Quantum Pattern Matching Fast on Average

The d -dimensional pattern matching problem is to find an occurrence of a pattern of length m × ⋯ × m within a text of length n × ⋯ × n , with n ≥ m . This task models various problems in text and image processing, among other application areas. This work describes a quantum algorithm which solves t...

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Bibliographic Details
Published in:Algorithmica Vol. 77; no. 1; pp. 16 - 39
Main Author: Montanaro, Ashley
Format: Journal Article
Language:English
Published: New York Springer US 01.01.2017
Springer Nature B.V
Subjects:
Algorithm Analysis and Problem Complexity
Algorithms
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Image processing
Mathematics of Computing
Pattern matching
Theory of Computation
Quantum algorithms
Average-case complexity
Pattern matching
Quantum query complexity
ISSN:0178-4617, 1432-0541
Online Access:Get full text
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Summary:The d -dimensional pattern matching problem is to find an occurrence of a pattern of length m × ⋯ × m within a text of length n × ⋯ × n , with n ≥ m . This task models various problems in text and image processing, among other application areas. This work describes a quantum algorithm which solves the pattern matching problem for random patterns and texts in time O ~ ( ( n / m ) d / 2 2 O ( d 3 / 2 log m ) ) . For large m this is super-polynomially faster than the best possible classical algorithm, which requires time Ω ~ ( n d / 2 + ( n / m ) d ) . The algorithm is based on the use of a quantum subroutine for finding hidden shifts in d dimensions, which is a variant of algorithms proposed by Kuperberg.
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-015-0060-4