Divide and Conquer Roadmap for Algebraic Sets
Let R be a real closed field and D ⊂ R an ordered domain. We describe an algorithm that given as input a polynomial P ∈ D [ X 1 , … , X k ] and a finite set, A = { p 1 , … , p m } , of points contained in V = Zer ( P , R k ) described by real univariate representations, computes a roadmap of V conta...
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| Vydáno v: | Discrete & computational geometry Ročník 52; číslo 2; s. 278 - 343 |
|---|---|
| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Boston
Springer US
01.09.2014
Springer Nature B.V Springer Verlag |
| Témata: | |
| ISSN: | 0179-5376, 1432-0444 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | Let
R
be a real closed field and
D
⊂
R
an ordered domain. We describe an algorithm that given as input a polynomial
P
∈
D
[
X
1
,
…
,
X
k
]
and a finite set,
A
=
{
p
1
,
…
,
p
m
}
, of points contained in
V
=
Zer
(
P
,
R
k
)
described by real univariate representations, computes a roadmap of
V
containing
A
. The complexity of the algorithm, measured by the number of arithmetic operations in
D
, is bounded by
(
∑
i
=
1
m
D
i
O
(
log
2
(
k
)
)
+
1
)
(
k
log
(
k
)
d
)
O
(
k
log
2
(
k
)
)
, where
d
=
deg
(
P
)
and
D
i
is the degree of the real univariate representation describing the point
p
i
. The best previous algorithm for this problem had complexity card
(
A
)
O
(
1
)
d
O
(
k
3
/
2
)
(Basu et al., ArXiv,
2012
), where it is assumed that the degrees of the polynomials appearing in the representations of the points in
A
are bounded by
d
O
(
k
)
. As an application of our result we prove that for any real algebraic subset
V
of
R
k
defined by a polynomial of degree
d
, any connected component
C
of
V
contained in the unit ball, and any two points of
C
, there exists a semi-algebraic path connecting them in
C
, of length at most
(
k
log
(
k
)
d
)
O
(
k
log
(
k
)
)
, consisting of at most
(
k
log
(
k
)
d
)
O
(
k
log
(
k
)
)
curve segments of degrees bounded by
(
k
log
(
k
)
d
)
O
(
k
log
(
k
)
)
. While it was known previously, by a result of D’Acunto and Kurdyka (Bull Lond Math Soc 38(6):951–965,
2006
), that there always exists a path of length
(
O
(
d
)
)
k
-
1
connecting two such points, there was no upper bound on the complexity of such a path. |
|---|---|
| Bibliografie: | SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 ObjectType-Article-1 ObjectType-Feature-2 content type line 23 |
| ISSN: | 0179-5376 1432-0444 |
| DOI: | 10.1007/s00454-014-9610-9 |