requirements above. We remark that there exist comparison-based data structures with history-independence but only with expected time complexity (e.g., [BGV08]). There also exist folklore data structures for integer coordinates that have historyindependence and worst-case time complexity (e.g., Lemma 3.15).…”

Section: Overviewmentioning

confidence: 99%

Near-Optimal Quantum Algorithms for String Problems

Akmal¹,

Jin²

2021

Preprint

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We study quantum algorithms for several fundamental string problems, including Longest Common Substring, Lexicographically Minimal String Rotation, and Longest Square Substring. These problems have been widely studied in the stringology literature since the 1970s, and are known to be solvable by near-linear time classical algorithms. In this work, we give quantum algorithms for these problems with near-optimal query complexities and time complexities. Specifically, we show that:• Longest Common Substring can be solved by a quantum algorithm in Õ(n 2/3 ) time, improving upon the recent Õ(n 5/6 )-time algorithm by Le Gall and Seddighin (2020). Our algorithm uses the MNRS quantum walk framework, together with a careful combination of string synchronizing sets (Kempa and Kociumaka, 2019) and generalized difference covers.• Lexicographically Minimal String Rotation can be solved by a quantum algorithm in n 1/2+o(1) time, improving upon the recent Õ(n 3/4 )-time algorithm by Wang and Ying (2020). We design our algorithm by first giving a new classical divide-and-conquer algorithm in near-linear time based on exclusion rules, and then speeding it up quadratically using nested Grover search and quantum minimum finding.• Longest Square Substring can be solved by a quantum algorithm in Õ( √ n) time. Our algorithm is an adaptation of the algorithm by Le Gall and Seddighin (2020) for the Longest Palindromic Substring problem, but uses additional techniques to overcome the difficulty that binary search no longer applies.Our techniques naturally extend to other related string problems, such as Longest Repeated Substring, Longest Lyndon Substring, and Minimal Suffix.

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“…History Independence in Persistent Storage vs. RAM History independence has been vigorously explored in the context of data structures that reside in RAM [20,21,23,36,43,46,47,61] (see Section 1.4), but significantly less so in external memory. Although there is some theoretical work on history-independent on-disk data structures [31][32][33] and experimental work on history-independent file systems [10][11][12]56], the area is substantially less explored.…”

Section: History Independence In Persistent Storagementioning

confidence: 99%

“…History-independent data structures are well-studied when the objective is to minimize RAM computations. Examples include SHI hashing [20,46,47], dictionaries and ordermaintenance [20], and history-independent data structures for computational geometry [21,61].…”

Section: Related Workmentioning

confidence: 99%

Anti-Persistence on Persistent Storage

Bender

¹

,

Berry

²

,

Johnson

³

et al. 2016

Proceedings of the 35th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems

11

0

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We present history-independent alternatives to a B-tree, the primary indexing data structure used in databases. A data structure is history independent (HI) if it is impossible to deduce any information by examining the bit representation of the data structure that is not already available through the API.We show how to build a history-independent cache-oblivious Btree and a history-independent external-memory skip list. One of the main contributions is a data structure we build on the way-a history-independent packed-memory array (PMA). The PMA supports efficient range queries, one of the most important operations for answering database queries.Our HI PMA matches the asymptotic bounds of prior non-HI packed-memory arrays and sparse tables. Specifically, a PMA maintains a dynamic set of elements in sorted order in a linearsized array. Inserts and deletes take an amortized O(log 2 N ) element moves with high probability. Simple experiments with our implementation of HI PMAs corroborate our theoretical analysis. Comparisons to regular PMAs give preliminary indications that the practical cost of adding history-independence is not too large.Our HI cache-oblivious B-tree bounds match those of prior non- * HI cache-oblivious B-trees. Searches take O(log B N ) I/Os; inserts and deletes take O( log 2 N B + log B N ) amortized I/Os with high probability; and range queries returning k elements take O(log B N + k/B) I/Os. Our HI external-memory skip list achieves optimal bounds with high probability, analogous to in-memory skip lists: O(log B N ) I/Os for point queries and amortized O(log B N ) I/Os for inserts/deletes.Range queries returning k elements run in O(log B N + k/B) I/Os. In contrast, the best possible highprobability bounds for inserting into the folklore B-skip list, which promotes elements with probability 1/B, is just Θ(log N ) I/Os. This is no better than the bounds one gets from running an inmemory skip list in external memory.

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Uniquely Represented Data Structures for Computational Geometry

Cited by 6 publications

References 25 publications

Near-Optimal Quantum Algorithms for String Problems

Near-Optimal Quantum Algorithms for String Problems

Near-Optimal Quantum Algorithms for String Problems

Anti-Persistence on Persistent Storage

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