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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| Veröffentlicht in: | Algorithmica Jg. 81; H. 9; S. 3630 - 3654 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
New York
Springer US
16.09.2019
Springer Nature B.V |
| Schlagworte: | |
| ISSN: | 0178-4617, 1432-0541 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | 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. |
|---|---|
| Bibliographie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0178-4617 1432-0541 |
| DOI: | 10.1007/s00453-019-00591-8 |