Tight Tradeoffs for Real-Time Approximation of Longest Palindromes in Streams

We consider computing a longest palindrome in the streaming model, where the symbols arrive one-by-one and we do not have random access to the input. While computing the answer exactly using sublinear space is not possible in such a setting, one can still hope for a good approximation guarantee. Our...

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Bibliographic Details
Published in:Algorithmica Vol. 81; no. 9; pp. 3630 - 3654
Main Authors: Gawrychowski, Paweł, Merkurev, Oleg, Shur, Arseny M., Uznański, Przemysław
Format: Journal Article
Language:English
Published: New York Springer US 16.09.2019
Springer Nature B.V
Subjects:
Algorithm Analysis and Problem Complexity
Algorithms
Approximation
Complexity
Computer Science
Computer simulation
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Errors
Lower bounds
Mathematical analysis
Mathematics of Computing
Random access
Real time
Theory of Computation
Streaming
Lower bound
68Q25
Palindrome
68W32
Real-time algorithm
68W25
ISSN:0178-4617, 1432-0541
Online Access:Get full text
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Summary:We consider computing a longest palindrome in the streaming model, where the symbols arrive one-by-one and we do not have random access to the input. While computing the answer exactly using sublinear space is not possible in such a setting, one can still hope for a good approximation guarantee. Our contribution is twofold. First, we provide lower bounds on the space requirements for randomized approximation algorithms processing inputs of length n . We rule out Las Vegas algorithms, as they cannot achieve sublinear space complexity. For Monte Carlo algorithms, we prove a lower bound of Ω ( M log min { | Σ | , M } ) bits of memory; here M = n / E for approximating the answer with additive error E , and M = log n / log ( 1 + ε ) for approximating the answer with multiplicative error ( 1 + ε ) . Second, we design four real-time algorithms for this problem. Three of them are Monte Carlo approximation algorithms for additive error, “small” and “big” multiplicative errors, respectively. Each algorithm uses O ( M ) words of memory. Thus the obtained lower bounds are asymptotically tight up to a logarithmic factor. The fourth algorithm is deterministic and finds a longest palindrome exactly if it is short. This algorithm can be run in parallel with a Monte Carlo algorithm to obtain better results in practice. Overall, both the time and space complexity of finding a longest palindrome in a stream are essentially settled.
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-019-00591-8