A Fast Algorithm for Computing the Number of Magic Series
We present a fast algorithm for computing the number of magic series, an enumeration problem of a certain integer partition. Kinnaes showed that the number appears as a coefficient in a Gaussian polynomial and that the exact value can be efficiently extracted with a finite variant of Cauchy’s integr...
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| Published in | Annals of combinatorics Vol. 26; no. 2; pp. 511 - 532 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
01.06.2022
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0218-0006 0219-3094 |
| DOI | 10.1007/s00026-022-00584-5 |
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| Summary: | We present a fast algorithm for computing the number of magic series, an enumeration problem of a certain integer partition. Kinnaes showed that the number appears as a coefficient in a Gaussian polynomial and that the exact value can be efficiently extracted with a finite variant of Cauchy’s integral formula. The algorithm requires a bit time complexity of
O
(
m
4
M
(
m
log
m
)
)
and
O
(
m
log
m
)
-bit space, where
m
is the order of the magic series and
M
(
n
)
is the time complexity of multiplying two
n
-bit numbers. Through our analysis, we confirm that this is the most efficient among previously reported algorithms. In addition, we show that the number can be computed with a bit time complexity of
O
(
m
3
log
m
M
(
m
log
m
)
)
by directly carrying out polynomial multiplication and division on the Gaussian polynomial. Though the space consumption increases to
O
(
m
3
log
m
)
bits, we demonstrate that our method actually computes the number faster for large orders. |
|---|---|
| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0218-0006 0219-3094 |
| DOI: | 10.1007/s00026-022-00584-5 |