Data Structures for Computing Unique Palindromes in Static and Non-Static Strings
A palindromic substring T [ i .. j ] of a string T is said to be a shortest unique palindromic substring (SUPS) in T for an interval [ p , q ] if T [ i .. j ] is a shortest palindromic substring such that T [ i .. j ] occurs only once in T , and [ i , j ] contains [ p , q ]. The SUPS problem is,...
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| Published in: | Algorithmica Vol. 86; no. 3; pp. 852 - 873 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
New York
Springer US
01.03.2024
Springer Nature B.V |
| Subjects: | |
| ISSN: | 0178-4617, 1432-0541 |
| Online Access: | Get full text |
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| Summary: | A palindromic substring
T
[
i
..
j
] of a string
T
is said to be a shortest unique palindromic substring (SUPS) in
T
for an interval [
p
,
q
] if
T
[
i
..
j
] is a shortest palindromic substring such that
T
[
i
..
j
] occurs only once in
T
, and [
i
,
j
] contains [
p
,
q
]. The SUPS problem is, given a string
T
of length
n
, to construct a data structure that can compute all the SUPSs for any given query interval. It is known that any SUPS query can be answered in
O
(
α
)
time after
O
(
n
)-time preprocessing, where
α
is the number of SUPSs to output (Inoue in J Discrete Algorithms 52-53:122–132, 2018). In this paper, we first show that
α
is at most 4, and the upper bound is tight. We also show that the total sum of lengths of minimal unique palindromic substrings of string
T
, which is strongly related to SUPSs, is
O
(
n
). Then, we present the first
O
(
n
)-bits data structures that can answer any SUPS query in constant time. Also, we present an algorithm to solve the SUPS problem for a sliding window that can answer any query in
O
(
log
log
W
)
time and update data structures in amortized
O
(
log
σ
+
log
log
W
)
time, where
W
is the size of the window, and
σ
is the alphabet size. Furthermore, we consider the SUPS problem in the after-edit model and present an efficient algorithm. Namely, we present an algorithm that uses
O
(
n
) time for preprocessing and answers any
k
SUPS queries in
O
(
log
n
log
log
n
+
k
log
log
n
)
time after single character substitution. Finally, as a by-product, we propose a fully-dynamic data structure for range minimum queries (RmQs) with a constraint where the width of each query range is limited to poly-logarithmic. The constrained RmQ data structure can answer such a query in constant time and support a single-element edit operation in amortized constant time. |
|---|---|
| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0178-4617 1432-0541 |
| DOI: | 10.1007/s00453-023-01170-8 |