Finding and Counting Permutations via CSPs
Permutation patterns and pattern avoidance have been intensively studied in combinatorics and computer science, going back at least to the seminal work of Knuth on stack-sorting (1968). Perhaps the most natural algorithmic question in this area is deciding whether a given permutation of length n con...
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| Vydáno v: | Algorithmica Ročník 83; číslo 8; s. 2552 - 2577 |
|---|---|
| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
New York
Springer US
01.08.2021
Springer Nature B.V |
| Témata: | |
| ISSN: | 0178-4617, 1432-0541 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | Permutation patterns and pattern avoidance have been intensively studied in combinatorics and computer science, going back at least to the seminal work of Knuth on stack-sorting (1968). Perhaps the most natural algorithmic question in this area is deciding whether a given permutation of length
n
contains a given pattern of length
k
. In this work we give two new algorithms for this well-studied problem, one whose running time is
n
k
/
4
+
o
(
k
)
, and a polynomial-space algorithm whose running time is the better of
O
(
1
.
6181
n
)
and
O
(
n
k
/
2
+
1
)
. These results improve the earlier best bounds of
n
0.47
k
+
o
(
k
)
and
O
(
1
.
79
n
)
due to Ahal and Rabinovich (2000) resp. Bruner and Lackner (2012) and are the fastest algorithms for the problem when
k
∈
Ω
(
log
n
)
. We show that both our new algorithms and the previous exponential-time algorithms in the literature can be viewed through the unifying lens of
constraint-satisfaction
. Our algorithms can also
count
, within the same running time, the number of occurrences of a pattern. We show that this result is close to optimal: solving the counting problem in time
f
(
k
)
·
n
o
(
k
/
log
k
)
would contradict the
exponential-time hypothesis
(ETH). For some special classes of patterns we obtain improved running times. We further prove that 3-
increasing
(4321-avoiding) and 3-
decreasing
(1234-avoiding) permutations can, in some sense,
embed
arbitrary permutations of almost linear length, which indicates that a sub-exponential running time is unlikely with the current techniques, even for patterns from these restricted classes. |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0178-4617 1432-0541 |
| DOI: | 10.1007/s00453-021-00812-z |