On the Expected Cost of Partial Match Queries in Random Quad-K-d Trees
Quad- K -d trees introduced by Bereckzy et al. (In: Proceedings of the 11th Latin merican Theoretical Informatics Conference (LATIN). Lecture Notes in Computer Science, vol. 8392, pp. 743–754, 2014) are a generalization of several well-known hierarchical multidimensional data structures. They provid...
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Published in | La matematica Vol. 3; no. 1; pp. 385 - 416 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
New York
Springer US
04.03.2024
|
Subjects | |
Online Access | Get full text |
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Summary: | Quad-
K
-d trees introduced by Bereckzy et al. (In: Proceedings of the 11th Latin merican Theoretical Informatics Conference (LATIN). Lecture Notes in Computer Science, vol. 8392, pp. 743–754, 2014) are a generalization of several well-known hierarchical multidimensional data structures. They provide a unified framework for the analysis of associative queries, and they are specially suitable to investigate the trade-offs between the cost of different operations and the memory needs (each node
x
of a quad-
K
-d tree has arity
2
m
(
x
)
for some
m
(
x
),
1
≤
m
(
x
)
≤
K
). Indeed, we consider here partial match—one of the fundamental associative queries for several families of quad-
K
-d trees including, among others, relaxed
K
-d trees and quadtrees. In particular, we prove that the expected cost
P
^
n
of a random partial match query that has
s
out of
K
specified coordinates in a random quad-
K
-d tree of size
n
is
P
^
n
∼
β
·
n
α
, where
α
and
β
are constants given in terms of
K
and
s
as well as additional parameters that characterize the specific family of quad-
K
-d trees under consideration. Additionally, we derive a precise asymptotic estimate for the main order term of the expected cost
P
n
,
q
of a fixed partial match with query
q
in a random quad-
K
-d tree of size
n
. The techniques used to derive the mentioned costs are those already applied successfully to derive analogous results in quadtrees and relaxed
K
-d trees; our results show that the previous results are just particular cases and prove the validity of the conjecture made in Duch et al. (In: Proceedings of the 12th Latin American Theoretical Informatics Conference (LATIN). Lecture Notes in Computer Science, vol. 9644, pp. 376–389, 2016) for a wider variety of multidimensional data structures. |
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ISSN: | 2730-9657 2730-9657 |
DOI: | 10.1007/s44007-024-00090-5 |