Shift Finding in Sub-linear Time
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Indyk_Shift.pdf
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Author(s)
Andoni, Alexandr
Hassanieh, Haitham
Indyk, Piotr
Katabi, Dina
Date Issued
January 2013
Journal
Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms
Publisher
Society for Industrial and Applied Mathematics
Citation
Andoni, Alexandr, Haitham Hassanieh, Piotr Indyk, and Dina Katabi. “Shift Finding in Sub-Linear Time.” Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms (January 6, 2013): 457–465. © 2013 SIAM.
Version
Final published version
Abstract
We study the following basic pattern matching problem. Consider a “code” sequence c consisting of n bits chosen uniformly at random, and a “signal” sequence x obtained by shifting c (modulo n) and adding noise. The goal is to efficiently recover the shift with high probability. The problem models tasks of interest in several applications, including GPS synchronization and motion estimation.
We present an algorithm that solves the problem in time Õ(n[superscript (f/(1+f)]), where Õ(N[superscript f]) is the running time of the best algorithm for finding the closest pair among N “random” sequences of length O(log N). A trivial bound of f = 2 leads to a simple algorithm with a running time of Õ(n[superscript 2/3]). The asymptotic running time can be further improved by plugging in recent more efficient algorithms for the closest pair problem.
Our results also yield a sub-linear time algorithm for approximate pattern matching algorithm for a random signal (text), even for the case when the error between the signal and the code (pattern) is asymptotically as large as the code size. This is the first sublinear time algorithm for such error rates.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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DOI of Published Version
https://doi.org/10.1137/1.9781611973105.33
